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Adjoint representation

In Lie theory, the adjoint representation of a Lie group G is a representation \operatorname{Ad}: G \to \mathrm{GL}(\mathfrak{g}) on its Lie algebra \mathfrak{g} = T_eG, defined as the differential at the identity of the conjugation action c_g(x) = gxg^{-1} for g \in G, which yields \operatorname{Ad}(g) \cdot X = gXg^{-1} explicitly for matrix Lie groups.^[1]^[2] For the associated Lie algebra, the adjoint representation \operatorname{ad}: \mathfrak{g} \to \mathrm{End}(\mathfrak{g}) is the Lie algebra homomorphism given by \operatorname{ad}_X(Y) = [X, Y], the Lie bracket, representing the infinitesimal action of \mathfrak{g} on itself via commutators.^[3]^[1] This representation plays a central role in the structure theory of Lie groups and algebras, as it encodes the inner automorphisms and facilitates the study of derivations, with the image of \operatorname{ad} consisting precisely of the inner derivations of \mathfrak{g}.^[1] The differential of \operatorname{Ad} at the identity recovers \operatorname{ad}, linking the group-level conjugation to the algebra-level bracket via one-parameter subgroups: [X, Y] = \frac{d}{dt}\big|_{t=0} \operatorname{Ad}(\exp(tX)) Y.^[2] Key properties include its linearity, preservation of the Lie bracket (making it a Lie algebra representation), and dimension equal to \dim \mathfrak{g}, often realized as matrices in a chosen basis.^[3] A fundamental invariant arising from the adjoint representation is the Killing form, a symmetric bilinear form K(X, Y) = \operatorname{Tr}(\operatorname{ad}_X \circ \operatorname{ad}_Y) on \mathfrak{g}, which is \operatorname{Ad}-invariant and non-degenerate for semisimple Lie algebras, enabling classifications like Cartan's criterion for solvability and semisimplicity.^[1] For compact semisimple Lie groups, the negative Killing form is positive definite, reflecting the orthogonal nature of the adjoint action with respect to suitable inner products.^[1] Examples include the adjoint representation of \mathfrak{su}(2), which is 3-dimensional and isomorphic to \mathfrak{so}(3), illustrating rotations in 3D space.^[3]

Adjoint Action on Lie Groups

Definition of the Adjoint Action

In Lie group theory, the adjoint action provides a fundamental way for a Lie group to act on itself through conjugation. For a Lie group G, the adjoint action is defined by the map \operatorname{Ad}: G \times G \to G given by \operatorname{Ad}_g(h) = g h g^{-1} for all g, h \in G.^[4] This construction equips G with a left action on itself, where the element g acts by conjugation on h. The adjoint action preserves the group structure of G, as \operatorname{Ad}_g(h_1 h_2) = g h_1 h_2 g^{-1} = (g h_1 g^{-1})(g h_2 g^{-1}) = \operatorname{Ad}_g(h_1) \operatorname{Ad}_g(h_2) for all h_1, h_2 \in G, and \operatorname{Ad}_g(e) = e where e is the identity element.^[4] Consequently, for each fixed g \in G, the map \operatorname{Ad}_g: G \to G is a group automorphism, meaning it is a bijective homomorphism from G to itself.^[5] This automorphism property highlights how the adjoint action encodes the inner symmetries of the group. The concept of the adjoint action originated in the late 19th century as part of Sophus Lie's foundational work on continuous transformation groups and their automorphisms, aimed at analyzing symmetries of differential equations. A concrete example arises in the matrix Lie group G = \mathrm{GL}(n, \mathbb{R}), where the adjoint action corresponds to matrix conjugation: for P \in \mathrm{GL}(n, \mathbb{R}) and A \in \mathrm{GL}(n, \mathbb{R}), \operatorname{Ad}_P(A) = P A P^{-1}.^[6] This operation preserves similarity classes of matrices and illustrates the action's role in linear algebraic contexts. The adjoint action thus generates the inner automorphisms of G.^[5]

