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Linear algebraic group

A linear algebraic group over a field k is a smooth affine algebraic variety over k that carries a group structure compatible with its algebraic structure, typically realized as a closed subgroup of the general linear group \mathrm{GL}_n(k) for some positive integer n, where the group operation is matrix multiplication and the defining equations are polynomial.^[1]^[2]^[3] These groups generalize classical Lie groups to an algebraic geometry setting, allowing study over arbitrary fields rather than just the reals or complexes, and they form the foundation for much of modern representation theory and arithmetic geometry.^[1]^[3] The theory of linear algebraic groups emerged in the mid-20th century, with foundational contributions from Claude Chevalley in the 1950s, who developed the structure theory over algebraically closed fields of characteristic zero, and Armand Borel, who extended it to positive characteristic.^[1] Key texts, such as James E. Humphreys' Linear Algebraic Groups (1975) and T.A. Springer's Linear Algebraic Groups (1968, revised 1998), formalized the subject by integrating algebraic geometry and Lie theory.^[2] Prominent examples include the general linear group \mathrm{GL}_n, consisting of all invertible n \times n matrices; the special linear group \mathrm{SL}_n, defined by determinant one; the orthogonal group \mathrm{O}_n; and the symplectic group \mathrm{Sp}_{2n}, all of which arise as subgroups preserving specific bilinear forms.^[1]^[2] Simpler cases are the additive group \mathbb{G}_a = (k, +) and the multiplicative group \mathbb{G}_m = k^\times, which embed into \mathrm{GL}_1.^[1] Structurally, every linear algebraic group G has a Lie algebra \mathfrak{g}, obtained as the tangent space at the identity equipped with a Lie bracket from the group law, and over algebraically closed fields of characteristic zero, G is closely tied to its Lie algebra via the exponential map.^[1] Groups are classified as solvable, semisimple, or reductive based on their unipotent radical and derived subgroup: reductive groups, such as \mathrm{GL}_n and \mathrm{SL}_n, have no nontrivial unipotent normal subgroups and have a structure theorem expressing them in terms of a maximal torus and unipotent root groups isomorphic to \mathbb{G}_a.^[2]^[1] Semisimple groups, like the simple groups \mathrm{SL}_n (for n \geq 2) or exceptional types such as E_8, are direct products of such simple components, classified by Dynkin diagrams into four infinite families (A, B, C, D) and five exceptional ones (E_6, E_7, E_8, F_4, G_2).^[2] Borel subgroups, maximal connected solvable subgroups, play a central role, being conjugate and parametrizing flag varieties like G/B.^[1]^[2] Linear algebraic groups underpin diverse applications, from the study of finite groups of Lie type (e.g., Chevalley groups) in finite group theory to Galois cohomology in number theory, where they model descent data and provide tools for solving Diophantine equations.^[3] In representation theory, their finite-dimensional representations decompose into irreducibles labeled by dominant weights, facilitating connections to quantum groups and modular forms.^[1] Over non-algebraically closed fields, phenomena like anisotropic tori and Galois actions introduce additional complexity, as explored in works on the Kneser-Tits problem.^[3]

Introduction and History

Overview and motivation

A linear algebraic group over a field k is defined as a closed subgroup of the general linear group \mathrm{GL}_n(k) for some positive integer n, where the subgroup is closed in the Zariski topology and carries a compatible group structure via matrix multiplication.^[4] This setup endows the group with the structure of an affine algebraic variety, allowing the application of algebraic geometry tools to study its properties.^[1] The Zariski topology on affine space \mathbb{A}^n_k, the spectrum of the polynomial ring k[x_1, \dots, x_n], is generated by the closed sets consisting of zeros of ideals of polynomials, making it coarser than classical topologies and suited to algebraic rather than analytic phenomena.^[1] Regular functions on such varieties are precisely the polynomial functions that restrict to the variety, forming the coordinate ring that encodes the geometry.^[4] This framework provides a purely algebraic setting for groups, contrasting with the differential structure of Lie groups over \mathbb{R} or \mathbb{C}. The study of linear algebraic groups is motivated by their role as algebraic counterparts to Lie groups, facilitating the analysis of representations and symmetries in both continuous and discrete settings.^[5] In particular, they arise naturally in the context of linear representations of abstract groups, where the image of a representation into \mathrm{GL}_n(k) often forms such a subgroup, bridging group theory with algebraic geometry.^[2] This connection is essential for understanding finite groups of Lie type through reduction modulo primes.^[5] The systematic development of linear algebraic groups began in the 1950s with Claude Chevalley's work, where they were first employed to construct Chevalley groups over arbitrary fields, providing a uniform algebraic framework for classical simple Lie groups.^[6]

Historical development

The theory of linear algebraic groups traces its roots to the late 19th century, when the classification of simple Lie algebras laid the groundwork for understanding the associated Lie groups. Wilhelm Killing initiated this effort in his seminal papers published between 1888 and 1890 in Mathematische Annalen, where he classified the finite-dimensional simple Lie algebras over the complex numbers, identifying four infinite families (corresponding to linear, orthogonal, symplectic, and exceptional types) and introducing the Cartan-Killing form as a key invariant.^[7] Élie Cartan refined and rigorously proved this classification in his 1894 doctoral thesis, Sur la structure des groupes de transformations finis et continus, establishing the complete list of simple Lie algebras over \mathbb{C} and extending the results to real forms by 1914, which provided the infinitesimal structure essential for later developments in continuous transformation groups. In the early 20th century, the focus shifted to continuous groups and their representations, with contributions from Ludwig Maurer, Élie Cartan, and Hermann Weyl. Maurer advanced the study of continuous transformation groups around 1900–1910, developing parameterizations and infinitesimal generators that bridged local and global aspects of Lie's original theory.^[8] Cartan extended his work on Lie algebras to infinite-dimensional continuous groups and their applications in geometry during the 1910s–1920s. Weyl, in a series of papers from 1925–1926, provided the first complete theory of representations of compact continuous groups, proving complete reducibility and deriving character formulas, which were pivotal for quantum mechanics and invariant theory; his 1939 book The Classical Groups further generalized these ideas to non-compact cases.^[9] The mid-20th century marked the transition to algebraic groups proper, beginning with Ellis Kolchin's 1948 paper "Algebraic Matric Groups and the Picard-Vessiot Theory of Homogeneous Linear Ordinary Differential Equations," which introduced foundational concepts for solvable algebraic matrix groups and proved the Lie-Kolchin theorem characterizing connected solvable subgroups of GL(n). Claude Chevalley advanced the theory in the 1950s, particularly through his 1955 work constructing simple algebraic groups over finite fields (now known as Chevalley groups), which linked Lie algebras to finite groups of Lie type, and his 1956–1958 seminar notes on semisimple groups.^[1] Armand Borel, collaborating with Chevalley, systematically developed the structure theory in the 1950s–1960s, including his 1956 paper on linear representations and Borel subgroups, culminating in his influential 1969 book Linear Algebraic Groups (revised 1991), which established key results on reductive groups and their cohomology. In the 1960s, Alexander Grothendieck integrated algebraic groups into his scheme-theoretic framework, as detailed in Séminaire de Géométrie Algébrique du Bois-Marie (SGA 3) (1963–1964), where group schemes were defined over arbitrary schemes, enabling the study of algebraic groups over rings and resolving issues with nilpotents and positive characteristic.^[1] This reformulation facilitated deeper connections to arithmetic geometry. Post-1960 developments included Jacques Tits and Borel's 1965 work on buildings, which provided a combinatorial framework for the structure of reductive groups over local fields via BN-pairs. Recent work continues to emphasize ties between linear algebraic groups and the Langlands program.^[10]

Definitions and Basic Concepts

Formal definition over algebraically closed fields

An affine algebraic variety over an algebraically closed field k is defined as the zero set V(I) of an ideal I in the polynomial ring k[x_1, \dots, x_n], where I is taken to be radical (the vanishing ideal I(V(I))), as justified by the Hilbert Nullstellensatz, which establishes a bijection between radical ideals and affine algebraic sets. The points of the variety correspond to maximal ideals in the coordinate ring k[V] = k[x_1, \dots, x_n]/I(V).^[11] The Zariski topology on such varieties is generated by the closed sets, which are themselves zero loci of polynomials, providing a coarse topology suited to algebraic geometry.^[12] A linear algebraic group G over k is a smooth affine algebraic variety that is a closed subgroup of \mathrm{GL}(n, k) for some n, where closedness is with respect to the Zariski topology on the ambient space \mathbb{A}^{n^2}.^[12] Equivalently, G is an affine variety equipped with morphisms of varieties m: G \times G \to G for multiplication and i: G \to G for inversion, satisfying the group axioms, with the identity morphism from the singleton variety to G.^[11] The coordinate ring k[G] of G is the finitely generated, reduced k-algebra k[x_1, \dots, x_n]/I(G), where I(G) is the vanishing ideal of G, and the group structure induces a comultiplication \Delta: k[G] \to k[G] \otimes_k k[G] making k[G] into a Hopf algebra, with the dual maps corresponding to the group operations.^[11] The multiplication m and inversion i are morphisms of varieties because they are given by polynomial functions on the ambient affine space, hence regular maps that restrict to G \times G and G, respectively, preserving the algebraic structure; this follows from the fact that the defining equations of G are preserved under these operations due to the subgroup property.^[12] For instance, if G is defined by polynomial equations f_j(g) = 0 for g \in \mathrm{GL}(n,k), then f_j(gh) = 0 and f_j(g^{-1}) = 0 whenever f_j(g) = f_j(h) = 0, ensuring the images lie in G.^[11] The dimension of G, as an algebraic variety, is the Krull dimension of k[G], equivalently the transcendence degree of the function field k(G) over k, or the dimension of the tangent space at the identity element.^[11] This dimension is finite since k[G] is finitely generated, and for closed embeddings into \mathrm{GL}(n,k), it satisfies \dim G \leq n^2, with equality for G = \mathrm{GL}(n,k).^[12]

