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Levi decomposition

The Levi decomposition is a fundamental theorem in Lie theory stating that every finite-dimensional Lie algebra over a field of characteristic zero can be expressed as a semidirect product of its radical—the unique maximal solvable ideal—and a semisimple subalgebra known as the Levi factor.^[1] This decomposition, denoted \mathfrak{g} = \mathfrak{r} \ltimes \mathfrak{s}, where \mathfrak{r} is the radical and \mathfrak{s} is the Levi factor, provides a canonical way to break down arbitrary Lie algebras into simpler components whose structures are better understood.^[2] The theorem was first proved by Eugenio Elia Levi in his 1905 paper "Sulla struttura dei gruppi finiti e continui," building on earlier conjectures by Wilhelm Killing and Élie Cartan regarding the structure of infinitesimal transformation groups.^[3] The existence of the decomposition follows from embedding the Lie algebra into a larger associative algebra and applying properties of semisimple representations, while the uniqueness of the Levi factor up to isomorphism was later established by Anatoly Maltsev in 1942.^[2] This result has profound implications for the classification and representation theory of Lie algebras, as semisimple Lie algebras admit complete classifications via root systems and Dynkin diagrams, while solvable ones can often be analyzed through derived series and nilpotent radicals.^[1] In the context of Lie groups, an analogous Levi decomposition exists for parabolic subgroups, decomposing them into a reductive Levi subgroup and a unipotent radical, which plays a key role in the study of algebraic groups and symmetric spaces.^[2] The theorem does not hold in general for fields of positive characteristic, though partial results exist under additional assumptions such as the presence of commuting semisimple derivations.^[4]

Preliminaries

Lie algebras

A Lie algebra over a field k of characteristic zero is a vector space \mathfrak{g} equipped with a bilinear operation [\cdot, \cdot]: \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}, called the Lie bracket, that is alternating (i.e., [x, x] = 0 for all x \in \mathfrak{g}, implying [x, y] = -[y, x]) and satisfies the Jacobi identity: [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0 for all x, y, z \in \mathfrak{g}.^[5] This structure captures infinitesimal symmetries and is central to the study of continuous transformation groups. Subalgebras and ideals of \mathfrak{g}, which are subspaces closed under the bracket in specific ways, provide building blocks for more advanced decompositions. Basic examples illustrate the diversity of Lie algebras. An abelian Lie algebra has trivial bracket [x, y] = 0 for all x, y \in \mathfrak{g}, making it commutative and suitable for modeling flat symmetries.^[5] The Heisenberg algebra is a 3-dimensional example over \mathbb{C} with basis \{X, Y, Z\} and nonzero bracket [X, Y] = Z, while all other brackets vanish; it arises in quantum mechanics as the algebra generated by position and momentum operators.^[6] In contrast, \mathfrak{sl}(2, \mathbb{C}), the Lie algebra of $2 \times 2 trace-zero complex matrices with bracket [A, B] = AB - BA, is simple: it has basis \{e = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, f = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, h = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\} and relations [h, e] = 2e, [h, f] = -2f, [e, f] = h, with no nontrivial ideals.^[7] The adjoint representation associates to each x \in \mathfrak{g} the linear map \mathrm{ad}_x: \mathfrak{g} \to \mathfrak{g} defined by \mathrm{ad}_x(y) = [x, y], turning \mathfrak{g} into a representation of itself; the image \mathrm{ad}(\mathfrak{g}) consists of all such maps and forms a Lie subalgebra of \mathrm{End}_k(\mathfrak{g}).^[8] This representation is a Lie algebra homomorphism \mathrm{ad}: \mathfrak{g} \to \mathrm{Der}(\mathfrak{g}), where \mathrm{Der}(\mathfrak{g}) is the Lie algebra of derivations of \mathfrak{g}. A derivation of \mathfrak{g} is a k-linear map D: \mathfrak{g} \to \mathfrak{g} satisfying D([x, y]) = [D(x), y] + [x, D(y)] for all x, y \in \mathfrak{g}; the set \mathrm{Der}(\mathfrak{g}) forms a Lie algebra under the commutator [D, E] = D \circ E - E \circ D.^[9] Inner derivations are those of the form \mathrm{ad}_x for x \in \mathfrak{g}, and they constitute an ideal in \mathrm{Der}(\mathfrak{g}), as the Jacobi identity ensures each \mathrm{ad}_x preserves the bracket.^[9]