Relation to Inner Automorphisms

The adjoint action of a Lie group G on itself induces a group homomorphism \operatorname{Ad}: G \to \operatorname{Aut}(G), where \operatorname{Aut}(G) denotes the automorphism group of G. This map sends each element g \in G to the inner automorphism \operatorname{Ad}_g: h \mapsto g h g^{-1} for h \in G.^[4] The image of \operatorname{Ad} is precisely the subgroup \operatorname{Int}(G) of inner automorphisms, which consists of all conjugations by elements of G.^[6] The kernel of \operatorname{Ad} is the center Z(G) = \{ z \in G \mid z g = g z \ \forall g \in G \}, as \operatorname{Ad}_z = \operatorname{id}_G if and only if z commutes with every element of G.^[4] Thus, \operatorname{Ad} factors through the quotient G / Z(G), yielding an isomorphism \operatorname{Int}(G) \cong G / Z(G).^[6] To see this, first note that each \operatorname{Ad}_g is indeed an automorphism of G, as conjugation preserves the group operation: \operatorname{Ad}_g(h_1 h_2) = g h_1 h_2 g^{-1} = (g h_1 g^{-1})(g h_2 g^{-1}) = \operatorname{Ad}_g(h_1) \operatorname{Ad}_g(h_2). The map \operatorname{Ad}: G \to \operatorname{Int}(G) is a surjective homomorphism because every inner automorphism arises as a conjugation, and its kernel is Z(G) by the definition above. By the first isomorphism theorem, G / Z(G) \cong \operatorname{Int}(G).^[4] The full automorphism group \operatorname{Aut}(G) contains \operatorname{Int}(G) as a normal subgroup, with the quotient \operatorname{Out}(G) = \operatorname{Aut}(G) / \operatorname{Int}(G) consisting of outer automorphisms, and \operatorname{Aut}(G) often decomposes as a semidirect product \operatorname{Int}(G) \rtimes \operatorname{Out}(G).^[7]

Adjoint Representation on Lie Algebras

The Lie Algebra Map ad

In the context of a Lie algebra \mathfrak{g} over a field of characteristic zero, the adjoint map \mathrm{ad}: \mathfrak{g} \to \mathrm{End}(\mathfrak{g}) is defined by \mathrm{ad}_x(y) = [x, y] for all x, y \in \mathfrak{g}, where [\cdot, \cdot] denotes the Lie bracket on \mathfrak{g}.^[8]^[4] This assignment yields a linear endomorphism \mathrm{ad}_x \in \mathrm{End}(\mathfrak{g}) for each fixed x \in \mathfrak{g}, making \mathrm{ad} a representation of \mathfrak{g} on itself.^[9] The map \mathrm{ad} arises as the infinitesimal counterpart to the adjoint action \mathrm{Ad} on the corresponding Lie group G, specifically as the differential of \mathrm{Ad} at the identity element e \in G. To see this explicitly, consider the curve g(t) = \exp(t x) in G for t \in \mathbb{R}, where \exp: \mathfrak{g} \to G is the exponential map. Then, \mathrm{ad}_x(y) = \frac{d}{dt}\Big|_{t=0} \mathrm{Ad}_{\exp(t x)}(y), which computes to [x, y] upon differentiating the conjugation formula \mathrm{Ad}_{g(t)}(y) = g(t) y g(t)^{-1} and evaluating at t=0 using the tangent space identification \mathfrak{g} \cong T_e G.^[8]^[4] This relation underscores \mathrm{ad} as the tangent space linearization of the group action.^[9] A key property of \mathrm{ad}_x is that it acts as a derivation on \mathfrak{g}: for all y, z \in \mathfrak{g},

\mathrm{ad}_x([y, z]) = [\mathrm{ad}_x(y), z] + [y, \mathrm{ad}_x(z)],

which follows directly from the bilinearity and skew-symmetry of the Lie bracket, placing \mathrm{ad}_x in the derivation algebra \mathrm{Der}(\mathfrak{g}).^[4]^[9] The kernel of \mathrm{ad}_x, consisting of those y \in \mathfrak{g} such that [x, y] = 0, characterizes the centralizer of x; in particular, \mathrm{ad}_x = 0 if and only if x lies in the center \mathfrak{z}(\mathfrak{g}) = \{ w \in \mathfrak{g} \mid [w, v] = 0 \ \forall v \in \mathfrak{g} \}, the set of elements commuting with all of \mathfrak{g}.^[4]^[8]