Extension to arbitrary fields and schemes

The notion of a linear algebraic group over an arbitrary field k extends the classical definition by considering the group of rational points G(k), which consists of the k-points of the scheme G satisfying the group law defined by the scheme morphisms. These points form an abstract group, but to capture the full algebraic structure, one uses Galois descent to relate G over k to its base change G_{k_s} over a separable closure k_s, where k_s / k is Galois. A k-form of a group H defined over k_s is a group G over k such that G_{k_s} \cong H, and such forms are classified by the Galois cohomology set H^1(k, \Aut(H)), where the action is via the Galois group \Gal(k_s / k). This descent ensures that properties like connectedness and dimension are preserved under base change.^[1]^[12]^[13] In the language of scheme theory, a linear algebraic group over k is an affine group scheme of finite type over \Spec(k), meaning it is a representable functor G: (\text{Comm } k\text{-Alg})^{\op} \to \text{Groups} from the opposite category of commutative k-algebras to groups, represented by an affine scheme \Spec(A) with A a finitely generated k-algebra. The group structure arises from compatible morphisms for multiplication, inversion, and the identity, satisfying the group axioms functorially. Over arbitrary k, including non-algebraically closed cases, this framework allows uniform treatment without assuming separability.^[1]^[14]^[12] For non-perfect fields (those where the Frobenius is not surjective), linear algebraic groups are defined to be smooth and geometrically reduced, ensuring that the relative tangent space at every point has dimension equal to the relative dimension of the scheme and that base change to the algebraic closure yields a reduced scheme with no nilpotent elements. Smoothness over such fields implies the existence of a maximal k-torus in connected reductive groups, as guaranteed by Grothendieck's theorem, and avoids pathologies like non-separated quotients. Reducedness excludes infinitesimal group schemes, focusing on those of classical interest.^[1]^[13]^[12] The coordinate ring O(G) = k[G] of an affine linear algebraic group G over k is a commutative Hopf k-algebra, equipped with a comultiplication \Delta: O(G) \to O(G) \otimes_k O(G) induced by the multiplication morphism m: G \times_k G \to G, a counit \varepsilon: O(G) \to k from the identity section, and an antipode S: O(G) \to O(G) from the inversion morphism. These structures encode the group law algebraically, with coassociativity and compatibility axioms holding functorially; for example, in \GL_n, \Delta(T_{ij}) = \sum_\ell T_{i\ell} \otimes T_{\ell j}. This Hopf algebra perspective dualizes the group scheme and facilitates representation theory over arbitrary k.^[14]^[1]^[13] A prominent example of a non-split form arises with anisotropic tori over the real numbers \mathbb{R}, such as the norm torus defined by the equation z \overline{z} = 1 in \mathbb{C}^\times, which is a form of \mathbb{G}_m over \mathbb{R} but contains no non-trivial split subtorus and is compact as a real Lie group. Such tori illustrate Galois descent in action, as their splitting fields are quadratic extensions of \mathbb{R}, and they appear in forms of orthogonal or unitary groups.^[1]^[13]

Group operations and homomorphisms

Linear algebraic groups are affine algebraic varieties equipped with group structure, where the multiplication map m: G \times G \to G, defined by m(g, h) = gh, and the inversion map i: G \to G, defined by i(g) = g^{-1}, are both morphisms of algebraic varieties. These operations ensure that the group law is compatible with the algebraic structure, allowing G to function as both a group and an algebraic variety.^[13] A homomorphism \phi: G \to H between linear algebraic groups G and H is a morphism of algebraic varieties that preserves the group operations, satisfying \phi(gh) = \phi(g)\phi(h) and \phi(g^{-1}) = \phi(g)^{-1} for all g, h \in G. The kernel of \phi, defined as \ker \phi = \{ g \in G \mid \phi(g) = e_H \}, where e_H is the identity in H, is a closed normal subgroup of G. The image \phi(G) is a closed subgroup of H.^[13] An isogeny is a surjective homomorphism \phi: G \to H with finite kernel; such maps are finite morphisms of degree equal to the order of the kernel. Isomorphisms are bijective homomorphisms whose inverses are also homomorphisms, equivalently bijective with bijective differential at the identity. In positive characteristic, an isogeny is separable if its kernel is reduced (étale) or if the induced map on tangent spaces is injective; inseparability arises when the kernel contains non-reduced components, as in the Frobenius morphism.^[13]^[15] The center Z(G) of a linear algebraic group G is the closed subgroup consisting of all elements that commute with every element of G, given by Z(G) = \{ z \in G \mid zg = gz \ \forall g \in G \}. The derived subgroup G', or commutator subgroup, is the smallest closed normal subgroup such that G/G' is abelian; it is generated by all commutators [g, h] = ghg^{-1}h^{-1} for g, h \in G. For connected semisimple groups, G' = G.^[13] Associated to any homomorphism \phi: G \to H is its differential at the identity, a linear map d\phi_e: \Lie(G) \to \Lie(H) between the Lie algebras, defined via the tangent space at the identity element. This map preserves the Lie bracket and is injective if \phi is separable; for isomorphisms, d\phi_e is an isomorphism of Lie algebras.^[13]

Examples

Classical matrix groups

The general linear group \mathrm{GL}(n,k) over a field k consists of all invertible n \times n matrices with entries in k, defined as the Zariski-open subset of the affine space of all n \times n matrices where the determinant is nonzero.^[16] This group has dimension n^2, as its Lie algebra is the full matrix algebra \mathfrak{gl}(n,k).^[16] It serves as the ambient space for many classical examples and is itself a fundamental linear algebraic group. The special linear group \mathrm{SL}(n,k) is the kernel of the determinant morphism \det: \mathrm{GL}(n,k) \to \mathbb{G}_m, where \mathbb{G}_m is the multiplicative group, consisting of all matrices in \mathrm{GL}(n,k) with determinant 1.^[16] Its defining equation is \det(A) = 1 for A \in M_n(k), and it has dimension n^2 - 1, reflecting the single polynomial constraint on \mathrm{GL}(n,k).^[16] For n \geq 2, \mathrm{SL}(n,k) is a simple linear algebraic group over algebraically closed fields, meaning it has no nontrivial proper connected normal subgroups.^[16] The orthogonal group \mathrm{O}(n,k) (assuming \mathrm{char}(k) \neq 2) preserves the standard nondegenerate symmetric bilinear form, defined by the equation X^T X = I_n for X \in M_n(k), where I_n is the n \times n identity matrix.^[17] This group has dimension n(n-1)/2, corresponding to the number of independent entries above the diagonal in an orthogonal matrix.^[17] The special orthogonal group \mathrm{SO}(n,k) is the kernel of the determinant map on \mathrm{O}(n,k), satisfying the same equation with the additional condition \det(X) = 1, and thus shares the same dimension n(n-1)/2.^[17] The symplectic group \mathrm{Sp}(2n,k) preserves a nondegenerate alternating bilinear form on k^{2n}, standardly given by the equation X^T J X = J, where J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix} and I_n is the n \times n identity.^[16] This defining relation ensures the form \omega(x,y) = x^T J y remains invariant under the group action. The dimension of \mathrm{Sp}(2n,k) is n(2n+1), arising from the constraints on the matrix entries that maintain the symplectic structure.^[16]

Elementary and other simple examples

One of the simplest non-trivial examples of a linear algebraic group is the multiplicative group \mathbb{G}_m, defined over a field k as the spectrum of the Hopf algebra k[T, T^{-1}], which corresponds to the affine line \mathbb{A}^1_k minus the origin.^[12] This group has dimension 1 and consists of invertible elements k^\times, with the group operation given by multiplication in k.^[12] It serves as the building block for more complex tori, which are split as direct products of copies of \mathbb{G}_m.^[18] Another elementary example is the additive group \mathbb{G}_a, the spectrum of the Hopf algebra k[T], representing the affine line \mathbb{A}^1_k itself.^[12] Here, the group operation is polynomial addition on k, and \mathbb{G}_a is unipotent of dimension 1, with all elements satisfying g - 1 nilpotent.^[12] Any connected one-dimensional linear algebraic group over an algebraically closed field is isomorphic to either \mathbb{G}_m or \mathbb{G}_a.^[19] For a higher-dimensional unipotent example, consider the Heisenberg group H_3 over k, a three-dimensional nilpotent algebraic group of class 2.^[20] It can be realized as the affine variety with coordinates (x, y, z) \in k^3 and multiplication (x, y, z) \cdot (x', y', z') = (x + x', y + y', z + z' + x y'), where the center is one-dimensional, spanned by elements with x = y = 0.^[20] This structure illustrates the nilpotency typical of unipotent groups, with the Lie algebra also Heisenberg-type.^[21] Among simple linear algebraic groups, the projective special linear group \mathrm{PSL}_n(k) for n \geq 2 provides a classical example, which is simple and of semisimple rank n-1.^[22] Exceptional simple groups include those of types E_6, E_7, and E_8, with ranks 6, 7, and 8, respectively, and corresponding dimensions 78, 133, and 248; these arise from exceptional Lie algebras and have no classical analogs.^[23] Their Dynkin diagrams distinguish them from series like A_n.^[22] Non-split examples highlight field dependence: over the real numbers \mathbb{R}, the multiplicative group of the Hamilton quaternion algebra \mathbb{H} (with basis $1, i, j, k and relations i^2 = j^2 = k^2 = ijk = -1) forms a non-split inner form of \mathrm{SL}_2(\mathbb{C}), where the norm-1 elements yield the compact group \mathrm{SU}(2).^[24] This contrasts with the split form \mathrm{M}_2(\mathbb{R}) over \mathbb{R}, demonstrating how quaternion algebras ramify at infinite places to produce anisotropic groups.^[24]

Structural Elements

Connected components and radical

In the Zariski topology on a linear algebraic group G, the connected components coincide with the irreducible components of G as an algebraic variety. These components all have the same dimension as G, and G is a finite disjoint union of them.^[13] The identity component G^0 is the unique connected component containing the identity element e; it is an irreducible affine algebraic variety that forms an open normal subgroup of G with finite index.^[13] As a normal subgroup, G^0 is stable under conjugation by elements of G, and the cosets gG^0 for g \in G partition G into its connected components. Over an algebraically closed field, G^0 is smooth, and it acts transitively on each irreducible component of homogeneous spaces under G. When G is defined over a subfield k, G^0 is also defined over k if k is perfect. Moreover, over perfect fields, G^0 is Zariski dense in G.^[13]^[25] The unipotent radical R_u(G) of G is defined as the largest normal unipotent subgroup of G; it is unique, characteristic (stable under automorphisms of G), closed, connected, nilpotent, and solvable.^[13] Equivalently, R_u(G) is the identity component of the intersection of the unipotent radicals of all Borel subgroups of G containing a fixed maximal torus. Over a perfect field k, if G is a k-group, then R_u(G) is defined over k. The quotient G / R_u(G) is a reductive linear algebraic group.^[13] The solvable radical R(G) of G is the largest connected normal solvable subgroup; it is unique, closed, and contains R_u(G) as its unipotent part.^[13] It can be characterized as the identity component of the intersection of all Borel subgroups of G. Over a perfect field, R(G) is defined over the base field and, in the reductive case, takes the form of a central torus. The quotient G / R(G) is then semisimple.^[13] A Levi subgroup L of G is a maximal reductive subgroup such that G = R_u(G) \rtimes [L](/page/Levi) via a semidirect product; it is unique up to conjugation and contains a maximal torus of G.^[13] The reductive quotient G / R_u(G) is isomorphic to L, providing a structural decomposition that reduces the study of general linear algebraic groups to reductive ones. Over algebraically closed fields, the existence of such Levi subgroups follows from the theory of parabolic subgroups.^[13]