Solvable and semisimple Lie algebras

A solvable Lie algebra \mathfrak{g} over a field F of characteristic zero is defined via its derived series, where \mathfrak{g}^{(0)} = \mathfrak{g} and \mathfrak{g}^{(k+1)} = [\mathfrak{g}^{(k)}, \mathfrak{g}^{(k)}] for k \geq 0; \mathfrak{g} is solvable if \mathfrak{g}^{(k)} = \{0\} for some positive integer k.^[10] This condition captures Lie algebras that can be "triangulated" in representations, analogous to solvable groups in group theory. Subalgebras and quotients of solvable Lie algebras remain solvable, providing a hereditary property useful in classification.^[10] Nilpotent Lie algebras form a subclass of solvable ones, characterized by the lower central series \mathfrak{g}_0 = \mathfrak{g} and \mathfrak{g}_{k+1} = [\mathfrak{g}, \mathfrak{g}_k] for k \geq 0; \mathfrak{g} is nilpotent if \mathfrak{g}_k = \{0\} for some positive integer k.^[11] Every nilpotent Lie algebra is solvable, since the lower central series refines the derived series, but the converse does not hold—for instance, the Lie algebra of strictly upper triangular matrices is nilpotent, while the Borel subalgebra of upper triangular matrices with constant diagonal is solvable but not nilpotent. A key structural result is Engel's theorem: over a field of characteristic zero, a Lie algebra \mathfrak{g} is nilpotent if and only if \mathrm{ad}_x is a nilpotent endomorphism of \mathfrak{g} for every x \in \mathfrak{g}.^[11] This implies the existence of a flag of ideals where the adjoint action acts nilpotently, facilitating simultaneous triangularization in representations.^[11] A semisimple Lie algebra \mathfrak{g} over a field F of characteristic zero has no nonzero solvable ideals. Equivalently, its radical is zero.^[12] Semisimplicity ensures that \mathfrak{g} is "rigid" with no proper extensions by solvable factors, contrasting with solvable algebras. Cartan's criterion provides a bilinear form characterization: \mathfrak{g} is semisimple if and only if its Killing form B(x,y) = \mathrm{tr}(\mathrm{ad}_x \mathrm{ad}_y) is nondegenerate.^[13] The Killing form, an invariant symmetric bilinear form, detects degeneracy precisely when abelian ideals exist, linking representation theory to intrinsic structure.^[13] The structure theorem for semisimple Lie algebras states that any finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero decomposes as a direct sum of simple ideals: \mathfrak{g} = \mathfrak{g}_1 \oplus \cdots \oplus \mathfrak{g}_k, where each \mathfrak{g}_i is simple (nonabelian with no nonzero proper ideals).^[14] This decomposition is unique up to isomorphism and permutation, and every ideal of \mathfrak{g} is a direct sum of some of the \mathfrak{g}_i.^[14] Simple Lie algebras form the building blocks, with their classifications underpinning much of representation theory. Prominent examples include the classical series: the special linear Lie algebra \mathfrak{sl}(n, \mathbb{C}) (type A_{n-1}) for n \geq 2, consisting of traceless n \times n matrices with bracket [X,Y] = XY - YX; the orthogonal Lie algebra \mathfrak{so}(2n+1, \mathbb{C}) (type B_n) for n \geq 1, preserving a nondegenerate symmetric bilinear form; and \mathfrak{so}(2n, \mathbb{C}) (type D_n) for n \geq 2.^[15] These arise as Lie algebras of special linear, orthogonal, and symplectic groups, respectively, and exhaust the finite-dimensional simple Lie algebras over \mathbb{C} alongside the exceptional types E_6, E_7, E_8, F_4, G_2.^[15]