The Representation Ad and Its Derivative

The adjoint representation of a Lie group G with Lie algebra \mathfrak{g} is the map \operatorname{Ad}: G \to \operatorname{GL}(\mathfrak{g}) defined by \operatorname{Ad}_g(X) = d(\operatorname{Conj}_g)|_e(X) for g \in G and X \in \mathfrak{g}, where \operatorname{Conj}_g(h) = g h g^{-1} is the conjugation map and d(\operatorname{Conj}_g)|_e denotes its differential at the identity element e \in G.^[2]^[1] This construction equips \operatorname{Ad} with the structure of a Lie group representation, as it acts linearly on the vector space \mathfrak{g}.^[2] The map \operatorname{Ad} is a Lie group homomorphism, satisfying \operatorname{Ad}_{gh} = \operatorname{Ad}_g \circ \operatorname{Ad}_h for all g, h \in G, which follows from the chain rule applied to the conjugation maps.^[2]^[1] Moreover, it intertwines the exponential maps via the relation \operatorname{Ad}_{\exp(X)} = \exp(\operatorname{ad}_X) for X \in \mathfrak{g}, where \operatorname{ad}: \mathfrak{g} \to \operatorname{gl}(\mathfrak{g}) is the adjoint map on the Lie algebra.^[2]^[1] The image of \operatorname{Ad} is the adjoint group \operatorname{Ad}(G) \subseteq \operatorname{GL}(\mathfrak{g}), which preserves the Lie bracket on \mathfrak{g}.^[1] The derivative of \operatorname{Ad} at the identity recovers the Lie algebra adjoint: d\operatorname{Ad}_e: \mathfrak{g} \to \operatorname{gl}(\mathfrak{g}) is given by d\operatorname{Ad}_e(X) = \operatorname{ad}_X, where \operatorname{ad}_X(Y) = [X, Y] is the Lie bracket action.^[2]^[1] The target space \operatorname{GL}(\mathfrak{g}) has dimension (\dim \mathfrak{g})^2, reflecting its identification with the general linear group on the finite-dimensional space \mathfrak{g}.^[2] This differential relationship underscores how the group-level representation \operatorname{Ad} linearizes to the infinitesimal action \operatorname{ad} near the identity.^[1]

Algebraic Aspects

Structure Constants

In a finite-dimensional Lie algebra \mathfrak{g} over a field of characteristic zero, choose a basis \{e_i\}_{i=1}^n. The Lie bracket is then expressed in coordinates by

[e_i, e_j] = \sum_{k=1}^n c_{ij}^k e_k,

where the scalars c_{ij}^k \in F are called the structure constants of \mathfrak{g} with respect to this basis. These constants fully encode the multiplication table of the Lie algebra and thus determine its isomorphism class up to the choice of basis.^[10] The adjoint map \mathrm{ad}: \mathfrak{g} \to \mathrm{End}(\mathfrak{g}) satisfies \mathrm{ad}_{e_i}(e_j) = [e_i, e_j] = \sum_{k=1}^n c_{ij}^k e_k, so the structure constants provide the coordinate expression for the adjoint action on basis elements. The antisymmetry of the Lie bracket [x, y] = -[y, x] implies that the structure constants are antisymmetric in the lower indices: c_{ij}^k = -c_{ji}^k for all i, j, k. The Jacobi identity [[x, y], z] + [[y, z], x] + [[z, x], y] = 0 imposes a quadratic relation on the constants:

\sum_m \left( c_{ij}^m c_{mk}^l + c_{jk}^m c_{mi}^l + c_{ki}^m c_{mj}^l \right) = 0

for all indices i, j, k, l. These relations ensure that the bracket defines a Lie algebra structure.^[11] In the adjoint representation, the endomorphisms \mathrm{ad}_{e_i} are represented by n \times n matrices whose entries are determined by the structure constants. With respect to the chosen basis, the matrix of \mathrm{ad}_{e_i} has entries (\mathrm{ad}_{e_i})_{jk} = -c_{ik}^j, following the sign convention that aligns the lower indices with the action on contravariant components (note that alternative conventions without the negative sign exist, depending on index placement). This matrix representation facilitates computations of the adjoint action, such as traces or determinants, in coordinate form.^[12] The structure constants are not intrinsic to the Lie algebra but depend on the basis; they transform under change of basis via the adjoint action of \mathrm{GL}(\mathfrak{g}). Specifically, if \{e'_p\} is a new basis related by e'_p = \sum_q g_p^q e_q with g \in \mathrm{GL}(n, F), the new constants

c'_{pq}^r

satisfy

c'_{pq}^r = \sum_{i,j,k} (g^{-1})^r_k \, g_p^i \, g_q^j \, c_{ij}^k,

reflecting the tensorial nature of the constants under basis transformations. This covariance ensures that properties like the Jacobi relations are preserved.^[10]

The Adjoint Operator as a Derivation

The adjoint operator \ad_x for a fixed x \in \mathfrak{g} is the linear map \ad_x: \mathfrak{g} \to \mathfrak{g} defined by \ad_x(y) = [x, y] for all y \in \mathfrak{g}. This map is a derivation of the Lie algebra \mathfrak{g}, meaning it preserves the Lie bracket in the sense of the Leibniz rule:

\ad_x([y, z]) = [\ad_x(y), z] + [y, \ad_x(z)]

for all y, z \in \mathfrak{g}. To verify this, apply the Jacobi identity to the left-hand side:

\ad_x([y, z]) = [x, [y, z]] = [[x, y], z] + [y, [x, z]] = [\ad_x(y), z] + [y, \ad_x(z)],

which establishes the required equality.^[13]^[14] The collection of all derivations of \mathfrak{g} forms a Lie subalgebra \Der(\mathfrak{g}) of \End(\mathfrak{g}) under the commutator bracket [D_1, D_2] = D_1 D_2 - D_2 D_1. The adjoint map \ad: \mathfrak{g} \to \Der(\mathfrak{g}) given by x \mapsto \ad_x is itself a Lie algebra homomorphism, since [\ad_x, \ad_{x'}] = \ad_{[x, x']}. The kernel of this map is the center \mathfrak{z}(\mathfrak{g}) = \{ x \in \mathfrak{g} \mid [x, y] = 0 \ \forall y \in \mathfrak{g} \}, so by the first isomorphism theorem, the image \ad(\mathfrak{g}), known as the inner derivations, is isomorphic to the quotient Lie algebra \mathfrak{g} / \mathfrak{z}(\mathfrak{g}).^[15]^[13] The inner derivations \ad(\mathfrak{g}) form an ideal in \Der(\mathfrak{g}). The outer derivations are the elements of the quotient \Der(\mathfrak{g}) / \ad(\mathfrak{g}), which classify derivations up to inner ones. For semisimple Lie algebras, this quotient is zero, so \Der(\mathfrak{g}) = \ad(\mathfrak{g}) and all derivations are inner.^[13]^[14] Solvable Lie algebras, in contrast, generally admit outer derivations. For instance, every nilpotent Lie algebra possesses at least one outer derivation; the three-dimensional Heisenberg algebra, with basis \{p, q, z\} and nonzero bracket [p, q] = z, provides such an example.^[16]

Key Properties

General Properties

The adjoint representation of a Lie group G on its Lie algebra \mathfrak{g} is faithful—that is, the homomorphism \mathrm{Ad}: G \to \mathrm{GL}(\mathfrak{g}) is injective—if and only if the center Z(G) of G is trivial.^[17] Equivalently, for the infinitesimal version, the adjoint map \mathrm{ad}: \mathfrak{g} \to \mathrm{End}(\mathfrak{g}) is injective if and only if the center of \mathfrak{g} is zero.^[17] This property highlights the adjoint representation's role in detecting the center, providing an embedding of the centerless quotient G/Z(G) into the general linear group. A key trace property arises from the Jacobi identity: for every x \in \mathfrak{g}, the trace \mathrm{tr}(\mathrm{ad}_x) = 0.^[18] This follows because \mathrm{ad}_x acts as a derivation, and in the adjoint representation over a field of characteristic zero, the trace vanishes on inner derivations, as commutators in \mathrm{End}(\mathfrak{g}) have zero trace.^[18] Consequently, for the group-level representation, \det(\mathrm{Ad}_g) = 1 for all g \in G, so the image of \mathrm{Ad} lies in the special linear group \mathrm{SL}(\dim \mathfrak{g}, \mathbb{R}).^[19] The adjoint action defines orbits in \mathfrak{g}: the G-orbit of an element x \in \mathfrak{g} under \mathrm{Ad} has dimension \dim(\mathrm{orbit}(x)) = \dim \mathfrak{g} - \dim \mathfrak{z}_\mathfrak{g}(x), where \mathfrak{z}_\mathfrak{g}(x) = \{ y \in \mathfrak{g} \mid [x, y] = 0 \} is the centralizer of x.^[20] At the Lie algebra level, the infinitesimal orbits under \mathrm{ad} satisfy a similar dimension formula, reflecting the stabilizer's codimension in the orbit-stabilizer theorem for the adjoint action.^[20] The adjoint representation extends, by the universal property of the universal enveloping algebra U(\mathfrak{g}), to a representation U(\mathfrak{g}) \to \mathrm{End}(\mathfrak{g}), making \mathfrak{g} into a left U(\mathfrak{g})-module (the adjoint module).^[21] This structure underlies many constructions in representation theory, such as the study of induced modules and Lie algebra cohomology.

Properties for Semisimple Lie Algebras

In semisimple Lie algebras over an algebraically closed field of characteristic zero, the adjoint representation exhibits distinctive structural properties tied to the algebra's decomposition. A semisimple Lie algebra \mathfrak{g} 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 ad-simple (i.e., simple as a Lie algebra). Under this decomposition, the adjoint representation of \mathfrak{g} restricts to the adjoint representation on each simple ideal \mathfrak{g}_i, and these restrictions are irreducible, meaning each \mathfrak{g}_i has no nontrivial \mathfrak{g}_i-invariant subspaces.^[22] A fundamental tool for analyzing these properties is the Killing form, defined on \mathfrak{g} by B(x, y) = \operatorname{tr}(\operatorname{ad}_x \circ \operatorname{ad}_y) for x, y \in \mathfrak{g}. For semisimple \mathfrak{g}, the Killing form is nondegenerate, meaning its radical is zero, which is equivalent to the semisimple condition and ensures that \mathfrak{g} can be faithfully represented via the adjoint action.^[22] Moreover, the Killing form is ad-invariant, satisfying B([x, z], y) + B(x, [z, y]) = 0 for all x, y, z \in \mathfrak{g}, a property derived from the trace's invariance under cyclic permutations and the Jacobi identity in the adjoint representation.^[23] This nondegeneracy restricts to each simple ideal, facilitating the orthogonal decomposition with respect to B. The semisimple structure also implies complete reducibility of representations: every finite-dimensional representation of \mathfrak{g}, including the adjoint representation on \mathfrak{g} itself, is completely reducible, decomposing as a direct sum of irreducible subrepresentations (Weyl's theorem).^[22] In the adjoint case, this yields the direct sum decomposition into the irreducible adjoint actions on the simple ideals, underscoring the absence of indecomposable but non-irreducible components.