Lie algebra and its properties

The Lie algebra of a linear algebraic group G over an algebraically closed field k, denoted \Lie(G), is defined as the tangent space at the identity element e of G.^[5] This vector space captures the infinitesimal structure of G, serving as a linear approximation to the group near the identity.^[13] In characteristic zero, \Lie(G) can also be identified with the space of k-derivations \Der_k(k[G], k) of the coordinate ring k[G], where derivations are left-invariant under the group action.^[5] This identification equips \Lie(G) with a Lie bracket given by the commutator of derivations, making it a Lie algebra over k.^[13] A key connection between the Lie algebra and the group is provided by the exponential map \exp: \Lie(G) \to G, which sends an element X \in \Lie(G) to the group element obtained via the formal power series \exp(X) = I + X + \frac{X^2}{2!} + \cdots, assuming a matrix representation of G.^[5] This map is well-defined for nilpotent elements and plays a central role in characteristic zero, where it relates the algebraic and infinitesimal structures.^[13] However, in positive characteristic, the exponential map is not always surjective onto unipotent elements, and its behavior is more restricted due to issues with formal power series convergence and p-nilpotency.^[5] The group G acts on \Lie(G) via the adjoint representation \Ad: G \to \Aut(\Lie(G)), defined by conjugation: for g \in G and X \in \Lie(G), \Ad(g)X = gXg^{-1}.^[13] The differential of this representation at the identity yields the adjoint action \ad: \Lie(G) \to \End(\Lie(G)), where \ad(X)Y = [X, Y] for Y \in \Lie(G).^[5] In characteristic zero, the Killing form \kappa(X, Y) = \Tr(\ad(X) \circ \ad(Y)) on \Lie(G) is an invariant bilinear form that provides important structural information, such as nondegeneracy for semisimple Lie algebras.^[5] This form arises naturally from the trace in the adjoint representation and aids in distinguishing solvable and semisimple components.^[13] For a closed subgroup H \leq G, the Lie algebra \Lie(H) is a Lie subalgebra of \Lie(G), obtained as the subspace of derivations vanishing on the ideal defining H.^[5] This inclusion preserves the Lie bracket and reflects the embedding of infinitesimal structures.^[13] Semisimple elements in \Lie(G) correspond to those whose adjoint action is diagonalizable over an algebraic closure.^[5] In positive characteristic p > 0, several classical properties of Lie algebras fail to hold for those arising from linear algebraic groups. For instance, Cartan's criterion for solvability or semisimplicity, which relies on the trace form in the adjoint representation, does not generally apply, as the Killing form may degenerate even for simple Lie algebras.^[5] Additionally, Lie's theorem on simultaneous triangularization of representations breaks down, exemplified by the Lie algebra \mathfrak{sl}_2(k) becoming solvable in characteristic 2 without upper-triangularizable representations.^[5] These limitations necessitate alternative approaches, such as restricted Lie algebras incorporating p-operations.^[13]

Element Decompositions

Unipotent elements

In a linear algebraic group G defined over an algebraically closed field k, an element g \in G is unipotent if all eigenvalues of g (with respect to any faithful rational representation of G) are equal to 1.^[1] This condition is independent of the choice of representation and characterizes unipotency intrinsically within the group.^[2] For the general linear group \mathrm{GL}(n, k), an element g is unipotent if and only if g - I is a nilpotent matrix, meaning there exists a positive integer m \leq n such that (g - I)^m = 0, where I denotes the identity matrix.^[1] In this setting, unipotent elements correspond to matrices that are conjugate to strictly upper triangular matrices with 1s on the diagonal.^[2] The set of all unipotent elements in a reductive linear algebraic group G forms a closed subvariety known as the unipotent cone.^[1] In characteristic zero, this cone corresponds closely to the image under the exponential map of the nilpotent cone in the Lie algebra of G, and the unipotent elements generate unipotent subgroups.^[1] The centralizer C_G(u) of a unipotent element u \in G in a connected reductive group is itself connected and has a structure given by a Levi subgroup together with its unipotent radical.^[1] The normalizer N_G(u) is finite over C_G(u), often involving components related to the Weyl group action on the centralizer.^[1] In positive characteristic, the definition of unipotency remains the same—all eigenvalues equal to 1—but the relationship to the Lie algebra differs significantly; for instance, not all unipotent elements arise as exponentials of nilpotent elements in the Lie algebra, due to the limited surjectivity of the exponential map.^[1]

Semisimple elements and tori

In the theory of linear algebraic groups over a field k, an element g \in G(k) of a linear algebraic group G is defined to be semisimple if, in every rational representation (V, \rho_V) of G, the linear operator \rho_V(g) is semisimple, meaning it is diagonalizable over the algebraic closure \overline{k}.^[26] This property is independent of the choice of representation and aligns with the semisimple part in the Jordan-Chevalley decomposition.^[27] Over perfect fields, semisimple elements form a subgroup G_s of G, though this subgroup is not necessarily closed in the Zariski topology; for example, in the group B_2 of upper triangular $2 \times 2 matrices with determinant 1 over an algebraically closed field of characteristic not 2, G_s is not closed.^[27] A fundamental property of semisimple elements is their containment in tori: every semisimple element s \in G_s lies in some maximal torus of G.^[26]^[27] Moreover, the centralizer Z_G(s) of a semisimple element s is connected and reductive.^[26] In connected solvable groups, the semisimple elements generate the derived subgroup, and their centralizers contain maximal tori.^[27] For reductive groups, regular semisimple elements—those whose centralizers are tori—form a Zariski-open dense subset of G, and their conjugates generate G.^[26] A torus in a linear algebraic group is a connected commutative algebraic subgroup T that becomes isomorphic, after base change to a finite separable field extension k'/k, to a finite direct product of copies of the multiplicative group \mathbb{G}_m.^[26] Equivalently, T is a connected subgroup consisting entirely of semisimple elements and is diagonalizable over \overline{k}, meaning its rational representations decompose as direct sums of one-dimensional representations.^[27] The character group X^*(T) of a torus T is a finitely generated free abelian group, and T is smooth with dimension equal to the rank of X^*(T).^[26] Split tori, which are isomorphic to \mathbb{G}_m^r directly over k, serve as building blocks; every torus is a quotient of an induced torus, constructed as a product of twists (\mathbb{G}_m)_{k_i/k} for separable extensions k_i/k.^[26] Maximal tori play a central role in the structure of reductive and semisimple groups. A maximal torus in a connected linear algebraic group G is a torus not properly contained in any larger torus.^[27] In a connected reductive group, any two maximal tori are conjugate under an element of G(k_s), where k_s is the separable closure of k, and the centralizer of a maximal torus T is T itself.^[26]^[27] For example, in GL_n(k), the group of invertible n \times n matrices, the diagonal matrices form a split maximal torus of dimension n, with normalizer the monomial matrices and Weyl group the symmetric group S_n.^[27] Tori normalize unipotent subgroups and appear in the Levi decomposition of parabolic subgroups, underscoring their role in classifying representations and root systems.^[26]

Jordan-Chevalley decomposition

In a linear algebraic group G over an algebraically closed field k, every element g \in G(k) admits a unique decomposition g = g_s g_u = g_u g_s, where g_s is semisimple and g_u is unipotent, with g_s, g_u \in G(k).^[28] This Jordan-Chevalley decomposition generalizes the classical Jordan canonical form for matrices and holds for any linear algebraic group, not just matrix groups like \mathrm{GL}_n(k).^[13] The proof proceeds by embedding G as a closed subgroup of \mathrm{GL}_n(k) and considering the regular representation or a faithful representation \phi: G \to \mathrm{GL}_V(k). For g \in G(k), the image \phi(g) decomposes uniquely as \phi(g) = \phi(g)_s \phi(g)_u = \phi(g)_u \phi(g)_s via the matrix Jordan form, where \phi(g)_s is diagonalizable over k and \phi(g)_u has all eigenvalues 1. Since the semisimple and unipotent parts are polynomials in \phi(g), they lie in \phi(G(k)), and by faithfulness of \phi, there exist unique g_s, g_u \in G(k) mapping to them. Uniqueness follows from the uniqueness in \mathrm{GL}_V(k) and the embedding properties.^[29]^[28] In characteristic 0, this relies directly on the Jordan canonical form over algebraically closed fields.^[13] The decomposition is preserved under group homomorphisms and automorphisms: if \psi: G \to H is a morphism of linear algebraic groups, then \psi(g_s) and \psi(g_u) are the semisimple and unipotent parts of \psi(g), respectively.^[28] This functoriality extends the abstract Jordan decomposition in the Lie algebra to the group level via the exponential map in characteristic 0.^[4] In positive characteristic p > 0, the decomposition still exists over algebraically closed k, but the proof adapts using the fact that semisimple elements are diagonalizable over k and unipotent elements satisfy g^{p^m} = 1 for some m, without relying on the full Jordan form, which may fail. Chevalley's version refines this for semisimple groups by incorporating restricted root systems in the Lie algebra, ensuring the decomposition aligns with the p-structure via the Frobenius map and ensures compatibility with the Chevalley basis.^[13]^[4] For rationality over non-closed fields, if G is defined over a perfect field F \subseteq k and g \in G(F), then g_s, g_u \in G(F), so the decomposition is defined over F. This fails over imperfect fields, where separability issues arise, but holds for perfect F due to the polynomial nature of the parts.^[28]^[13] This property facilitates descent and Galois cohomology applications in the study of forms of algebraic groups.^[4]

