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Lie algebra representation

In mathematics, a Lie algebra representation (or simply a representation of a Lie algebra \mathfrak{g}) is a vector space V equipped with a Lie algebra homomorphism \rho: \mathfrak{g} \to \mathfrak{gl}(V), where \mathfrak{gl}(V) denotes the Lie algebra of all linear endomorphisms of V, such that the Lie bracket is preserved: \rho([X, Y]) = [\rho(X), \rho(Y)] for all X, Y \in \mathfrak{g}. This structure equivalently defines a module over the universal enveloping algebra U(\mathfrak{g}), providing a linear algebraic framework to study the infinitesimal actions of Lie algebras on vector spaces. Lie algebra representations are intimately connected to representations of Lie groups: for a Lie group G with Lie algebra \mathfrak{g}, any finite-dimensional representation \pi: G \to \mathrm{GL}(V) differentiates to yield a Lie algebra representation \rho = d\pi_e: \mathfrak{g} \to \mathfrak{gl}(V) at the , and this process establishes an for connected, simply connected Lie groups. Key concepts include irreducible representations, which admit no nontrivial subspaces (subrepresentations), and completely reducible representations, which decompose as direct sums of irreducibles; over the complex numbers, finite-dimensional representations of semisimple Lie algebras are completely reducible by Weyl's theorem. Schur's lemma further characterizes endomorphisms of irreducible representations as scalar multiples of the . Representations of Lie algebras form the cornerstone of representation theory, enabling the classification and decomposition of symmetry actions in algebraic structures, with profound implications for solving differential equations and understanding geometric invariants. In physics, they underpin the modeling of continuous symmetries in quantum mechanics and particle physics, such as the SU(2) representations describing isospin for nucleons (protons and neutrons as spin-1/2 doublets) and SU(3) representations organizing hadrons into multiplets via the eightfold way, which predicted particles like the \Omega^- baryon. These tools extend to broader applications in topology, symmetric spaces, and quantum field theory, where they facilitate the analysis of conserved quantities and particle interactions under group symmetries.

Definition

Formal Definition

A representation of a \mathfrak{g} over a k on a vector space V is a Lie algebra homomorphism \rho: \mathfrak{g} \to \mathfrak{gl}(V), where \mathfrak{gl}(V) denotes the Lie algebra of all linear endomorphisms of V equipped with the commutator bracket [A, B] = AB - BA for A, B \in \mathfrak{gl}(V). This homomorphism assigns to each element X \in \mathfrak{g} a linear map \rho(X): V \to V, thereby defining an action of \mathfrak{g} on V by linear transformations. The k is typically \mathbb{C} or \mathbb{R}, though the definition holds more generally. The defining property of the homomorphism requires that \rho preserves the Lie bracket structure of \mathfrak{g}, satisfying \rho([X, Y]) = [\rho(X), \rho(Y)] for all X, Y \in \mathfrak{g}. Explicitly, this compatibility condition is \rho([X, Y]) = \rho(X) \rho(Y) - \rho(Y) \rho(X), ensuring that the induced action on V respects the algebraic relations in \mathfrak{g}. The V, known as the representation space, may be finite-dimensional or infinite-dimensional over k. While finite-dimensional representations are central to classical —particularly for semisimple Lie algebras over \mathbb{C}—infinite-dimensional cases arise in contexts like universal enveloping algebras and , where \mathfrak{g} itself is often finite-dimensional but acts on larger spaces.