Examples

Classical Matrix Lie Algebras

The classical matrix Lie algebras provide explicit finite-dimensional realizations of Lie algebras over the complex numbers, where the adjoint representation can be concretely described using matrix operations. These algebras are subalgebras of \mathfrak{gl}(n,\mathbb{C}), the general linear Lie algebra consisting of all n \times n complex matrices equipped with the commutator Lie bracket [X,Y] = XY - YX.^[18] The dimension of \mathfrak{gl}(n,\mathbb{C}) is n^2, and its adjoint representation is the natural action on itself via the Lie bracket: for X,Y \in \mathfrak{gl}(n,\mathbb{C}), the infinitesimal adjoint map is \mathrm{ad}_X(Y) = [X,Y] = XY - YX, which yields a representation \mathrm{ad}: \mathfrak{gl}(n,\mathbb{C}) \to \mathfrak{gl}(\mathfrak{gl}(n,\mathbb{C})) of dimension n^2 \times n^2.^[10] Correspondingly, the adjoint representation of the Lie group \mathrm{GL}(n,\mathbb{C}) acts by conjugation: \mathrm{Ad}_A(B) = A B A^{-1} for A \in \mathrm{GL}(n,\mathbb{C}) and B \in \mathfrak{gl}(n,\mathbb{C}).^[18] The special linear Lie algebra \mathfrak{sl}(n,\mathbb{C}) is the subalgebra of \mathfrak{gl}(n,\mathbb{C}) consisting of trace-zero matrices, with dimension n^2 - 1.^[10] The adjoint representation restricts naturally to \mathfrak{sl}(n,\mathbb{C}), as the Lie bracket preserves the trace-zero condition: \mathrm{tr}([X,Y]) = \mathrm{tr}(XY - YX) = 0 for X,Y \in \mathfrak{sl}(n,\mathbb{C}).^[18] Thus, \mathrm{ad}_X(Y) = [X,Y] maps \mathfrak{sl}(n,\mathbb{C}) to itself, giving a representation \mathrm{ad}: \mathfrak{sl}(n,\mathbb{C}) \to \mathfrak{gl}(\mathfrak{sl}(n,\mathbb{C})) of dimension (n^2 - 1) \times (n^2 - 1). The group-level action \mathrm{Ad}_A(B) = A B A^{-1} for A \in \mathrm{SL}(n,\mathbb{C}) similarly preserves \mathfrak{sl}(n,\mathbb{C}), since \mathrm{tr}(A B A^{-1}) = \mathrm{tr}(B) = 0.^[10] For the orthogonal case, the special orthogonal Lie algebra \mathfrak{so}(n,\mathbb{C}) consists of n \times n skew-symmetric matrices satisfying X^T = -X, with dimension n(n-1)/2.^[18] The Lie bracket is again [X,Y] = XY - YX, and the adjoint representation is \mathrm{ad}_X(Y) = [X,Y] for X,Y \in \mathfrak{so}(n,\mathbb{C}), which preserves skew-symmetry because if X^T = -X and Y^T = -Y, then [X,Y]^T = (XY - YX)^T = Y^T X^T - X^T Y^T = (-Y)(-X) - (-X)(-Y) = YX - XY = -[X,Y].^[10] This yields matrices of size [n(n-1)/2] \times [n(n-1)/2]. At the group level, elements R \in \mathrm{SO}(n,\mathbb{C}) satisfy R^T R = I, so R^{-1} = R^T, and the adjoint action is \mathrm{Ad}_R(X) = R X R^T for X \in \mathfrak{so}(n,\mathbb{C}), which preserves the skew-symmetric condition.^[18] The symplectic Lie algebra \mathfrak{sp}(2n,\mathbb{C}) is the subalgebra of \mathfrak{gl}(2n,\mathbb{C}) consisting of $2n \times 2n matrices X that preserve the symplectic form, satisfying X^T J + J X = 0 where J = \begin{pmatrix} 0 & I_n \\ -I_n & 0 \end{pmatrix}, with dimension n(2n+1).^[10] The adjoint representation uses the commutator bracket [X,Y] = XY - YX, and \mathrm{ad}_X(Y) = [X,Y] preserves the symplectic condition, as the form is bilinear and the bracket maintains the defining relation.^[18] This gives representation matrices of size [n(2n+1)] \times [n(2n+1)]. For the group \mathrm{Sp}(2n,\mathbb{C}), elements g satisfy g^T J g = J, implying g^{-1} = -J^{-1} g^T J = J^{-1} g^T J (since J^{-1} = -J), and the adjoint action is \mathrm{Ad}_g(X) = g X g^{-1}, which preserves the Lie algebra.^[10]