Key Subgroups

Maximal tori

A torus T in a linear algebraic group G defined over a field k is a connected diagonalizable subgroup, meaning it is isomorphic to a closed subgroup of the diagonal matrices in \mathrm{GL}_n(k) for some n. Over an algebraically closed field, every torus T is isomorphic to (\mathbb{G}_m)^r for some integer r \geq 0, where \mathbb{G}_m = k^\times is the multiplicative group and r = \dim T is the dimension of the torus.^[2] The character lattice X^*(T) = \mathrm{Hom}(T, \mathbb{G}_m) of a torus T is then a free abelian group of rank r, consisting of the algebraic group homomorphisms from T to \mathbb{G}_m.^[25] A maximal torus in G is a torus that is maximal among all tori with respect to inclusion, or equivalently, a maximal connected abelian subgroup consisting entirely of semisimple elements. In a connected linear algebraic group G over an algebraically closed field k, every maximal torus has the same dimension, called the rank of G, and any two maximal tori are conjugate under an element of G(k).^[30] Moreover, every semisimple element of G lies in some maximal torus.^[30] Over a general field k, a torus T may not be split, meaning it need not be isomorphic to (\mathbb{G}_m)^r over k itself; instead, there exists a finite Galois extension L/k, called a splitting field for T, over which T_L becomes isomorphic to (\mathbb{G}_m)^r. The absolute rank or split rank of T is this dimension r, while the k-rank of G is the dimension of a maximal k-split torus in G, i.e., the largest r such that G contains a subgroup isomorphic to (\mathbb{G}_m)^r over k. Any torus decomposes uniquely as a product of its maximal k-split part and its k-anisotropic kernel.^[25] For a maximal torus T in a connected reductive group G, the normalizer N_G(T) and centralizer C_G(T) satisfy C_G(T)^0 = T, and the Weyl group W(G, T) = N_G(T) / C_G(T) is a finite group that acts faithfully on the character lattice X^*(T) by permutation of the basis elements corresponding to the roots. This action is independent of the choice of maximal torus up to isomorphism.^[25]^[30]

Unipotent subgroups

A unipotent subgroup U of a linear algebraic group G is a closed subgroup consisting entirely of unipotent elements, i.e., elements whose eigenvalues are all 1 in every rational representation of G.^[16] Such subgroups arise as the unipotent radicals of parabolic subgroups and are generated by unipotent elements of G.^[13] In characteristic 0, every unipotent subgroup is connected.^[16] Unipotent subgroups are nilpotent algebraic groups, meaning that the lower central series U = \gamma_1(U) \supset \gamma_2(U) \supset \cdots, defined by \gamma_{i+1}(U) = [U, \gamma_i(U)], terminates at the identity after finitely many steps.^[16] The commutator subgroup [U, U] is itself a unipotent subgroup, and this property holds over any field.^[13] Nilpotency implies the existence of a central series $1 = Z_0(U) \subset Z_1(U) \subset \cdots \subset Z_m(U) = U, where each Z_{i+1}/Z_i(U) is central in U/Z_i(U).^[16] Over a perfect field, a connected unipotent subgroup U admits a composition series with successive quotients isomorphic to the additive group \mathbb{G}_a, the one-dimensional unipotent group. Over a perfect field, a connected unipotent subgroup U of dimension d admits a composition series with successive quotients isomorphic to ℊ_a, making it a nilpotent group of dimension d. In particular, it is unipotent and, in characteristic 0, isomorphic as a variety to affine space 𝔸^d via the exponential map.^[16]^[13] In characteristic 0, the exponential map \exp: \operatorname{Lie}(U) \to U is an isomorphism of algebraic varieties, identifying the Lie algebra with the group via a polynomial bijection.^[16] The group multiplication on U corresponds to the Baker-Campbell-Hausdorff formula on \operatorname{Lie}(U), which expresses the product \exp(x) \exp(y) = \exp(Z(x,y)) for x, y \in \operatorname{Lie}(U), where Z(x,y) is a convergent power series in the nilpotent Lie algebra.^[16] This isomorphism facilitates the study of representations and structure theorems for unipotent subgroups.^[13]

Borel subgroups and parabolic subgroups

In a linear algebraic group G defined over an algebraically closed field k, a Borel subgroup B is a maximal connected closed solvable subgroup.^[2] Such a subgroup B contains a maximal torus T (as detailed in the section on maximal tori) and can be decomposed as a semidirect product B = T \ltimes U, where U is the unipotent radical of B consisting of unipotent elements (as covered in the section on unipotent subgroups).^[31] All Borel subgroups of G are conjugate under the action of G(k).^[31] A fundamental consequence of this structure is the Bruhat decomposition, which expresses G as a disjoint union G = \bigcup_{w \in W} B w B, where W = N_G(T)/T is the Weyl group of G consisting of cosets of the normalizer N_G(T) of T in G modulo T.^[25] Each double coset B w B is a locally closed subvariety of G, known as a Bruhat cell, with dimension \dim(B w B) = \dim B + \ell(w), where \ell(w) denotes the length of w in W with respect to a set of representatives for the simple reflections in W.^[32] A parabolic subgroup P of G is a proper closed connected subgroup that contains some Borel subgroup B.^[33] Equivalently, P is parabolic if the homogeneous space G/P is a projective variety.^[18] Every parabolic subgroup P admits a Levi decomposition P = L \ltimes R_u(P), where L is a connected reductive Levi subgroup (a factor of P) and R_u(P) is the unipotent radical of P, a normal unipotent subgroup.^[34] All Levi factors of P defined over k are conjugate by elements of R_u(P)(k).^[34] The quotient G/P, known as a (partial) flag variety, is a smooth projective variety parameterizing flags of subspaces stabilized by P.^[18] Its dimension is given by \dim(G/P) = \dim G - \dim P.^[25] For a Borel subgroup B, the full flag variety G/B has dimension equal to the dimension of the unipotent radical of the opposite Borel.^[35] Given a Borel subgroup B = T \ltimes U containing the maximal torus T, the opposite Borel subgroup B^- is the unique Borel subgroup containing T such that B \cap B^- = T.^[35] It decomposes as B^- = T \ltimes U^-, where U^- is the unipotent radical opposite to U, generated by the unipotent subgroups corresponding to the negative roots relative to those in U. In a connected reductive group G, the subgroup generated by U and U^- equals the derived subgroup [G, G]. In general, the opposite Borels B and B^- play symmetric roles in decompositions of G.^[35]^[25]

Reductive and Semisimple Groups

Definitions and characterizations

A reductive linear algebraic group G over a field k is a smooth connected affine group such that its unipotent radical R_u(G) is trivial, meaning R_u(G) = \{e\}.^[26] Equivalently, G is reductive if the unipotent radical of G / Z(G)^0 is trivial, where Z(G)^0 denotes the connected component of the center of G.^[26] A semisimple linear algebraic group is a reductive group with finite center, or equivalently, one whose solvable radical R(G) is trivial.^[26] Thus, semisimple groups form a subclass of reductive groups, distinguished by the absence of nontrivial connected solvable normal subgroups and a finite center. Reductive groups admit a structural characterization as G = Z(G)^0 \cdot [G, G], an almost direct product where [G, G] is the derived subgroup, which is semisimple, with finite intersection.^[26] Over a field of characteristic zero, a connected linear algebraic group G is reductive if and only if its Lie algebra \mathfrak{g} = \mathrm{Lie}(G) is reductive.^[26] Prominent examples of reductive groups include the general linear group \mathrm{GL}_n, which has a one-dimensional center and thus is not semisimple, and classical groups such as the orthogonal group \mathrm{SO}_n and symplectic group \mathrm{Sp}_{2n}.^[26] The special linear group \mathrm{SL}_n exemplifies a semisimple group, possessing finite center for n \geq 2.^[26]

Structure theorems including Levi decomposition

One of the central structure theorems for linear algebraic groups concerns the Levi decomposition of parabolic subgroups. For a parabolic subgroup P of a reductive linear algebraic group G over a field k, there exists a reductive subgroup L of P, called a Levi factor, such that P = L \rtimes R_u(P), where R_u(P) is the unipotent radical of P and the semidirect product is split.^[26] This decomposition holds over algebraically closed fields of characteristic zero for parabolic subgroups of reductive groups, and Levi factors are unique up to conjugation by elements of the unipotent radical.^[36] In positive characteristic, existence may fail for certain group schemes, such as \mathrm{[SL](/page/SL)}_n(W_2(k)), and conjugacy of Levi factors is not always guaranteed, though it holds when the dimension of the unipotent radical is less than the characteristic p.^[34] The proof of the Levi decomposition proceeds by induction on the dimension of G, using centralizers of maximal tori. For a parabolic P containing a maximal torus T, the centralizer C_G(T) intersects P in a Levi factor L, and the unipotent radical R_u(P) acts faithfully on L via the semidirect product structure, ensuring the splitting.^[26] In characteristic zero, cohomological vanishing conditions, such as H^1(G, V) = 0 and H^2(G, V) = 0 for relevant modules V, guarantee the existence and conjugacy of Levi factors.^[34] Over fields of positive characteristic, restrictions arise from the Frobenius endomorphism and p-envelopes in the Lie algebra, which control the unipotent structure but may prevent splitting in non-reduced cases.^[36] A related theorem addresses solvable normal subgroups. For a connected reductive group G over a field of characteristic zero and any connected solvable normal subgroup N, the group decomposes as G = C_G(N) \rtimes N, where C_G(N) is the centralizer of N in G.^[26] This follows by induction on dimension: the centralizer C_G(N) is reductive, normalizes N, and the action is faithful, yielding the semidirect product. In positive characteristic, the decomposition holds for smooth groups over perfect fields but requires adjustments for the unipotent part using p-envelopes to handle inseparability.^[26] Reductive groups also admit an almost direct product decomposition involving their center and derived subgroup. For a connected reductive group G over an algebraically closed field, G = Z(G)^0 \cdot [G, G], where Z(G)^0 is the connected component of the center (serving as the radical, a torus), and [G, G] is the derived subgroup, with finite intersection.^[26] Here, [G, G] is semisimple, and the product is almost direct in the sense that the central torus Z(G)^0 intersects the semisimple part finitely. The proof uses induction on dimension and centralizers: the centralizer of a maximal torus yields the torus component, while the derived subgroup captures the semisimple structure, with finiteness ensured by dimension counts.^[26] In positive characteristic, the decomposition persists for reductive groups, but the derived subgroup may involve Frobenius kernels, and intersections remain controlled by p-envelopes.^[36]