The radical of a Lie algebra

In Lie algebra theory, the radical of a finite-dimensional Lie algebra \mathfrak{g} over a field of characteristic zero, denoted \mathrm{rad}(\mathfrak{g}), is defined as the unique maximal solvable ideal of \mathfrak{g}.^[8] This ideal captures the "solvable part" of \mathfrak{g} and plays a central role in structural decompositions. Equivalently, \mathrm{rad}(\mathfrak{g}) can be expressed as the sum of all solvable ideals of \mathfrak{g}.^[8] The existence of \mathrm{rad}(\mathfrak{g}) follows from the finite-dimensionality of \mathfrak{g}. The set of solvable ideals is nonempty, as it includes the zero ideal, which is solvable. The sum of any two solvable ideals \mathfrak{a} and \mathfrak{b} is again a solvable ideal: if \mathfrak{a}^{(k)} = 0 and \mathfrak{b}^{(m)} = 0 for the derived series, then (\mathfrak{a} + \mathfrak{b})^{(r)} = \mathfrak{a}^{(r)} + \mathfrak{b}^{(r)} = 0 for r = \max(k, m), and the sum inherits the ideal property under the Lie bracket with \mathfrak{g}. Since \mathfrak{g} has finite dimension, the sum of all solvable ideals is a finite (hence well-defined) solvable ideal that contains every solvable ideal and thus is maximal.^[8] Alternatively, in the algebraic geometry context over an algebraically closed field, the solvable ideals form a closed subvariety of the Grassmannian of subspaces of \mathfrak{g} that are ideals, ensuring their intersection (or a maximal one) is solvable and nonempty.^[16] A key property of \mathrm{rad}(\mathfrak{g}) is its uniqueness: suppose \mathfrak{r}_1 and \mathfrak{r}_2 are two maximal solvable ideals; then \mathfrak{r}_1 + \mathfrak{r}_2 would be a larger solvable ideal, contradicting maximality unless \mathfrak{r}_1 = \mathfrak{r}_2. Consequently, every solvable ideal \mathfrak{h} of \mathfrak{g} satisfies \mathfrak{h} \subseteq \mathrm{rad}(\mathfrak{g}).^[16] Moreover, \mathrm{rad}(\mathfrak{g}) is characteristic, meaning it is invariant under all automorphisms of \mathfrak{g}, and the quotient \mathfrak{g} / \mathrm{rad}(\mathfrak{g}) is semisimple.^[8] The nilradical \mathrm{nil}(\mathfrak{g}), defined as the maximal nilpotent ideal of \mathfrak{g}, satisfies \mathrm{nil}(\mathfrak{g}) \subseteq \mathrm{rad}(\mathfrak{g}), because every nilpotent Lie algebra is solvable (its lower central series terminates, implying the derived series does as well). Equality holds in specific classes, such as when \mathfrak{g} is the Lie algebra of a solvable algebraic group over a field of characteristic zero, or for certain nilpotent-by-abelian structures like filiform Lie algebras.^[8] In general, the inclusion is proper if \mathfrak{g} admits solvable ideals that are not nilpotent. A representative example is the affine Lie algebra \mathfrak{g} = \mathfrak{sl}_n(k) \ltimes k^n over a field k of characteristic zero, where \mathfrak{sl}_n(k) acts on the abelian ideal k^n by the standard matrix representation. Here, \mathrm{rad}(\mathfrak{g}) = k^n, the abelian factor, as it is the unique maximal solvable ideal (noting that \mathfrak{sl}_n(k) is semisimple). This illustrates how the radical isolates the translation component in affine structures.^[17]

The theorem

Statement

The Levi decomposition theorem asserts that if \mathfrak{g} is a finite-dimensional Lie algebra over a field K of characteristic zero, then there exists a semisimple Lie subalgebra \mathfrak{s} of \mathfrak{g}, called a Levi subalgebra, such that \mathfrak{g} = \mathrm{rad}(\mathfrak{g}) \ltimes \mathfrak{s}, where \ltimes denotes the semidirect product and \mathrm{rad}(\mathfrak{g}) is the radical of \mathfrak{g}.^[18] This means that \mathfrak{g} = \mathrm{rad}(\mathfrak{g}) + \mathfrak{s} as vector spaces, the sum is direct (i.e., \mathrm{rad}(\mathfrak{g}) \cap \mathfrak{s} = \{0\}), and [\mathrm{rad}(\mathfrak{g}), \mathfrak{s}] \subseteq \mathrm{rad}(\mathfrak{g}).^[18] This decomposition was first established by Eugenio Elia Levi in 1905 and was subsequently extended to arbitrary fields of characteristic zero.^[19]^[18] A direct consequence of the theorem is that \mathfrak{g} is semisimple if and only if \mathrm{rad}(\mathfrak{g}) = \{0\}, and \mathfrak{g} is solvable if and only if \mathfrak{s} = \{0\}.^[18]

Uniqueness

The Levi decomposition of a finite-dimensional Lie algebra \mathfrak{g} over a field of characteristic zero is unique in the sense that if \mathfrak{g} = \mathfrak{r} \ltimes \mathfrak{s} = \mathfrak{r} \ltimes \mathfrak{s}', where \mathfrak{r} = \mathrm{rad}(\mathfrak{g}) is the solvable radical and \mathfrak{s}, \mathfrak{s}' are semisimple Levi subalgebras, then \mathfrak{s} \cong \mathfrak{s}' as Lie algebras.^[8] This isomorphism follows from the fact that the natural projection \pi: \mathfrak{g} \to \mathfrak{g}/\mathfrak{r} restricts to Lie algebra isomorphisms \mathfrak{s} \to \mathfrak{g}/\mathfrak{r} and \mathfrak{s}' \to \mathfrak{g}/\mathfrak{r}, since \mathfrak{s} \cap \mathfrak{r} = \{0\} = \mathfrak{s}' \cap \mathfrak{r} and \dim \mathfrak{s} = \dim(\mathfrak{g}/\mathfrak{r}); the semisimple quotient \mathfrak{g}/\mathfrak{r} is thus canonically isomorphic to both \mathfrak{s} and \mathfrak{s}'.^[8] Moreover, any two Levi subalgebras are conjugate under the action of the inner automorphism group of \mathfrak{g}: there exists z \in \mathfrak{n}(\mathfrak{g}), the nilradical of \mathfrak{g}, such that \exp(\mathrm{ad} z) conjugates \mathfrak{s} to \mathfrak{s}'. This conjugacy ensures that while the specific subset comprising a Levi subalgebra is not unique, all such subalgebras are equivalent under inner automorphisms of \mathfrak{g}.^[8] For instance, different embeddings of \mathfrak{sl}(2, k) as a Levi subalgebra in a given \mathfrak{g} (such as in extensions by solvable modules) are related by such conjugations, preserving the abstract structure.^[20]