sl(2,ℝ) and sl(2,ℂ)

The Lie algebra \mathfrak{sl}(2,\mathbb{C}) consists of $2 \times 2 traceless matrices over \mathbb{C} and admits a standard basis

h = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \quad e = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \quad f = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix},

satisfying the commutation relations [h, e] = 2e, [h, f] = -2f, and [e, f] = h.^[24] In this basis \{h, e, f\}, the adjoint representation maps elements of \mathfrak{sl}(2,\mathbb{C}) to $3 \times 3 matrices acting on the Lie algebra itself via the Lie bracket. The explicit matrices are

\mathrm{ad}_h = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & -2 \end{pmatrix}, \quad \mathrm{ad}_e = \begin{pmatrix} 0 & 0 & 1 \\ -2 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \quad \mathrm{ad}_f = \begin{pmatrix} 0 & -1 & 0 \\ 0 & 0 & 0 \\ 2 & 0 & 0 \end{pmatrix}.

These matrices arise directly from computing \mathrm{ad}_z(w) = [z, w] for each basis element z and expressing the result in coordinates with respect to \{h, e, f\}.^[24] The adjoint representation of \mathfrak{sl}(2,\mathbb{C}) is irreducible, providing the unique 3-dimensional irreducible representation up to equivalence.^[25] It is isomorphic to the irreducible representation of highest weight 2 (often denoted V_2), which has dimension $2+1=3 and corresponds to the spin-1 representation in physics terminology; equivalently, it is the symmetric square \mathrm{Sym}^2(\mathbb{C}^2) of the fundamental 2-dimensional representation.^[26] The real Lie algebra \mathfrak{sl}(2,\mathbb{R}) consists of $2 \times 2 traceless matrices over \mathbb{R} and shares the same standard basis \{h, e, f\} as above, now viewed over \mathbb{R}, with identical commutation relations. The adjoint representation is thus realized by the same $3 \times 3 real matrices as in the complex case.^[24] Since \mathfrak{sl}(2,\mathbb{R}) is a simple Lie algebra, its adjoint representation is irreducible over \mathbb{R}.^[26] The complexification \mathfrak{sl}(2,\mathbb{R}) \otimes \mathbb{C} \cong \mathfrak{sl}(2,\mathbb{C}) identifies the two adjoint representations upon extension of scalars, underscoring their structural similarity despite the differing real geometries.^[26]

Roots and Advanced Structures

Roots in Semisimple Lie Algebras

In semisimple Lie algebras over the complex numbers, a Cartan subalgebra \mathfrak{[h](/page/H+)} \subseteq \mathfrak{g} is a maximal toral subalgebra, meaning it is abelian and consists of semisimple elements whose joint eigenspaces decompose \mathfrak{g}. The adjoint representation of \mathfrak{g} restricted to \mathfrak{[h](/page/H+)} yields the root space decomposition \mathfrak{g} = \mathfrak{[h](/page/H+)} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_\alpha, where \Phi \subset \mathfrak{[h](/page/H+)}^* is the root system and each root space is defined as \mathfrak{g}_\alpha = \{ x \in \mathfrak{g} \mid \mathrm{ad}_h x = \alpha([h](/page/H+)) [x](/page/x) \ \forall [h](/page/H+) \in \mathfrak{[h](/page/H+)} \}. This decomposition arises because the adjoint operators \mathrm{ad}_[h](/page/H+) for [h](/page/H+) \in \mathfrak{[h](/page/H+)} are simultaneously diagonalizable, with \mathfrak{[h](/page/H+)} as the zero eigenspace and the \mathfrak{g}_\alpha as the nonzero eigenspaces. The adjoint action on each root space is scalar: for h \in \mathfrak{h} and x \in \mathfrak{g}_\alpha, \mathrm{ad}_h|_{\mathfrak{g}_\alpha} = \alpha(h) \mathrm{Id}, where \alpha \in \mathfrak{h}^* is the corresponding root functional. The Lie bracket respects the grading: [\mathfrak{g}_\alpha, \mathfrak{g}_\beta] \subseteq \mathfrak{g}_{\alpha + \beta} for \alpha, \beta \in \Phi \cup \{0\}, with equality holding when \alpha + \beta is also a root. In the classical finite-dimensional semisimple case, each root space is one-dimensional, \dim \mathfrak{g}_\alpha = 1 for all \alpha \in \Phi. As an \mathfrak{h}-module via the adjoint representation, \mathfrak{g} decomposes into weight spaces: the adjoint representation is the direct sum \mathfrak{g} \cong \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi} \mathbb{C}_\alpha, where each \mathbb{C}_\alpha is the one-dimensional representation with weight \alpha. The number of nonzero roots equals the corank, |\Phi| = \dim \mathfrak{g} - \dim \mathfrak{h}, reflecting the structure of the root system \Phi.