Classification

Root systems and Dynkin diagrams

In a semisimple linear algebraic group G over an algebraically closed field of characteristic zero, with respect to a maximal torus T, the Lie algebra \mathfrak{g} = \Lie(G) admits a root space decomposition \mathfrak{g} = \Lie(T) \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_\alpha, where \Phi is the set of roots, each root space \mathfrak{g}_\alpha is one-dimensional, and for basis elements e_\alpha \in \mathfrak{g}_\alpha with \alpha > 0 and f_\alpha = e_{-\alpha} \in \mathfrak{g}_{-\alpha}, the Lie bracket satisfies [e_\alpha, f_\alpha] = h_\alpha with h_\alpha \in \Lie(T).^[37] This decomposition arises from the adjoint action of T on G, where roots correspond to the non-trivial weight spaces.^[37] The roots \Phi form a finite reduced root system in the real vector space X^*(T) \otimes \mathbb{R}, where X^*(T) is the character group of T, consisting of nonzero elements \alpha \in X^*(T) \otimes \mathbb{Q} such that \mathfrak{g}_\alpha \neq \{0\}.^[37] A subset \Delta \subset \Phi of simple roots is a basis for the root lattice over \mathbb{Z}, linearly independent over \mathbb{R}, such that every root is an integer linear combination of elements in \Delta with all coefficients nonnegative or all nonpositive.^[38] The Cartan matrix A = (a_{ij}) associated to \Delta = \{\alpha_i\} has entries a_{ij} = 2 (\alpha_i, \alpha_j) / (\alpha_j, \alpha_j), where (\cdot, \cdot) is the invariant bilinear form on X^*(T) \otimes \mathbb{R} induced from the Killing form on \mathfrak{g}; the diagonal entries are 2, and off-diagonal entries are nonpositive integers.^[39] This matrix is independent of the choice of simple roots and determines the root system up to isomorphism.^[37] The Dynkin diagram encodes the Cartan matrix as a graph with vertices corresponding to simple roots and edges (single, double, or triple) indicating the absolute values of off-diagonal entries, oriented arrows distinguishing root lengths when unequal.^[40] The irreducible root systems of rank n are classified by the following diagrams: A_n (linear chain of n vertices, corresponding to \mathfrak{sl}_{n+1}); B_n (chain of n vertices with a double bond and arrow at the short root end); C_n (chain with double bond and arrow at the long root end); D_n (chain of n-2 vertices branching to two at the end); E_6, E_7, E_8 (chain with a branch at the third vertex from the end, extending to lengths 6, 7, 8); F_4 (chain with double and triple bonds); and G_2 (two vertices connected by a triple bond with arrow).^[38] Extended Dynkin diagrams append an additional vertex connected to the highest root, yielding affine types used in the study of affine Kac-Moody algebras.^[39] The Weyl group W is the finite group generated by reflections s_\alpha: \lambda \mapsto \lambda - \langle \lambda, \alpha^\vee \rangle \alpha for \alpha \in \Phi, where \alpha^\vee = 2\alpha / (\alpha, \alpha) is the coroot; it acts faithfully on X^*(T) \otimes \mathbb{R} and permutes the roots.^[37] The longest element w_0 \in W is the unique element sending all simple roots to negative roots, with length equal to the number of positive roots.^[38] A choice of Borel subgroup B containing T determines a set of positive roots \Phi^+ = \{\alpha \in \Phi \mid \mathfrak{g}_\alpha \subset \Lie(U)\}, where U is the unipotent radical of B, consisting of those roots with nonnegative coefficients in the basis \Delta.^[37] The height function on \Phi^+ is defined by \ht(\alpha) = \sum k_i for \alpha = \sum k_i \alpha_i with \alpha_i \in \Delta and k_i \in \mathbb{N}, attaining its maximum at the highest root.^[39]

Chevalley groups and split forms

Chevalley introduced a basis for the Lie algebra of a split semisimple algebraic group that allows the construction of integral models over the integers. For a root system with simple roots \alpha_i, the Chevalley basis consists of elements h_i (corresponding to simple coroots), e_\alpha for positive roots \alpha, and f_\alpha = e_{-\alpha} for negative roots, satisfying [e_\alpha, f_\alpha] = h_\alpha where h_\alpha = \sum k_i h_i with integer coefficients k_i, and the structure constants in the Lie brackets [e_\alpha, e_\beta] = N_{\alpha,\beta} e_{\alpha+\beta} (with N_{\alpha,\beta} \in \mathbb{Z}) are integers determined by the root system.^[41] This basis ensures that the Lie algebra admits a \mathbb{Z}-form, enabling the extension to characteristic zero and positive characteristics. Using this basis, Chevalley constructed a split reductive group scheme G over \mathbb{Z} for each root datum, generated by a maximal split torus T and unipotent subgroups U_\alpha (one for each positive root \alpha), subject to relations mirroring the Lie algebra brackets, such as [x_\alpha(t), x_\beta(u)] = \prod x_{\alpha+\beta}(c t^k u^m) with integer constants c, k, m.^[42] The group G is smooth and affine over \Spec(\mathbb{Z}), with generic fiber the split reductive group over \mathbb{Q}, and it possesses a BN-pair structure: a Borel subgroup B = T U (with U the product of U_\alpha) and a normalizer N generated by T and Weyl group elements n_\alpha satisfying the Tits axioms, including the Bruhat decomposition G = \bigcup B w B for w in the Weyl group. Specializing to finite fields \mathbb{F}_q (with q = p^n, p prime not dividing certain denominators from the root system), the groups G(\mathbb{F}_q) are the finite Chevalley groups, which coincide with the fixed points of a Frobenius endomorphism on the split form over the algebraic closure. For adjoint types (where the root datum has trivial center), these groups are simple except in small cases like A_1(\mathbb{F}_2), A_1(\mathbb{F}_3), and B_2(\mathbb{F}_2).^[42] The universal Chevalley group corresponds to the simply connected form (minimal kernel in the isogeny to the adjoint form), while the adjoint form is the quotient by the center; over \mathbb{F}_q, the simply connected versions may have non-trivial Schur multipliers (the second homology H_2(G(\mathbb{F}_q), \mathbb{Z})), which are typically cyclic of order 1, 2, 3, or 8 depending on the type and q, as computed for all irreducible root systems.^[43] Twisted forms arise by applying a Galois automorphism \sigma of the algebraic closure that acts on the Dynkin diagram via a graph automorphism (of order 2 or 3 for non-exceptional types), yielding groups like ^2A_n(\mathbb{F}_{q^2}) or ^2B_2(\mathbb{F}_{q^2}) as fixed points under the twisted Frobenius \text{Fr}_\sigma, which permutes roots accordingly while preserving the BN-pair structure up to isomorphism.^[44]

Classification over non-algebraically closed fields

Over a non-algebraically closed field k, the isomorphism classes of reductive linear algebraic groups G with a given root datum are classified using Galois cohomology. Specifically, the forms of a fixed split reductive group \hat{G} over the algebraic closure \bar{k} are parametrized by the first cohomology set H^1(k, \hat{G}), which captures the k-twists of \hat{G}.^[26] This cohomology measures the extent to which \hat{G} fails to be split over k, with trivial elements corresponding to the split form. Inner forms arise from the center of the simply connected cover, classified by H^1(k, Z(\hat{[G](/page/G)})), where Z(\hat{[G](/page/G)}) is the center of the dual group \hat{[G](/page/G)}. These are isogenous to the split form and preserve the adjoint quotient. Outer forms, in contrast, are classified by H^1(k, \Aut(\hat{[G](/page/G)})), reflecting automorphisms of the root datum that do not arise from inner automorphisms; they include more general twists and are parametrized by torsors under the outer automorphism group \Out(\hat{[G](/page/G)}). For semisimple groups, the full set of forms is the image of H^1(k, \Aut(\hat{[G](/page/G)})) \to H^1(k, \hat{[G](/page/G)}).^[26]^[45] A reductive group G over k is quasi-split if it contains a Borel subgroup defined over k, equivalently, if it admits a maximal split torus over k. In this case, the minimal parabolic subgroup over k is unique up to conjugation, and the relative root system is generated by a base of simple roots with no distinguished roots (i.e., \Delta_0 = \emptyset). Quasi-split forms represent the "most split" non-split possibilities, with a unique inner form up to isomorphism.^[26]^[45] The anisotropic kernel of G with respect to a maximal split torus S is the derived subgroup of the centralizer Z_G(S); it is an anisotropic semisimple group. For semisimple G, the anisotropic kernel is anisotropic semisimple. Dimension bounds follow from the structure: the dimension of a maximal split torus S satisfies \dim S \geq \# k\Delta, where k\Delta is the set of k-simple roots, and the dimension of a Borel subgroup is \frac{1}{2}(\dim G + \dim S). Anisotropic groups over local fields like \mathbb{R} have compact groups of k-points.^[26]^[45] Tits provided a complete classification of semisimple groups over arbitrary fields using the geometry of buildings and apartments associated to maximal split tori. The building of G is a simplicial complex whose apartments correspond to maximal split tori, with chambers labeled by Weyl group elements; the k-structure is captured by the Galois action on the root system via the Frobenius or Galois group \Gamma = \Gal(\bar{k}/k). The invariant is the index I(G) = (D, \Delta_0, \star), where D is the Dynkin diagram, \Delta_0 the set of distinguished simple roots (fixed by \Gamma), and \star indicates diagram automorphisms. This classifies all forms up to isomorphism, extending the split case.^[26]^[45] For example, over \mathbb{R}, the special linear group \SL_n has two real forms up to isomorphism: the split form \SL_n(\mathbb{R}), which is non-compact, and the compact form \SU(n), arising as an inner twist via the quaternion algebra (for even n) or outer automorphism. The compact form has trivial \mathbb{R}-points beyond \{\pm I\} in certain cases, illustrating anisotropy, while the split form has maximal rank tori of dimension n-1.^[26]^[45]

Representations

Irreducible representations and weights

For a semisimple linear algebraic group G over an algebraically closed field k of characteristic zero, the finite-dimensional irreducible rational representations are parametrized by dominant weights in the character lattice X^*(T) of a maximal torus T \subset G. In characteristic zero, all finite-dimensional rational representations of semisimple groups are completely reducible into direct sums of irreducibles. A weight \Lambda \in X^*(T) is dominant if it lies in the closed fundamental Weyl chamber X^*(T)^+, defined as the set of \lambda such that \langle \lambda, \alpha^\vee \rangle \geq 0 for all simple coroots \alpha^\vee corresponding to a choice of positive roots. This parametrization follows from the highest weight theorem, which asserts that every irreducible representation has a unique highest weight that is dominant. Given a dominant weight \Lambda \in X^*(T)^+, the corresponding irreducible representation is denoted V(\Lambda), realized as the quotient of the induced module from the Borel subgroup by its unique simple submodule. It admits a highest weight vector v_\Lambda such that t \cdot v_\Lambda = \Lambda(t) v_\Lambda for t \in T and the unipotent radical U_\alpha of a positive root subgroup acts trivially on v_\Lambda for each positive root \alpha > 0, i.e., U_\alpha v_\Lambda = 0. The weights of V(\Lambda) all lie in the convex hull of the Weyl group orbit W \cdot \Lambda, and the representation is completely determined by the action on this cyclic vector generated by v_\Lambda. The character \chi_\Lambda of V(\Lambda), which encodes the formal sum of weights with multiplicities, is given by the Weyl character formula:

\chi_\Lambda = \frac{\sum_{w \in W} \varepsilon(w) e^{w(\Lambda + \rho) - \rho}}{\sum_{w \in W} \varepsilon(w) e^{w(\rho) - \rho}},

where W is the Weyl group, \varepsilon(w) is the sign of w, \rho is half the sum of the positive roots, and e^\mu denotes the basis element for weight \mu. This formula, originally due to Weyl for compact Lie groups but valid in the algebraic setting via analytic continuation of characters, allows explicit computation of dimensions and weight multiplicities for classical types. To compute the multiplicity m(\mu) of a weight \mu in V(\Lambda), the recursive Freudenthal multiplicity formula provides an efficient algorithm: for \mu \neq \Lambda,

m(\Lambda, \mu) = \sum_{\alpha > 0} \frac{(\Lambda + \mu + 2\rho, \alpha)}{(\alpha, \alpha)} m(\Lambda, \mu + \alpha),

with m(\Lambda, \Lambda) = 1 and m(\Lambda, \nu) = 0 if \nu \not\leq \Lambda in the dominance order. This formula, derived from the adjoint action of the Lie algebra on the tensor product with the adjoint representation, is particularly useful for verifying branching rules and decomposing tensor products in low ranks. In positive characteristic p > 0, the highest weight theory simplifies for restricted dominant weights \Lambda \in X_1(T)^+, where $0 \leq \langle \Lambda, \alpha^\vee \rangle < p for all simple coroots; the modules L(\Lambda) with such \Lambda are irreducible and form a complete set of simple rational G-modules of restricted highest weight. For general dominant \Lambda, the Frobenius endomorphism F: G \to G induces kernels G_r = \ker(F^r) for r \geq 1, which are infinitesimal group schemes whose representations parametrize the composition factors of L(\Lambda) via the Steinberg tensor product theorem: L(\Lambda) \cong L(\Lambda_0) \otimes L(\Lambda_1)^{(1)} \otimes \cdots \otimes L(\Lambda_s)^{(s)}, where \Lambda = \Lambda_0 + p \Lambda_1 + \cdots + p^s \Lambda_s with each \Lambda_i restricted, and V^{(j)} denotes the j-th Frobenius twist. This decomposition highlights the p-adic structure but introduces complications like non-vanishing cohomology absent in characteristic zero.^[16]

Induced and tensor representations

One fundamental construction of representations for a linear algebraic group G over an algebraically closed field k is induction from a subgroup H \leq G. Given a rational representation \rho: H \to \mathrm{GL}(V) of H, the induced representation \mathrm{Ind}_H^G V is the k[G] \otimes_{k[H]} V, the tensor product of the coordinate ring of G over that of H with V, equipped with the natural G-action. For parabolic subgroups P, this yields a rational G-representation whose underlying vector space includes the finite-dimensional space of global sections of the associated vector bundle on G/P.^[16] Induction from Borel subgroups B is especially significant for reductive groups, as it realizes finite-dimensional irreducible representations. For a character \chi_\lambda: B \to k^\times corresponding to a weight \lambda \in X(T) \otimes \mathbb{Z} (with T a maximal torus in B), the induced module \mathrm{Ind}_B^G k_\lambda (often denoted E(\lambda)) has nonzero global sections precisely when \lambda is dominant, and it has a unique simple quotient L(\lambda) of highest weight \lambda. This follows from the structure of parabolic inductions and the semisimplicity of representations in characteristic zero.^[16] Frobenius reciprocity provides a duality between induction and restriction functors, facilitating the study of these constructions. Specifically, for rational G-representations W and H-representations V, there is a natural isomorphism \mathrm{Hom}_G(W, \mathrm{Ind}_H^G V) \cong \mathrm{Hom}_H(\mathrm{Res}_H^G W, V), which preserves exactness and dimensions. This adjunction is central to decomposing induced modules and computing multiplicities in representation theory.^[16] A geometric incarnation of induction from Borel subgroups is given by the Borel–Weil theorem, which realizes irreducible representations as cohomology of line bundles on the flag variety. For a complex semisimple group G with Borel subgroup B and dominant integral weight \lambda, let L_\mu denote the line bundle on G/B associated to weight \mu, and let w_0 be the longest element of the Weyl group W. Then, the contragredient of the induced representation (\mathrm{Ind}_B^G L_{-w_0 \lambda})^\vee is isomorphic to the irreducible representation V(\lambda) of highest weight \lambda, where higher cohomology vanishes. This theorem, extended by Bott to account for Weyl group action in general cases, underscores the interplay between algebraic and geometric methods.^[46] Tensor products offer another construction, decomposing into direct sums of irreducibles via combinatorial coefficients. For the general linear group \mathrm{GL}_n(\mathbb{C}), if V(\lambda) and V(\mu) are irreducible polynomial representations labeled by partitions \lambda, \mu, their tensor product decomposes as

V(\lambda) \otimes V(\mu) = \bigoplus_\nu c^\nu_{\lambda \mu} V(\nu),

where the Littlewood–Richardson coefficients c^\nu_{\lambda \mu} are nonnegative integers counting semi-standard Young tableaux of skew shape \nu / \lambda with content \mu that are lattice permutations. These coefficients determine the representation ring of \mathrm{GL}_n and extend to other classical groups via branching rules.^[47] Restriction of representations to Levi subgroups of parabolic subgroups P = L U (with L the Levi factor) yields modules that decompose into direct sums over characters of the center of L, often analyzable as Harish-Chandra modules. These are finitely generated modules over the universal enveloping algebra of the Lie algebra of L, with finite-dimensional generalized eigenspaces for the center, finite multiplicity for each generalized weight, and L-finite action. Such modules facilitate parabolic induction, where \mathrm{Ind}_P^G (V \otimes \chi) for V a representation of L and \chi a character of L produces representations with explicit Langlands quotient structure in characteristic zero.^[16] In positive characteristic p > 0, the Steinberg tensor product theorem provides a canonical decomposition for irreducible rational representations of semisimple algebraic groups. For a dominant weight \lambda \in X(T)^+, write \lambda = \lambda_0 + p \lambda_1 + \cdots + p^r \lambda_r in its p-adic expansion, with each $0 \leq \lambda_i < p. Then, the irreducible module L(\lambda) is isomorphic to the tensor product

L(\lambda) \cong L(\lambda_0) \otimes L(\lambda_1)^{(1)} \otimes \cdots \otimes L(\lambda_r)^{(r)},

where each L(\lambda_i) is the irreducible module of restricted highest weight \lambda_i. This theorem, due to Steinberg, exploits Frobenius morphisms and controls the complexity of representations in modular settings.^[16]

Applications

Geometric invariant theory and quotients

Geometric invariant theory (GIT) provides a framework for constructing quotients of algebraic varieties under actions of linear algebraic groups, particularly reductive ones, by associating invariants to orbits and forming moduli spaces that parametrize isomorphism classes. For a reductive linear algebraic group G acting on an affine variety X over an algebraically closed field k, the ring of invariants k[X]^G consists of regular functions on X that are fixed by the G-action.^[48] This ring is finitely generated as a k-algebra when G is reductive, ensuring that the categorical quotient X // G = \operatorname{Spec} k[X]^G exists as an affine variety.^[48] The natural projection \pi: X \to X // G is a G-invariant morphism that is constant on G-orbits, separates closed G-invariant subsets, and is universal among such morphisms with these properties.^[48] To extend this to projective varieties and ensure well-behaved quotients, GIT introduces notions of stability via linearizations of ample line bundles. A key tool is the Hilbert-Mumford criterion, which characterizes semistable points in terms of one-parameter subgroups (1-PS). For a linearized action of G on a projective variety X = \mathbb{P}(V), a point x \in X is semistable if there exists no 1-PS \lambda of G such that \lim_{t \to 0} \lambda(t) \cdot x = 0.^[48] Points satisfying the stricter condition that the limit exists and the stabilizer is finite-dimensional are stable. The semistable locus X^{ss} admits a good quotient X^{ss} // G, which is a projective geometric quotient for the stable locus X^s, meaning orbits are closed and separated in the quotient.^[48] David Mumford developed GIT in 1965 to construct such quotients systematically, resolving issues with orbit closures and providing projective moduli spaces for stable objects under group actions.^[48] For affine actions, the existence of finite generation of invariants in positive characteristic relies on the geometric reductivity of reductive groups, established by Haboush's theorem: over any field, a reductive group is geometrically reductive, implying that the invariants k[X]^G are finitely generated for any affine X and thus affine quotients exist.^[49] This result, originally conjectured by Mumford, confirms that semisimple groups (hence reductive ones) yield well-defined categorical quotients without pathological behavior in mixed characteristic.^[49] A classic example arises from the action of \mathrm{SL}(2) on the space of binary forms of degree d, viewed as an affine variety in coefficients. The ring of invariants is generated by the discriminant for d=2 (a quadratic form) and more generally by covariants like the Aronhold invariants for higher degrees, allowing the categorical quotient to parametrize isomorphism classes of such forms up to \mathrm{SL}(2)-equivalence.^[48] This construction illustrates how GIT quotients capture moduli of plane curves or conics, with semistable points excluding those with repeated roots via the Hilbert-Mumford criterion applied to diagonal 1-PS.^[48]