Module Formulation

A Lie algebra representation can equivalently be formulated in terms of modules over the Lie algebra \mathfrak{g}. In this view, a vector space V over a k (of characteristic zero) is a \mathfrak{g}-module if there exists a bilinear map \mathfrak{g} \times V \to V, denoted (X, v) \mapsto X \cdot v, satisfying linearity in V: X \cdot (a v + b w) = a (X \cdot v) + b (X \cdot w) for all X \in \mathfrak{g}, v, w \in V, and a, b \in k, and compatibility with the Lie bracket: [X, Y] \cdot v = X \cdot (Y \cdot v) - Y \cdot (X \cdot v) for all X, Y \in \mathfrak{g} and v \in V. This action ties the adjoint representation of \mathfrak{g} on itself—where \mathrm{ad}_X(Y) = [X, Y]—to the module structure by ensuring the induced map \rho: \mathfrak{g} \to \mathfrak{gl}(V) defined by \rho(X)(v) = X \cdot v preserves the bracket: \rho([X, Y]) = [\rho(X), \rho(Y)]. In the literature, \mathfrak{g}-modules are conventionally left modules, with the action written on the left as X \cdot v; right modules appear less frequently and typically require adjusting the bracket condition by a sign change, such as [X, Y] \cdot v = - (Y \cdot (X \cdot v) - X \cdot (Y \cdot v)), to preserve compatibility, though most texts standardize on the left convention for consistency with enveloping algebra actions. This module perspective gained prominence in the mid-20th century alongside advances in , which provided tools for studying extensions, , and derived functors in the category of \mathfrak{g}-modules.

Examples

Adjoint Representation

The of a \mathfrak{g} over a k (of zero) is the \mathrm{ad}: \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g}) defined by \mathrm{ad}_X(Y) = [X, Y] for all X, Y \in \mathfrak{g}, where \mathfrak{gl}(\mathfrak{g}) denotes the Lie algebra of endomorphisms of \mathfrak{g}. This map equips \mathfrak{g} with a natural action on itself via the Lie bracket, realizing the algebra's structure as a representation. The satisfies the defining property of a Lie algebra representation due to the : \mathrm{ad}_{[X,Y]} = [\mathrm{ad}_X, \mathrm{ad}_Y], where the bracket on the right is the [\cdot, \cdot]_{\mathfrak{gl}(\mathfrak{g})}(A, B) = AB - BA in \mathfrak{gl}(\mathfrak{g}). Each \mathrm{ad}_X is an inner of \mathfrak{g}, meaning it preserves the Lie bracket via the derivation property \mathrm{ad}_X([Y, Z]) = [\mathrm{ad}_X(Y), Z] + [Y, \mathrm{ad}_X(Z)]. The kernel of \mathrm{ad} is the center Z(\mathfrak{g}) = \{X \in \mathfrak{g} \mid [X, Y] = 0 \ \forall Y \in \mathfrak{g}\}, which is an of \mathfrak{g}. The image \mathrm{ad}(\mathfrak{g}) consists of the inner derivations \mathrm{Inn}(\mathfrak{g}), a Lie subalgebra of the derivation algebra \mathrm{Der}(\mathfrak{g}), while the subspace generated by the action \mathrm{ad}(\mathfrak{g})(\mathfrak{g}) is the derived algebra [\mathfrak{g}, \mathfrak{g}]. In the structure theory of algebras, the classifies inner derivations as those arising from the bracket, distinguishing them from outer derivations, and induces an \mathfrak{g}/Z(\mathfrak{g}) \cong \mathrm{Inn}(\mathfrak{g}). For semisimple algebras, \mathrm{ad} is faithful (with trivial center) and realizes all derivations as inner, providing a foundational tool for decomposition into simple ideals and root space analysis. This also connects to the Killing form, which is the trace form associated to \mathrm{ad}.