Weyl Group and Adjoint Orbits

In semisimple Lie algebras, the Weyl group W is defined as the quotient N_G(\mathfrak{h}) / Z_G(\mathfrak{h}), where \mathfrak{h} is a Cartan subalgebra, G is the adjoint group of the Lie algebra \mathfrak{g}, N_G(\mathfrak{h}) is the normalizer of \mathfrak{h} in G, and Z_G(\mathfrak{h}) is the centralizer of \mathfrak{h} in G. This finite group acts on the Cartan subalgebra \mathfrak{h} by conjugation: for w \in W and h \in \mathfrak{h}, the action is given by w \cdot h = w h w^{-1}.^[27] The action extends to the root system associated with \mathfrak{g}, inducing permutations on the roots via w \cdot \alpha(h) = \alpha(w^{-1} h w) for a root \alpha.^[27] Adjoint orbits arise from the action of G on \mathfrak{g} via the adjoint representation. For a regular element x \in \mathfrak{g}, the stabilizer Z_G(x) is the center of G, so the orbit G \cdot x is isomorphic to G / Z_G(x), and its dimension is \dim \mathfrak{g} - \dim \mathfrak{z}_\mathfrak{g}(x), where \mathfrak{z}_\mathfrak{g}(x) = \{ y \in \mathfrak{g} \mid [y, x] = 0 \} is the centralizer of x in \mathfrak{g}.^[28] For a semisimple element x \in \mathfrak{h}, the adjoint orbit G \cdot x intersects \mathfrak{h} precisely in the Weyl group orbit W \cdot x \subseteq \mathfrak{h}, reflecting the discrete symmetries preserved by the conjugation action.^[27] These adjoint orbits possess a canonical symplectic structure known as the Kirillov-Kostant-Souriau form, which endows them with the geometry of a symplectic manifold and plays a key role in geometric quantization and representation theory.^[28] The slice theorem for the adjoint action provides a local normal form near each orbit: around a point x \in \mathfrak{g}, there exists a G-invariant neighborhood modeled as a product of the orbit G \cdot x and a slice S_x, a submanifold transverse to the orbit lying in the centralizer \mathfrak{z}_\mathfrak{g}(x) and of dimension equal to the codimension of the orbit.^[29] This decomposition facilitates the study of the local geometry and stratification of \mathfrak{g} under the adjoint action.^[29]