Connections to Lie theory and number theory

Linear algebraic groups defined over the real or complex numbers give rise to Lie groups through their points over these fields. For a linear algebraic group G defined over \mathbb{C}, the group G(\mathbb{C}) is a complex Lie group, and when G is simply connected, semisimple, and connected, it coincides with the simply connected complex Lie group having the same Lie algebra.^[5] Similarly, for G defined over \mathbb{R}, the real points G(\mathbb{R}) form a real Lie group, which in the semisimple case is a covering group of the real points of an algebraic group, with the covering related via the universal cover of the corresponding complex group.^[5] These identifications allow the structure theory of Lie groups to inform the analytic properties of algebraic groups over \mathbb{R} and \mathbb{C}. A key structural feature for real semisimple Lie groups arising from linear algebraic groups is the Cartan decomposition. For such a group G(\mathbb{R}), there exists a Cartan involution \theta on the Lie algebra \mathfrak{g}_\mathbb{R}, yielding a decomposition \mathfrak{g}_\mathbb{R} = \mathfrak{k} \oplus \mathfrak{p}, where \mathfrak{k} is the Lie algebra of a maximal compact subgroup K and \mathfrak{p} is the orthogonal complement with respect to the Killing form made positive definite by \theta. At the group level, this extends to G(\mathbb{R}) = K \exp(\mathfrak{p}), a diffeomorphism that generalizes the polar decomposition of matrices and facilitates the study of invariant metrics and representations.^[50] For complex semisimple Lie groups G(\mathbb{C}) from linear algebraic groups, the Iwasawa decomposition provides a useful uniformization. It decomposes G(\mathbb{C}) = K A N, where K is a maximal compact subgroup, A is a maximal split torus, and N is the unipotent radical of a Borel subgroup, with the decomposition being a diffeomorphism. This structure, analogous to the real case but adapted to the complex setting, aids in analyzing the topology and harmonic analysis on these groups, reducing problems to the compact factor K.^[50] In the specific case of G = \mathrm{GL}_n, the decomposition takes the form \mathrm{GL}_n(\mathbb{R}) = N A K with K = \mathrm{O}_n(\mathbb{R}), A the diagonal positive matrices, and N upper triangular unipotent.^[5] Turning to number theory, arithmetic subgroups of linear algebraic groups play a central role in the study of automorphic forms and related arithmetic objects. For a linear algebraic group G defined over the rationals \mathbb{Q}, an arithmetic subgroup \Gamma is the intersection of G(\mathbb{Q}) with a lattice in G(\mathbb{R}), such as \Gamma = G(\mathcal{O}) for a ring of integers \mathcal{O}. These subgroups are discrete in G(\mathbb{R}) and act properly discontinuously on symmetric spaces associated to G, yielding quotients that parametrize arithmetic data like modular curves. A prototypical example is G = \mathrm{SL}_2, where \Gamma = \mathrm{SL}_2(\mathbb{Z}) is an arithmetic subgroup whose action on the upper half-plane produces the modular curve X(1), and modular forms of weight $2k for \Gamma are holomorphic functions f: \mathbb{H} \to \mathbb{C} satisfying f\left( \frac{az + b}{cz + d} \right) = (cz + d)^{2k} f(z) for \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma, holomorphic at cusps.^[51] These forms generate L-functions with Euler products, linking to Dirichlet series and arithmetic invariants like class numbers.^[51] The Langlands program establishes profound connections between linear algebraic groups and number theory by conjecturing a correspondence between automorphic representations and Galois representations. For a reductive linear algebraic group G over a number field F, the program posits that irreducible automorphic representations \pi of the adelic group G(\mathbb{A}_F) (factorizing as tensors \pi = \otimes_v \pi_v over places v) correspond to continuous homomorphisms \rho: \mathrm{Gal}(\overline{F}/F) \to {}^L G(\mathbb{C}), where {}^L G is the Langlands dual group. The local components \pi_v at finite places match the Frobenius semisimplification of \rho via L-parameters, with global compatibility ensured by matching Artin and automorphic L-functions L(s, \pi) = L(s, \rho).^[52] For G = \mathrm{GL}_n, this reduces to a bijection between cuspidal automorphic representations and n-dimensional Galois representations, realizing number-theoretic objects like modular forms as geometric Galois data.^[52] In the local setting over p-adic fields, points G(F) for a non-archimedean local field F (residue characteristic p) form p-adic Lie groups central to the local Langlands correspondence. These groups admit the Bruhat-Tits building B(G, F), a simplicial complex on which G(F) acts, with vertices parametrized by special parahoric subgroups and edges reflecting valuations on the root datum.^[53] The building encodes the structure of hyperspecial maximal compact subgroups and filtration spaces, facilitating the classification of smooth irreducible representations of G(F). The local Langlands conjecture, proven for certain groups like \mathrm{GL}_n, asserts a bijection between such representations and irreducible representations of the Weil-Deligne group of F, with the building providing a geometric framework for supercuspidal induction and depth-zero types in the correspondence.^[53] This local picture underpins the global Langlands program by providing the place-by-place matching required for the adelic automorphic forms.

Computational aspects and modern uses

Computational software systems such as GAP and Magma provide essential tools for constructing and manipulating linear algebraic groups, particularly Chevalley groups and root systems. In GAP, the CHEVIE package implements algorithms for finite reflection groups and reductive algebraic groups over finite fields, enabling the computation of Weyl group representations and character tables for groups like SL_n(q).^[54] Similarly, Magma's intrinsic functions support the creation of root systems and root data for semisimple groups, allowing users to compute Dynkin diagrams, positive roots, and Galois actions on groups defined over number fields.^[55] These capabilities facilitate the explicit construction of split forms of linear algebraic groups, such as the special linear group SL_n over finite fields, by generating generators and relations from root system data.^[56] Recognition algorithms for black-box presentations of linear algebraic groups, where elements are accessed only via multiplication oracles, have advanced significantly for classical types. These methods identify groups isomorphic to PSL_n(q) or other projective special linear groups by probabilistic testing of subgroup structures, often running in polynomial time relative to the group order.^[57] For classical groups, the Aschbacher-O'Nan-Scott theorem provides a structural foundation, classifying maximal subgroups into geometric, irreducible, or almost simple classes, which guides constructive recognition by reducing the problem to verifying specific embedding dimensions and field characteristics.^[58] Such algorithms output standard generators, enabling further computations like order calculation or representation construction, with Las Vegas guarantees for correctness.^[59] In modern machine learning, representations of linear algebraic groups underpin equivariant neural networks, which enforce symmetry in models for physical systems. Post-2020 developments include E(3)-equivariant architectures that use irreducible representations of rotation and translation groups to process molecular structures, improving generalization in density functional theory predictions.^[60] Similarly, Clifford group equivariant networks leverage spin representations for O(3)-invariant tasks, such as protein folding, by parameterizing layers with group actions to preserve equivariance under transformations.^[61] These approaches draw on highest weight theory for decomposing tensor products, linking back to irreducible representations while enhancing efficiency in high-dimensional data.^[62] Quantum computing applications exploit Lie group structures from linear algebraic groups for algorithm design and complexity analysis. Recent work employs Lie algebras to characterize variational quantum circuits, deriving explicit generators for the unitary group and analyzing barren plateaus through concentration of measure on group manifolds.^[63] For instance, free-fermion encodings map to Lie subalgebras, enabling simulation of symmetry-protected phases with reduced qubit overhead.^[64] These methods also inform quantum advantage proofs by quantifying entanglement via representation theory of reductive groups.^[65] In particle physics, reductive linear algebraic groups model gauge symmetries, with spontaneous symmetry breaking generating masses via Higgs mechanisms. The Standard Model's SU(2) × U(1) electroweak sector, a reductive group over the complexes, breaks to U(1) electromagnetism, yielding the W and Z boson masses while preserving photon masslessness.^[66] Grand unified theories extend this to larger reductive groups like SU(5), where breaking patterns unify forces and predict proton decay, though unobserved, constraining model parameters.^[67] Numerical methods for matrix groups over the reals emphasize stability in Lie group integrators to preserve manifold structure. Lie-group exponential maps ensure solutions remain in subgroups like SO(n), avoiding drift from rounding errors in Euler discretizations.^[68] For GL(n, ℝ), symmetry-adapted solvers use Cayley transforms for orthogonal projections, maintaining positive definiteness and conditioning in optimization tasks.^[69] These techniques achieve backward stability, with error bounds scaling as machine epsilon times the condition number of the representation.^[70]

Relation to Lie groups

Linear algebraic groups defined over the real numbers \mathbb{R} or complex numbers \mathbb{C} yield Lie groups via their points over these fields: the set G(\mathbb{R}) forms a real Lie group, and G(\mathbb{C}) forms a complex Lie group, embedded as closed subgroups of the general linear group \mathrm{GL}(n, \mathbb{R}) or \mathrm{GL}(n, \mathbb{C}).^[5] Over \mathbb{C}, every connected semisimple Lie group is algebraic, meaning it arises precisely as the complex points of a linear algebraic group.^[5] Over \mathbb{R}, connected semisimple real Lie groups are covering groups of real algebraic groups, with the algebraic structure providing a polynomial description of the group's defining relations.^[5] These Lie groups admit a maximal compact subgroup K, which is compact and lies in the center of the group's unitary representations; for instance, in the special linear group \mathrm{SL}(n, \mathbb{R}), K = \mathrm{SO}(n).^[71] Linear algebraic groups serve as an "integral form" or algebraic backbone for their associated Lie groups, capturing the structure in terms of polynomial equations over the base field. A prime example is \mathrm{SL}(n, \mathbb{R}), which acts as a real form of the complex Lie group \mathrm{SL}(n, \mathbb{C}): the former consists of real matrices of determinant 1, providing an algebraic model that restricts the complex group's analytic freedom to real coefficients while preserving the Lie algebra \mathfrak{sl}(n, \mathbb{R}).^[71] This real form is obtained via complex conjugation as an involution on \mathrm{SL}(n, \mathbb{C}), yielding fixed points that form the real algebraic group.^[71] Such forms highlight how algebraic groups encode the "discrete" or polynomial aspects of Lie groups, facilitating arithmetic and geometric applications. The representation theory of these real Lie groups draws heavily on the algebraic data, as pioneered by Harish-Chandra. Specifically, the Harish-Chandra embedding theorem allows the construction of discrete series representations—square-integrable irreducible unitary representations—from the algebraic structure of the group, embedding the representation into induced modules over the complexified algebraic group.^[72] For reductive real algebraic groups, this approach shows that discrete series exist if and only if the group admits a compact Cartan subgroup, linking analytic unitarity to algebraic compactness conditions.^[72] Infinitesimally, linear algebraic groups and their Lie group realizations share the same Lie algebra, defined as the tangent space at the identity equipped with the commutator bracket, ensuring local equivalence near the identity element.^[5] However, the algebraic framework is more restrictive: linear algebraic groups are Zariski-closed subgroups defined by polynomial ideals, imposing global polynomial constraints absent in general Lie groups.^[5] Consequently, not all Lie subgroups of algebraic groups are algebraic; a canonical counterexample is the one-parameter subgroup of the 2-torus T^2 = \mathbb{R}^2 / \mathbb{Z}^2 generated by the flow (t \mapsto (t, \alpha t \mod 1)) for irrational \alpha, which is dense and non-closed, hence non-algebraic despite being a Lie subgroup.^[73] This illustrates the analytic flexibility of Lie groups beyond algebraic varieties.