Proof

Key lemmas

Lie's theorem provides a fundamental characterization of representations of solvable Lie algebras over algebraically closed fields of characteristic zero. Specifically, for a solvable Lie algebra \mathfrak{g} and a finite-dimensional \mathfrak{g}-module V, there exists a flag of submodules $0 = V_0 \subset V_1 \subset \cdots \subset V_n = V such that each successive quotient V_{i+1}/V_i is one-dimensional, and the action of \mathfrak{g} on each quotient is via a character, implying that the adjoint representation of \mathfrak{g} on itself admits a basis in which all elements act by upper triangular matrices.^[8] This result, proved using induction on the dimension and the existence of common eigenvectors, underpins the triangulability of solvable actions and is essential for analyzing the structure of the radical in the Levi decomposition.^[8] Weyl's theorem establishes the complete reducibility of finite-dimensional representations for semisimple Lie algebras. For a semisimple Lie algebra \mathfrak{g} over an algebraically closed field of characteristic zero, every finite-dimensional \mathfrak{g}-module decomposes as a direct sum of irreducible submodules.^[8] The proof relies on the existence of a nondegenerate invariant bilinear form on the module, constructed using the Killing form and Casimir operators, which allows the construction of a complementary submodule to any given invariant subspace.^[8] This property highlights the "rigid" representation theory of semisimple algebras, contrasting with the non-completely reducible representations typical of solvable ones.^[14] A key property is that the ideal [\mathfrak{g}, \mathfrak{r}], where \mathfrak{r} is the radical, is nilpotent (the nilradical), and every element x \in [\mathfrak{g}, \mathfrak{r}] acts nilpotently via the adjoint representation, meaning \mathrm{ad}_x is a nilpotent endomorphism of \mathfrak{g}.^[8] This follows from the fact that [\mathfrak{g}, \mathfrak{r}] acts nilpotently in any finite-dimensional representation, including the adjoint, by Engel's theorem applied to its action.^[21] Consequently, the nilpotent radical [\mathfrak{g}, \mathfrak{r}] is contained within the Fitting subgroup of the adjoint action, reinforcing the separation of solvable and semisimple components.^[8] The Jacobson density theorem plays a crucial role in the representation theory of enveloping algebras of Lie algebras. For a primitive ring R (such as the universal enveloping algebra U(\mathfrak{g}) acting faithfully on a simple module), the centralizer of the action on the simple module is a dense subring in the sense of containing all rational functions on matrix entries that are fixed by the action.^[22] In the context of Lie algebras over algebraically closed fields of characteristic zero, this theorem implies that for an irreducible finite-dimensional representation of \mathfrak{g}, the image of U(\mathfrak{s}) (where \mathfrak{s} is semisimple) acts densely in the endomorphism ring, facilitating the identification of Levi subalgebras via their representation-theoretic properties.^[22] The Killing form \kappa(X, Y) = \operatorname{tr}(\mathrm{ad}_X \circ \mathrm{ad}_Y) on a finite-dimensional Lie algebra \mathfrak{g} over a field of characteristic zero exhibits distinct behaviors on semisimple and radical components. For semisimple \mathfrak{g}, the Killing form is nondegenerate, serving as an invariant symmetric bilinear form that induces an isomorphism with the dual space and supports the Cartan-Weyl decomposition into root spaces.^[8] In contrast, the Killing form vanishes identically on the radical \mathfrak{r} of \mathfrak{g}, as \kappa(\mathfrak{r}, \mathfrak{g}) = 0 due to the solvability of \mathfrak{r} and the orthogonality to the derived algebra, which implies \kappa(\mathfrak{r}, [\mathfrak{g}, \mathfrak{g}]) = 0.^[8] This degeneracy on the radical provides a criterion for semisimplicity and aids in decomposing \mathfrak{g} orthogonally with respect to \kappa.^[8]

Construction of the Levi subalgebra

The existence of the Levi subalgebra is established by induction on the dimension of the Lie algebra g. For the base case, if \dim g = 1, then g is abelian, so \rad(g) = g and the Levi subalgebra is the trivial subalgebra \{0\}, which is semisimple.^[18] Assume the result holds for all finite-dimensional Lie algebras over a field of characteristic zero with dimension less than \dim g. Let r = \rad(g). The quotient \bar{g} = g / r is semisimple, as the radical is the maximal solvable ideal.^[23] To lift a semisimple complement from the quotient, consider the short exact sequence of Lie algebras