Variants and Generalizations

Real vs. Complex Forms

The complexification of a real Lie algebra \mathfrak{g}_\mathbb{R} is the complex Lie algebra \mathfrak{g}_\mathbb{C} = \mathfrak{g}_\mathbb{R} \otimes_\mathbb{R} \mathbb{C}, which extends the bracket structure linearly over \mathbb{C}.^[27] The adjoint representation extends accordingly: for Z = X + iY \in \mathfrak{g}_\mathbb{C} with X, Y \in \mathfrak{g}_\mathbb{R}, the complex adjoint operator is \mathrm{ad}_\mathbb{C}(Z) = \mathrm{ad}_X + i \mathrm{ad}_Y, where \mathrm{ad}_X and \mathrm{ad}_Y are the real adjoints.^[30] This extension decomposes the action into holomorphic and anti-holomorphic components, facilitating the analysis of representations over \mathbb{C} while preserving properties like solvability from the real case.^[27] Real forms of a complex semisimple Lie algebra \mathfrak{g}_\mathbb{C} are real subalgebras \mathfrak{g}_\mathbb{R} \subset \mathfrak{g}_\mathbb{C} such that \mathfrak{g}_\mathbb{C} = \mathfrak{g}_\mathbb{R} \otimes_\mathbb{R} \mathbb{C}, fixed by an antilinear involution.^[27] These forms are classified as compact or non-compact based on the corresponding Lie group. For compact real forms, such as \mathfrak{su}(2), the adjoint representation preserves a negative definite invariant bilinear form (the Killing form), rendering it orthogonal with respect to that form.^[30] In contrast, non-compact forms like \mathfrak{sl}(2, \mathbb{R}) yield adjoint orbits with hyperbolic geometry, such as hyperboloids, reflecting the indefinite nature of the underlying structure.^[31] A key tool for distinguishing real forms is the Cartan involution \theta: \mathfrak{g}_\mathbb{R} \to \mathfrak{g}_\mathbb{R}, a Lie algebra automorphism satisfying \theta^2 = \mathrm{id} such that the bilinear form B_\theta(X, Y) = -B(X, \theta(Y)) is positive definite, where B is the Killing form.^[32] This involution decomposes \mathfrak{g}_\mathbb{R} = \mathfrak{k} \oplus \mathfrak{p}, where \mathfrak{k} is the +1-eigenspace (a compact subalgebra) and \mathfrak{p} is the -1-eigenspace, with Lie bracket relations [\mathfrak{k}, \mathfrak{k}] \subset \mathfrak{k}, [\mathfrak{k}, \mathfrak{p}] \subset \mathfrak{p}, and [\mathfrak{p}, \mathfrak{p}] \subset \mathfrak{k}.^[32] The decomposition is invariant under the adjoint action of the maximal compact subgroup \exp(\mathfrak{k}).^[32] The signature of the Killing form B(X, Y) = \mathrm{tr}(\mathrm{ad}_X \mathrm{ad}_Y) further differentiates forms: it is negative definite on compact real forms like \mathfrak{su}(2), ensuring complete reducibility of the adjoint representation, while indefinite on non-compact forms like \mathfrak{sl}(2, \mathbb{R}), where B is negative definite on \mathfrak{k} and positive definite on \mathfrak{p}.^[30]^[32] This property, central to semisimple Lie algebras, underscores the analytic differences between real and complex settings.^[27]

Infinite-Dimensional Analogues

The adjoint representation extends to infinite-dimensional Lie algebras, where the finite-dimensional structure gives way to more complex decompositions involving infinite root systems and central extensions. In Kac-Moody algebras, defined via a generalized Cartan matrix A = (a_{ij}) with a_{ii} = 2 and a_{ij} \leq 0 for i \neq j, the algebra \mathfrak{[g](/page/G)}(A) decomposes as \mathfrak{[g](/page/G)}(A) = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Delta} \mathfrak{g}_\alpha, where \mathfrak{h} is the Cartan subalgebra spanned by h_i and a central element, and \Delta is an infinite root system.^[33] The adjoint action \mathrm{ad}_h for h \in \mathfrak{h} acts diagonally on the root spaces \mathfrak{g}_\alpha, with [\ h_i, e_j\ ] = a_{ij} e_j and [\ h_i, f_j\ ] = -a_{ij} f_j, mirroring the finite-dimensional case but extended over infinitely many roots due to the loop-like structure.^[33]^[34] A prominent example is the Virasoro algebra, the universal central extension of the Witt algebra of vector fields on the circle, with generators L_n satisfying the commutation relations

[L_m, L_n] = (m - n) L_{m+n} + \frac{c}{12} m (m^2 - 1) \delta_{m+n, 0},

where c is the central charge, a scalar invariant labeling representations.^[35] Here, the adjoint representation is realized by these brackets, so \mathrm{ad}_{L_n}(L_m) = [L_n, L_m] = (n - m) L_{n+m} + \frac{c}{12} n (n^2 - 1) \delta_{n+m, 0}, introducing a central term absent in finite-dimensional semisimple cases.^[35] This structure arises in two-dimensional conformal field theory, where the Virasoro algebra governs symmetries of stress-energy tensors. Unlike finite-dimensional semisimple Lie algebras, where the adjoint representation is finite-dimensional and irreducible, infinite-dimensional analogues lack finite-dimensional irreducible representations beyond the trivial one, leading to challenges in classification and unitarity.^[36] Representations are typically studied as highest-weight modules, Verma modules, or integrable modules at positive integer levels, with the adjoint action preserving gradings but requiring analytic continuation or regularization for convergence.^[36]^[34] Loop algebras provide another analogue, formed as \mathfrak{g} \otimes \mathbb{C}[t, t^{-1}] for a finite-dimensional Lie algebra \mathfrak{g}, where the adjoint representation acts on this tensor product space.^[37] The algebra admits a \mathbb{Z}-gradation by Laurent degree, with components \bigoplus_{n \in \mathbb{Z}} \mathfrak{g} \otimes t^n, and the adjoint action preserves this grading, as [X \otimes t^k, Y \otimes t^l] \in \mathfrak{g} \otimes t^{k+l}.^[37] Affine Kac-Moody algebras arise as central extensions of these loop algebras, enhancing the adjoint structure with derivation and central elements.^[37]

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