From Lie Groups

Lie algebra representations arise naturally from representations of Lie groups through differentiation at the identity element. Given a Lie group G with Lie algebra \mathfrak{g} and a finite-dimensional representation \pi: G \to \mathrm{GL}(V) on a vector space V, the induced Lie algebra representation d\pi: \mathfrak{g} \to \mathfrak{gl}(V) is defined by its action on elements X \in \mathfrak{g}. Specifically, for a smooth curve \gamma: (-\epsilon, \epsilon) \to G with \gamma(0) = e (the identity) and \gamma'(0) = X, the differential is given by d\pi(X) v = \left. \frac{d}{dt} \right|_{t=0} \pi(\gamma(t)) v for all v \in V. This construction ensures that d\pi is a Lie algebra homomorphism, preserving the Lie bracket [X, Y] \mapsto [d\pi(X), d\pi(Y)]. For matrix Lie groups, where G \subseteq \mathrm{GL}(n, \mathbb{R}) or \mathrm{GL}(n, \mathbb{C}), the infinitesimal generators can be expressed explicitly using the exponential map. The action of d\pi(X) on v \in V is d\pi(X) v = \lim_{t \to 0} \frac{\pi(\exp(tX)) v - v}{t}, which linearizes the group action infinitesimally. This limit corresponds to the derivative along the one-parameter subgroup generated by X. Furthermore, for connected Lie groups, the compatibility with the exponential map yields \pi(\exp(X)) = \exp(d\pi(X)), linking the global group structure to the local Lie algebra behavior. A key uniqueness result holds for connected Lie groups: every finite-dimensional representation of G arises uniquely from a representation of its \mathfrak{g} via this differentiation process. The differential at the identity fully determines the group representation on the connected component of the identity, ensuring that the induced d\pi captures the essential infinitesimal symmetries. This correspondence facilitates the study of representations by reducing global properties to algebraic ones.

In Physics

In quantum mechanics, representations of the Lie algebra \mathfrak{su}(2) form the foundation for describing angular momentum, where the generators J_x, J_y, J_z satisfy the commutation relations [J_i, J_j] = i \hbar \epsilon_{ijk} J_k. The irreducible representations are labeled by the spin quantum number j = 0, 1/2, 1, \dots, each with dimension $2j + 1, corresponding to the possible eigenvalues of the total angular momentum operator J^2 = j(j+1)\hbar^2. These representations underpin the quantization of orbital and spin angular momentum in atomic and molecular systems. In particle physics, representations of the Lie algebra \mathfrak{su}(3) play a central role in the Eightfold Way classification scheme for hadrons, proposed by Murray Gell-Mann. Quarks transform under the fundamental representation of dimension 3, while mesons and baryons often occupy the adjoint representation of dimension 8, such as the octet of pseudoscalar mesons (\pi, K, \eta) and vector mesons (\rho, K^*, \omega, \phi). This structure predicted the existence of the \Omega^- baryon before its discovery, validating the quark model interpretation of strong interactions. Lie algebra representations supply the infinitesimal generators for continuous symmetries in physical theories, enabling the analysis of . For instance, the Lie algebra \mathfrak{so}(3,1) of the generates transformations in , with representations classifying particles by their spin and parity under boosts and rotations. In , representations of conformal algebras like \mathfrak{so}(2,d) in d spatial dimensions describe scale-invariant fixed points, crucial for understanding and holographic dualities.

Fundamental Concepts

Invariant Subspaces and Irreducibility

In the context of Lie algebra representations, an plays a central role in decomposing representations into simpler components. Given a representation \rho: \mathfrak{g} \to \mathfrak{gl}(V) of a \mathfrak{g} on a vector space V, a subspace W \subseteq V is said to be invariant if \rho(X)W \subseteq W for all X \in \mathfrak{g}. Equivalently, in the module formulation, where V is a \mathfrak{g}-module, W is a submodule if X \cdot w \in W for all X \in \mathfrak{g} and w \in W. Such subspaces inherit the representation structure, forming subrepresentations that allow for the analysis of the overall action of \mathfrak{g} on V. A representation \rho is called irreducible if the only subspaces of V are the trivial ones: \{0\} and V itself. This means there are no proper nontrivial subrepresentations, ensuring that the action of \mathfrak{g} cannot be restricted to a smaller while preserving the structure. In the language of modules, an corresponds to a simple module, which has no nontrivial submodules. serve as the fundamental indecomposable units in the study of representations, particularly for finite-dimensional cases over algebraically closed fields of characteristic zero. For finite-dimensional representations over the complex numbers \mathbb{C}, irreducible representations act as the key building blocks for more general representations, as anticipated by results like Maschke's theorem (detailed in subsequent sections on complete reducibility). This decomposition property underscores their importance in classifying representations of semisimple Lie algebras.