$0 \to r \to g \to \bar{g} \to 0,

where the action of g on r is the adjoint action. Since \bar{g} is semisimple, Whitehead's second lemma implies that the Lie algebra cohomology group H^2(\bar{g}, r) = 0, where r is viewed as a \bar{g}-module via the induced action. Thus, the extension splits: there exists a Lie algebra homomorphism \sigma: \bar{g} \to g that is a section of the projection \pi: g \to \bar{g}, i.e., \pi \circ \sigma = \id_{\bar{g}}.^[24] The image s = \sigma(\bar{g}) is then a semisimple subalgebra of g isomorphic to \bar{g}, serving as the Levi subalgebra. This yields the semidirect product decomposition g = r \rtimes s.^[23] An explicit construction of s can be obtained by pulling back a basis of \bar{g} via \sigma: if \{\bar{x}_1, \dots, \bar{x}_k\} is a basis for \bar{g}, then \{ \sigma(\bar{x}_1), \dots, \sigma(\bar{x}_k) \} spans s, with Lie bracket relations preserved modulo r. In practice, \sigma is found by solving for a linear section that respects the Lie structure, often via successive approximations in a basis adapted to a composition series of r when r is nilpotent.^[25] To verify the properties, first note that [r, s] \subseteq r, since the action of s on r is the induced module action from \bar{g} on r, and r is an ideal. Second, s is semisimple because it is isomorphic to the semisimple algebra \bar{g}. Finally, g = r + s and r \cap s = \{0\}, as \sigma is injective and \pi maps s isomorphically onto \bar{g}, ensuring s spans a complement to r. If the induction hypothesis is needed (e.g., when constructing \sigma explicitly for non-abelian r), reduce to the case where [r, r] \neq r by quotienting by [r, r], applying the hypothesis to obtain a partial complement, and lifting step-by-step.^[18]

Applications

To Lie groups

For a connected Lie group G with Lie algebra \mathfrak{g}, the Levi decomposition \mathfrak{g} = \mathrm{rad}(\mathfrak{g}) \ltimes \mathfrak{s} of the Lie algebra, where \mathrm{rad}(\mathfrak{g}) is the solvable radical and \mathfrak{s} is a semisimple Levi subalgebra, lifts to a corresponding semidirect product decomposition at the group level. Specifically, G admits a unique maximal connected normal solvable subgroup R, called the solvable radical of G, and a connected semisimple Lie subgroup S such that G = R \ltimes S. The subgroup R contains the image of the exponential map \exp: \mathrm{rad}(\mathfrak{g}) \to G and coincides with it when G is simply connected; in general, R is the connected component of the normal solvable subgroup generated by \exp(\mathrm{rad}(\mathfrak{g})).^[26] The semisimple subgroup S complements R in the sense that the projection G \to G/R has a section with image S, and S is unique up to conjugation by elements of G. This decomposition reduces the study of general connected Lie groups to the cases of solvable and semisimple groups. An analogous result holds for connected linear algebraic groups over fields of characteristic zero, as stated in the Levi–Malcev theorem. For such a group G defined over a field k of characteristic zero, G decomposes as G = R_u(G) \ltimes L, where R_u(G) is the unipotent radical (the maximal normal unipotent subgroup) and L is a reductive Levi subgroup isomorphic to the quotient G / R_u(G). The Levi subgroup L exists and is unique up to conjugation by elements of R_u(G)(k).^[8] As an illustrative example, consider the general linear group \mathrm{GL}_n(k), which is reductive and thus has trivial unipotent radical R_u(\mathrm{GL}_n(k)) = \{1\}, with the Levi subgroup being \mathrm{GL}_n(k) itself. For a parabolic subgroup P of \mathrm{GL}_n(k), such as the block upper triangular matrices corresponding to a composition of n, the decomposition refines to P = R_u(P) \ltimes L, where R_u(P) consists of the unipotent block upper triangular matrices with identity blocks on the diagonal, and L is the corresponding block diagonal reductive subgroup (a product of general linear groups on the blocks). In particular, the Borel subgroup of upper triangular matrices in \mathrm{GL}_n(k) decomposes as the semidirect product of the strict upper triangular unipotents and the diagonal torus.^[27]