Group schemes and Hopf algebras

Linear algebraic groups can be generalized to the broader framework of affine group schemes over a base field k, where an affine group scheme G is defined as the spectrum of a Hopf algebra A over k. The Hopf algebra structure on A encodes the group operations: the comultiplication \Delta: A \to A \otimes_k A corresponds to the multiplication map, the counit \varepsilon: A \to k to the unit map, and the antipode S: A \to A to the inversion map.^[74] This construction allows for a functorial description of the group, where G represents the functor from the category of k-algebras to groups given by G(R) = \Hom_{k\text{-alg}}(A, R) for any commutative k-algebra R.^[74] Linear algebraic groups correspond precisely to those affine group schemes that are of finite type over k, meaning A is finitely generated as a k-algebra.^[75] The representable functor perspective unifies the geometric and algebraic viewpoints: morphisms from an affine scheme \Spec R to G in the category of schemes over k are in natural isomorphism with algebra homomorphisms A \to R, preserving the group structure via the Hopf algebra maps.^[74] This equivalence extends the classical notion of points of an algebraic group, allowing group schemes to capture infinitesimal or non-reduced structures that linear algebraic groups over algebraically closed fields may not exhibit. For instance, in positive characteristic, affine group schemes of finite type include linear algebraic groups but also more general objects like finite flat group schemes, which are flat and of finite presentation over k.^[74] Finite flat group schemes provide concrete examples beyond smooth linear algebraic groups. The scheme of nth roots of unity, denoted \mu_n, is the affine group scheme \Spec k/(x^n - 1), representing the functor R \mapsto \{r \in R^\times \mid r^n = 1\} on k-algebras R.^[74] In characteristic p > 0, the Frobenius kernel \alpha_p is another finite flat group scheme, given by \Spec k/(x^p), which represents the additive group functor R \mapsto (R, +) but truncated at p-torsion elements, as it is the kernel of the Frobenius endomorphism on the additive group \mathbb{G}_a.^[74] These schemes are flat over k and of order p or n, illustrating how Hopf algebras can model group-like objects with nilpotent elements. Representations of an affine group scheme G = \Spec A are equivalently comodules over the Hopf algebra A, where a finite-dimensional representation corresponds to a right comodule structure \rho: V \to V \otimes_k A satisfying coassociativity and counit properties derived from the coalgebra structure of A.^[74] The category of finite-dimensional comodules over A is thus tensorial and equivalent to the representation category of G, generalizing the module category for linear algebraic groups.^[74] This coalgebra perspective highlights the duality between the group scheme and its Hopf algebra, where coactions encode how G acts on vector spaces. Over non-algebraically closed fields, defining linear algebraic groups requires descent data to ensure consistency under base change. For an affine group scheme G over a Galois extension K/k with Galois group \Gamma, descent data consists of a \Gamma-action on the Hopf algebra A_K compatible with the Hopf structure, allowing reconstruction of a Hopf algebra A over k such that G = \Spec A descends properly.^[74] More generally, faithfully flat descent applies to arbitrary base changes, ensuring that group schemes over k can be obtained from those over a faithfully flat extension via cocycle conditions on the Hopf algebra maps.^[74] This framework is essential for studying linear algebraic groups over fields like the rationals or finite fields, where Galois cohomology classifies torsors and forms.^[74]

Tannakian categories and abelian varieties

A Tannakian category over a field k is defined as a rigid abelian tensor category \mathcal{C} equipped with a fiber functor \omega: \mathcal{C} \to \mathrm{Vec}_k, which is an exact faithful k-linear tensor functor to the category of finite-dimensional k-vector spaces, such that the endomorphism ring of the unit object is k.^[76] The automorphism group scheme \mathrm{Aut}^\otimes(\omega) of the fiber functor is a pro-algebraic group scheme over k, representing the tensor automorphisms of \omega.^[76] The reconstruction theorem states that for a neutral Tannakian category \mathcal{C} over k, there exists an affine group scheme G over k such that \mathcal{C} \cong \mathrm{Rep}_k(G) as tensor categories, via the equivalence induced by \omega, where G = \mathrm{Aut}^\otimes(\omega).^[76] This duality recovers the linear algebraic group G from its representation category, generalizing Tannaka's original theorem for compact groups to the algebraic setting.^[76] Abelian varieties provide a commutative specialization of linear algebraic groups, serving as the projective analogues of the additive group \mathbb{G}_a. An abelian variety A over a field k is a complete connected commutative group variety, meaning it is a projective algebraic variety equipped with a group structure given by morphisms of varieties.^[77] Each abelian variety A of dimension g admits a dual abelian variety \hat{A}, which parametrizes the degree-zero Picard group \mathrm{Pic}^0(A) consisting of translation-invariant invertible sheaves on A.^[77] The Picard variety of A is the component of the identity in the Picard scheme, often isomorphic to \hat{A}, and encodes the algebraic structure of line bundles on A.^[77] For a smooth projective curve C of genus g \geq 1 over k, the Jacobian variety \mathrm{Jac}(C) is the abelian variety of dimension g parametrizing the degree-zero line bundles on C, with a canonical principal polarization.^[77] The group law on \mathrm{Jac}(C) can be realized using theta divisors: ample line bundles correspond to effective theta divisors \Theta_L, and the addition of points translates via the relation \Theta_{L \otimes M} = t_{a+b}^* \Theta_L + t_a^* \Theta_M - 2\Theta_L for points a, b representing L, M, up to linear equivalence.^[77] Isogenies between abelian varieties are surjective group homomorphisms with finite kernels, preserving the group structure and having degree a power of the dimension, such as \deg() = n^{2g} for the multiplication-by-n map.^[77] For a prime \ell \neq \mathrm{char}(k), the \ell-adic Tate module T_\ell A = \varinjlim A[\ell^n](k^{\mathrm{sep}}) is a free \mathbb{Z}_\ell-module of rank $2g, carrying a continuous representation of the absolute Galois group \mathrm{Gal}(\bar{k}/k) that encodes p-adic étale cohomology and arithmetic properties of A.^[77]

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This article provides a review of the elementary theory of semisimple Lie al- gebras and Lie groups. It is essentially a summary of much of [K3]. The four.
[53]
[PDF] Modular Functions and Modular Forms
It is easy to define modular functions and forms, but less easy to say why they are important, especially to number theorists. Thus I will begin with a rather ...
[54]
AN ELEMENTARY INTRODUCTION TO THE LANGLANDS ...
This program emerged around 1967 in the form of a series of conjectures, and it has subsequently influenced recent research in number theory in much the same ...
[55]
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### Summary of p-Adic Groups, Bruhat-Tits Buildings, and Local Langlands Correspondence
[56]
The development version of the CHEVIE package of GAP3
I describe the current state of the development version of the CHEVIE package, which allows to transform the theories of Coxeter groups, reductive algebraic ...
[57]
Root Systems and Root Data - Magma Computational Algebra System
Sep 26, 2025 · Root systems are used to classify the semisimple Lie algebras. The closely related concept of a root datum is used to classify the groups of Lie ...Missing: linear | Show results with:linear
[58]
Introduction - Magma Computational Algebra System
This chapter describes Magma functions for computing with finite real root systems. A root system describes the reflections in a reflection group (Chapter ...Missing: software | Show results with:software
[59]
Effective black-box constructive recognition of classical groups
We describe a black-box Las Vegas algorithm to construct standard generators for classical groups defined over finite fields. We assume that the field has ...
[60]
[PDF] Black box classical groups - University of Oregon
If a black box simple group is known to be isomorphic to a classical group over a field of known characteristic, a Las Vegas algorithm is used to produce.
[61]
[PDF] Black-box recognition of finite simple groups of Lie type by statistics ...
Given a black-box group G isomorphic to some finite simple group of Lie type and the characteristic of G, we compute the standard name of G by a Monte.
[62]
General framework for E(3)-equivariant neural network ... - Nature
May 18, 2023 · Here we propose an E(3)-equivariant deep-learning framework to represent density functional theory (DFT) Hamiltonian as a function of material structure.Results · Equivariance Of The... · The Neural Network...Missing: post- | Show results with:post-
[63]
[PDF] Clifford Group Equivariant Neural Networks
We introduce Clifford Group Equivariant Neural Networks: a novel approach for constructing O(n)- and E(n)-equivariant models. We identify and study the.
[64]
Theoretical Aspects of Group Equivariant Neural Networks - arXiv
Apr 10, 2020 · Group equivariant neural networks have been explored in the past few years and are interesting from theoretical and practical standpoints.Missing: post- | Show results with:post-
[65]
A Lie algebraic theory of barren plateaus for deep parameterized ...
Aug 22, 2024 · In this work, we present a general Lie algebraic theory for BPs that can be applied to any deep, unitary, parametrized quantum circuit ...
[66]
Characterization of variational quantum algorithms using free fermions
Mar 30, 2023 · We study variational quantum algorithms from the perspective of free fermions. By deriving the explicit structure of the associated Lie algebras.
[67]
Lie groups for quantum complexity and barren plateau theory - arXiv
Jul 30, 2025 · ... quantum algorithms. In this review, we introduce the theory of Lie groups and their algebras to analyze two fundamental problems in quantum ...
[68]
[PDF] Introduction to Symmetry and Broken Symmetry in Physics - HAL
Mar 13, 2025 · In particle physics this is the Higgs mechanism, in condensed matter its non-relativistic version is known as the Anderson-Higgs model. ...Missing: reductive | Show results with:reductive
[69]
[PDF] Symmetries and Group Theory in Particle Physics
... symmetry, now called flavor SU(3) invariance, which allowed the inclusion of ... breaking, which we illustrate in the frame of a couple of simple ...
[70]
[PDF] Lie-group methods - HAL
Jun 8, 2016 · Definition 2.13 A real matrix Lie group is a smooth subset G ⊆ Rn×n, closed ... is more stable when numerical methods are implemented with small ...
[71]
[PDF] computational-aspects-of-lie-groups-algorithms-and-applications-in ...
Jan 30, 2025 · In quantum information science, efficient algorithms for computing Lie group exponentials help in designing quantum gates and simulating ...
[72]
A method of Lie-symmetry GL(n,R) for solving non-linear dynamical ...
The Lie-group numerical method has a stabilizing effect to retain the solution on the associated manifold, which thus naturally has a regularization effect to ...
[73]
[PDF] Lie Groups and Algebraic Groups - UCSD Math
We show how to put a Lie group structure on a closed subgroup of the general linear group and determine the Lie algebras of the classical groups. We develop the ...
[74]
[PDF] Author(s): Armand Borel and Harish—Chandra
The following proposition strengthens 1.7 , and extends to reductive real algebraic groups well-known properties of semi-simple Lie groups. 1.10. PROPOSITION.
[75]
[PDF] Lie Groups. Representation Theory and Symmetric Spaces
In this Chapter we discuss elementary properties of Lie groups, Lie algebras and their relationship. We will assume a good knowledge of manifolds, vector.
[76]
[PDF] Basic Theory of Affine Group Schemes - James Milne
Mar 11, 2012 · This is a modern exposition of the basic theory of affine group schemes. Although the emphasis is on affine group schemes of finite type ...
[77]
Introduction to Affine Group Schemes - SpringerLink
These turn out to be equivalent to the ring-theoretic objects called Hopf algebras, with which we can then con struct new examples. Study of their ...
[78]
[PDF] Tannakian Categories
May 13, 2011 · The functor A. SpecA defines an equivalence of categories between the category of k-bialgebras and the category of affine group schemes over k.
[79]
[PDF] Abelian Varieties - James Milne
Mar 16, 2008 · These notes are an introduction to the theory of abelian varieties, including the arithmetic of abelian varieties and Faltings's proof of ...