Homomorphisms and Isomorphisms

A homomorphism between two representations \rho: \mathfrak{g} \to \mathfrak{gl}(V) and \sigma: \mathfrak{g} \to \mathfrak{gl}(W) of a Lie algebra \mathfrak{g} over a field k is a linear map \phi: V \to W that commutes with the \mathfrak{g}-action, satisfying \phi(\rho(X)v) = \sigma(X)\phi(v) for all X \in \mathfrak{g} and v \in V. Equivalently, in module notation, \phi(X \cdot v) = X \cdot \phi(v). Such a homomorphism \phi is called an isomorphism if it is bijective as a linear map between vector spaces; in this case, the representations \rho and \sigma are said to be equivalent, meaning there exists an invertible linear change of basis intertwining the actions. The inverse map \phi^{-1}: W \to V automatically satisfies the homomorphism condition, preserving the structure. For any homomorphism \phi: V \to W, the kernel \ker \phi = \{v \in V \mid \phi(v) = 0\} is an invariant subspace of V under the \rho-action, and the image \operatorname{im} \phi = \{\phi(v) \mid v \in V\} is an invariant subspace of W under the \sigma-action; thus, both induce subrepresentations. The term intertwiner is often used as a synonym for a representation homomorphism, emphasizing the commutation property that "intertwines" the two actions.

Schur's Lemma

Schur's lemma is a fundamental result in the of , characterizing the endomorphisms of irreducible representations. Let \mathfrak{g} be a over a k, and let \rho: \mathfrak{g} \to \mathfrak{gl}(V) be a finite-dimensional irreducible representation on a vector space V. Then, any endomorphism \phi: V \to V that commutes with \rho, meaning [\rho(X), \phi] = 0 or equivalently \phi \rho(X) = \rho(X) \phi for all X \in \mathfrak{g}, is either zero or invertible. To sketch the proof, consider the kernel \ker \phi and image \operatorname{im} \phi, both of which are invariant subspaces under \rho. By irreducibility of V, \ker \phi = \{0\} or \ker \phi = V, so \phi is either zero or injective. Similarly, \operatorname{im} \phi = V or \operatorname{im} \phi = \{0\}, so \phi is either zero or surjective. For finite-dimensional V, injectivity implies surjectivity and vice versa, hence \phi is invertible if nonzero. Over an algebraically closed field like \mathbb{C}, the commutant consists precisely of scalar multiples \lambda I for \lambda \in \mathbb{C}, since \phi has an eigenvalue \lambda whose eigenspace is invariant, forcing \phi = \lambda I by irreducibility. In the complex case for semisimple Lie algebras, the scalars lie in \mathbb{C}, ensuring the endomorphism ring is exactly the scalars. Over the reals, for irreducible representations of semisimple real Lie algebras, the endomorphism algebra is a real division algebra, which could be \mathbb{R}, \mathbb{C}, or the quaternions \mathbb{H}, introducing additional structure beyond mere scalars. This lemma underpins complete reducibility for semisimple Lie algebras, as explored further in that section.