In representation theory

The Levi decomposition \mathfrak{g} = \mathfrak{r} \ltimes \mathfrak{s}, where \mathfrak{s} is a semisimple Levi subalgebra and \mathfrak{r} is the solvable radical, provides a framework for analyzing finite-dimensional representations of \mathfrak{g} over \mathbb{C}. A representation \rho: \mathfrak{g} \to \mathfrak{gl}(V) restricts to representations of \mathfrak{s} and \mathfrak{r} on V. By Weyl's theorem, the \mathfrak{s}-representation is completely reducible: V decomposes as a direct sum of irreducible \mathfrak{s}-submodules. Since \mathfrak{r} is solvable, Lie's theorem guarantees the existence of a complete flag of \mathfrak{r}-invariant subspaces of V on which \mathfrak{r} acts by upper-triangular matrices with respect to a suitable basis. The semidirect product structure ensures compatibility between these actions, as elements of \mathfrak{r} act as \mathfrak{s}-module derivations on V. For infinite-dimensional representations, particularly generalized Harish-Chandra modules—those finitely generated over the universal enveloping algebra U(\mathfrak{g}) and locally finite-dimensional under the action of \mathfrak{s}—the Levi decomposition enables a decomposition into generalized weight spaces relative to \mathfrak{s}. For a Lie algebra \mathfrak{l} with Levi decomposition \mathfrak{l} = \mathfrak{r} \ltimes \mathfrak{s} where \mathfrak{s} is semisimple, every simple \mathfrak{l}-module locally finite over \mathfrak{s} has finite length, and such modules decompose as direct sums of generalized eigenspaces for the action of a Cartan subalgebra of \mathfrak{s}, with finite multiplicities determined by the \mathfrak{s}-structure.^[28] This decomposition simplifies the classification and study of these modules, reducing problems to the semisimple case modulated by the solvable radical. A concrete example arises with \mathfrak{g} = V \ltimes \mathfrak{sl}(2,\mathbb{C}), where V is a finite-dimensional irreducible \mathfrak{sl}(2,\mathbb{C})-module (the "affine" case, with V as the radical). The irreducible representations of \mathfrak{g} are classified using a \mu-invariant tied to the action of \mathfrak{sl}(2,\mathbb{C}) on V, where finite-dimensional irreducibles correspond to those where the \mathfrak{sl}(2,\mathbb{C})-action on the representation space aligns with the module structure on V, often via extensions or twists by characters of V.^[17] In the context of Lie groups, the Levi decomposition of parabolic subgroups P = U \ltimes L (with Levi subgroup L) underpins the construction of induced representations. For an irreducible representation \sigma of L, the induced representation \mathrm{Ind}_P^G \sigma on the reductive group G typically decomposes into irreducibles in a manner controlled by intertwining operators, forming the basis for the Langlands classification of representations.^[29] This induction preserves key analytic properties, such as unitarity, when \sigma is unitary.

Generalizations and extensions

Positive characteristic

In fields of positive characteristic p > 0, the Levi decomposition theorem fails to hold in general for finite-dimensional Lie algebras over algebraically closed fields. Unlike the characteristic zero case, where every Lie algebra decomposes as a semidirect product of its solvable radical and a semisimple subalgebra, no such semisimple complement to the radical need exist in characteristic p. This failure arises primarily because representations of solvable Lie algebras are not completely reducible, preventing the existence of invariant complements for ideals like the radical. A concrete counterexample is the Lie algebra of the algebraic group \mathrm{[SL](/page/SL)}_2 over the ring of 2-truncated Witt vectors W_2(k) in characteristic p, where the unipotent radical admits no reductive complement, as there is no homomorphic section to the reduction map onto the base field.^[20] To address this limitation, alternatives involving restricted structures have been developed for Lie algebras in characteristic p. A restricted Lie algebra (or p-Lie algebra) is equipped with a p-operation satisfying certain axioms, allowing for a p-power map analogous to the p-th power in the group setting. The p-envelope of a Lie algebra L is the smallest restricted Lie algebra containing L as a p-ideal, constructed by adjoining formal p-th powers if necessary. The restricted radical of L, denoted \mathrm{Rad}_p(L), is the largest restricted solvable ideal contained in the (ordinary) radical \mathrm{Rad}(L).^[30] These notions provide a partial analogue to the Levi decomposition, though a full semisimple complement may still be absent. The Tits-Witt theorem relates to the classification of simple restricted Lie algebras in characteristic p, showing that they arise as p-envelopes of classical types or Cartan-Witt types, offering a framework for understanding semisimple components without direct complements.^[30] Under additional assumptions, partial decompositions are possible. Jacobson's theorem establishes that if a solvable Lie algebra L in characteristic p has the property that the quotients in its derived series have dimension less than p, then it admits an analogue of the Levi decomposition, where the radical is complemented by a semisimple subalgebra, provided the algebra is p-nilpotent (meaning the p-th powers generate a nilpotent ideal). This condition ensures sufficient complete reducibility in representations to allow complements. A prominent example illustrating the failure is the modular Witt algebra W(1) in characteristic p > 0, which is simple and restricted but admits no semisimple complement to its (zero) radical in broader extensions, as its representations lack the necessary splitting properties.^[31] In computational settings, exact Levi decompositions may not exist, but algorithms for approximations or checks in positive characteristic have been implemented in computer algebra systems. These typically involve computing the p-envelope and restricted radical via Gröbner bases on structure constants or nilpotency tests on the lower central series, providing bounds on potential complements when they exist. For instance, in systems like MAGMA or GAP, one can approximate decompositions by embedding into the p-envelope and testing for restricted semisimple factors, with polynomial-time complexity in the dimension for restricted cases under dimension-p bounds. Such methods are crucial for studying modular representations, where full decompositions fail but structural insights via p-structures persist.

Infinite-dimensional cases

In contrast to the finite-dimensional setting, where every Lie algebra over a field of characteristic zero admits a Levi decomposition into a semisimple subalgebra and its radical, no such general decomposition exists for infinite-dimensional Lie algebras. This failure arises because the second cohomology group H^2(\mathfrak{g}, \mathbb{C}) may be non-zero, obstructing the required extension properties that hold in finite dimensions via Whitehead's lemma. For instance, the Virasoro algebra, an infinite-dimensional central extension of the Witt algebra, lacks any Levi decomposition, as its non-vanishing second cohomology prevents splitting into a semisimple part and a solvable radical.^[24] Kac–Moody algebras provide an important class where partial analogs of Levi decompositions appear, particularly in the context of parabolic subalgebras. These subalgebras often admit a Levi decomposition \mathfrak{p} = \mathfrak{l} \oplus \mathfrak{n}, but the Levi factor \mathfrak{l} is typically infinite-dimensional, reflecting the overall infinite structure of the algebra. Such generalized Levi factors play a role in representation theory and modular forms on Kac–Moody groups.^[32] For locally finite Lie algebras over fields of characteristic zero—those where every finitely generated subalgebra is finite-dimensional—block decompositions exist that extend the finite-dimensional Levi framework, allowing the algebra to be expressed as a direct sum of finite-dimensional blocks with controlled radical behavior. Finitary Lie algebras, a subclass of locally finite ones acting on infinite-dimensional vector spaces while preserving finite-dimensional subspaces, always possess a Levi component, which is a maximal semisimple subalgebra.^[33]^[34] In the case of \mathbb{Z}-graded Lie algebras over characteristic zero, a compatible Levi decomposition exists that preserves the grading: if \mathfrak{g} = \bigoplus_{i \in \mathbb{Z}} \mathfrak{g}_i admits a Levi decomposition \mathfrak{g} = \mathfrak{s} \ltimes \mathfrak{r}, then there are graded subalgebras \mathfrak{s}' \subseteq \mathfrak{s} and \mathfrak{r}' \subseteq \mathfrak{r} such that \mathfrak{g} = \mathfrak{s}' \ltimes \mathfrak{r}' with \mathfrak{s}' semisimple and \mathfrak{r}' the radical, both graded. This result, established in 2017, ensures the decomposition respects the algebraic grading structure.^[35] Computational approaches to Levi decompositions in low-dimensional infinite-dimensional cases leverage software like GAP and Maple, which implement algorithms for structure detection in finitely presented or graded Lie algebras, though full infinite-dimensional computations remain challenging due to dimensionality issues. For example, Maple's DifferentialGeometry package computes Levi decompositions for finite-dimensional Lie algebras, applicable in cases with finite-dimensional components.^[36]

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May 5, 2013 · For a semisimple Lie algebra, we can take the adjoint representation, and for a one-dimensional Lie algebra we can take the identity map. 2.
[9]
[PDF] Lecture 2 — Some Sources of Lie Algebras
Sep 14, 2010 · Lie algebras are often constructed as the algebra of derivations of a given algebra. This corresponds to the use of vector fields in geometry. ...
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[PDF] Lecture 2 - Fundamental definitions, and Engel's Theorem - Penn Math
Sep 11, 2012 · We call L solvable if L(n) = {0} for some n. Proposition 2.1 (Humphreys 3.1) Let L be a Lie algebra. 2. Page 3. a) If L is solvable, so are ...
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[PDF] Nilpotent Lie Algebras and Engel's Theorem
(Engel's Theorem) Let g be a Lie algebra over F. Then g is nilpotent if and only if, for all x ∈ g, ad x is a nilpotent linear operator on g. It ...
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[PDF] Semisimple Lie Algebras and the Root Space Decomposition
May 1, 2015 · Show that a Lie algebra is semisimple iff it has no nonzero abelian ideals. Show that g/Rad(g) is semisimple. This last fact suggests that we ...
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[PDF] Cartan's Criteria for Solvability and Semisimplicity
(Cartan's Criterion for Semisimplicity) Let g be a Lie algebra over F. Then g is semisimple if and only if the Killing form B is nondegenerate. We will ...
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[PDF] Semisimple Lie Algebras: Basic Structure and Representations
A semisimple Lie algebra is a direct sum of simple ideals, and its study reduces to the study of simple Lie algebras.
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[PDF] Complex Semisimple Lie Algebras - Pierre Clare
For n ~ 1, An= sl(n + 1) is the Lie algebra of the special linear group in n + 1 variables, SL(n + 1). For n ~ 2, Bn = so(2n + 1) is the Lie algebra of the ...
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[PDF] Lecture 11 — The Radical and Semisimple Lie Algebras
Oct 14, 2010 · Definition 11.4. A Lie algebra g is called simple if its only ideals are 0 and g and g is not abelian. Corollary 11.1. Any semisimple, finite- ...
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[PDF] Representations of Lie algebras - UCI Mathematics
As any such Lie algebra decomposes into a semisimple and an abelian Lie algebra, the chapter is roughly divided into three parts. First, we illustrate how a ...
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[PDF] Lecture 23 — Decomposition of Semisimple Lie Algebras
Decomposition of Semisimple Lie Algebras. Prof. Victor Kac. Scribe: William ...Missing: original | Show results with:original
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[PDF] The Levi decomposition of a graded Lie algebra
We assume throughout that Lie algebras are real and finite-dimensional. In. 1905, E. E. Levi [3] showed that every Lie algebra may be decomposed as a.
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[PDF] Math 249B. Levi subgroups 1. Introduction In the theory of Lie ...
Details are given in Bourbaki LIE I, §6.8. Such an h is usually called a “Levi factor” of g. In the context of linear algebraic groups, we have a related ...
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Lie algebras : Jacobson, Nathan - Internet Archive
Apr 13, 2022 · Lie algebras. by: Jacobson, Nathan. Publication date: 1962. Topics: Algebre di Lie. Publisher: New York ; London : Interscience. Collection ...
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https://archive.org/details/liealgebras0000jaco_o3t8
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[PDF] On Levi Decompositions in Finite and Infinite Dimensional Lie ...
Jan 28, 2021 · ... decomposition L = RadL ⊕ S, where. S is semisimple. As we will see, such a decomposition can be made for every finite dimensional Lie algebra.Missing: original | Show results with:original
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[PDF] Computing Levi Decompositions in Lie algebras
[3]). In this paper we show that a Levi decomposition for Lie algebras over the rational numbers Q can be obtained in polynomial time. As a ...
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[PDF] Levi decompositions Let g be a Lie algebra. The radical of g is the ...
The radical of g is the maximal solvable ideal of g. The Levi decomposition of g expresses g as the direct sum of its radical and a semisimple subalgebra of g.<|control11|><|separator|>
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[PDF] LEVI DECOMPOSITIONS OF A LINEAR ALGEBRAIC GROUP
In summary: linear algebraic groups in characteristic zero always have a Levi decomposition. Moreover, it follows from Theorem 5.1 below that any two Levi ...
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Lie algebra modules which are locally finite over the semi-simple part
Jan 9, 2020 · ... Levi decomposition \mathfrak{L} = \mathfrak{g} \oplus \mathfrak{r} where \mathfrak{g} is semi-simple, we investigate \mathfrak{L}-modules ...
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[PDF] Induced representations of reductive p-adic groups. I - Numdam
(b) The studying of induced representations. More precisely, let P be a parabolic subgroup in G, M its Levi subgroup, p a cuspidal irreducible representation of ...
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Restricted Lie Algebras and Their Envelopes | Canadian Journal of ...
Nov 20, 2018 · Let L be a restricted Lie algebra over a field of characteristic p. Denote by u(L) its restricted enveloping algebra and by ωu(L) the ...
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Lie p-algebra - Encyclopedia of Mathematics
Dec 17, 2019 · For any Lie p - algebra L there is a p - universal (restricted universal) enveloping associative algebra Up(L) . If dimk L=n ...
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Non-singular derivations of solvable Lie algebras in prime ...
Feb 1, 2020 · We show that Jacobson's Theorem remains true if the quotients of the derived series have dimension less than p. We also study the structure of ...
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Structure of parabolically induced modules for affine Kac–Moody ...
Apr 15, 2018 · Let P ⊂ G be a parabolic subalgebra of G such that P = l ⊕ n is a Levi decomposition and l is infinite dimensional Levi factor. Denote by l 0 ...
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On the Jacobson Radicals of Infinite Dimensional Lie Algebras
Jul 10, 1978 · The purpose of this paper is to study the Jacobson radicals of infinite-dimensional Lie algebras. We employ the notation and terminology in [1].
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[PDF] Principal series representations of infinite-dimensional Lie groups, I
In this section we discuss Levi components of complex parabolic subalgebras, recalling results from [8], [9], [4], [10], [5] and [25]. We start with the ...<|control11|><|separator|>
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[1705.06727] The Levi Decomposition of a Graded Lie Algebra - arXiv
Abstract:We show that a graded Lie algebra admits a Levi decomposition that is compatible with the grading. Subjects: Group Theory (math.GR).
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https://www.maplesoft.com/support/help/maple/view.aspx?path=DifferentialGeometry%2FLieAlgebras%2FLeviDecomposition