Complete Reducibility

A representation \rho: \mathfrak{g} \to \mathfrak{gl}(V) of a \mathfrak{g} on a finite-dimensional V over a k is said to be completely reducible if V decomposes as a of irreducible \mathfrak{g}-subrepresentations, i.e., V = \bigoplus_{i=1}^m V_i where each V_i is irreducible. Equivalently, every \mathfrak{g}- W \subseteq V admits a complementary invariant subspace W' \subseteq V such that V = W \oplus W'. For finite-dimensional representations over \mathbb{C} of a finite-dimensional \mathfrak{g}, an analogue of Maschke's theorem holds in characteristic zero: every such representation is completely reducible provided \mathfrak{g} is reductive. This result ensures that the category of finite-dimensional representations behaves semisimple-like under these conditions. In particular, when \mathfrak{g} is semisimple, every finite-dimensional is completely reducible; this is Weyl's theorem on complete reducibility, whose proof relies on the non-degeneracy of the Killing form and the existence of invariant complements via the Casimir operator (detailed in the section on Weyl's theorems). However, complete reducibility fails for non-reductive Lie algebras. For example, consider the Borel subalgebra \mathfrak{b}_2 of upper triangular $2 \times 2 matrices over \mathbb{C}, which is solvable (hence in the broad sense of having a non-trivial ). Its defining on \mathbb{C}^2 has the line spanned by the first basis vector as an without a complementary invariant subspace, so it is indecomposable but not irreducible, hence not completely reducible. More generally, Lie algebras over \mathbb{C}, such as the Heisenberg algebra, admit finite-dimensional representations that are not completely reducible due to the existence of non-split extensions of irreducibles.

Invariant Theory

In the context of Lie algebra representations, invariants are elements of the representation space that remain fixed under the action of the . Given a representation \rho: \mathfrak{g} \to \mathfrak{gl}(V) on a V, an is a vector v \in V such that \rho(X)v = 0 for all X \in \mathfrak{g}. Equivalently, v lies in the of every \rho(X), forming the of fixed points under the \mathfrak{g}-action. A prominent example arises in polynomial representations, where \mathfrak{g} acts on the S(V^*) over a V via the : for X \in \mathfrak{g} and f \in S(V^*), the action is X \cdot f = \rho(X) f, where \rho(X) differentiates f along the direction corresponding to X. An invariant polynomial f satisfies X \cdot f = 0 for all X \in \mathfrak{g}, generating the ring of invariants S(V^*)^\mathfrak{g}. For finite-dimensional representations of semisimple Lie algebras over \mathbb{C}, this ring encodes symmetry properties and is central to classifying orbits under the action. Casimir operators provide key invariants in the universal enveloping algebra U(\mathfrak{g}). These are elements C \in Z(U(\mathfrak{g})), the center of U(\mathfrak{g}), which act on any finite-dimensional irreducible representation \rho by scalar multiplication: \rho(C) = \lambda I for some \lambda \in \mathbb{C} depending on the representation. The quadratic Casimir, constructed using an invariant bilinear form like the Killing form, is fundamental; for semisimple \mathfrak{g}, it distinguishes irreducible modules up to isomorphism in many cases. Detailed constructions and eigenvalues appear in the theory of enveloping algebras. The of a finite-dimensional \rho: \mathfrak{[g](/page/G)} \to \mathfrak{gl}(V) is the function \chi_\rho: \mathfrak{[g](/page/G)} \to \mathbb{C} defined by \chi_\rho(X) = \operatorname{Tr}(\rho(X)). This trace is Ad-invariant, meaning \chi_\rho(\operatorname{Ad}_Y X) = \chi_\rho(X) for all Y \in \mathfrak{[g](/page/G)}, hence constant on adjoint orbits. For irreducible representations of semisimple algebras, characters separate distinct modules and relate to formal characters via multiplicities, facilitating decomposition of representations. Hilbert's 14th problem addresses the finiteness of invariant rings in . For reductive algebras over \mathbb{C}, acting linearly on a , the ring of invariants is finitely generated as a , analogous to the result for reductive algebraic groups proved by Hilbert for \mathrm{[SL](/page/SL)}_n and extended by Weyl to semisimple cases. This finiteness holds for completely reducible representations, ensuring computational tractability in , though counterexamples exist for non-reductive actions.

Constructions

Tensor Products and Direct Sums

In the context of representations, the provides a way to combine two representations into a single one on the of their underlying vector spaces. Given a \mathfrak{g} over a field k (typically \mathbb{C}), and finite-dimensional representations \rho: \mathfrak{g} \to \mathfrak{gl}(V) on a vector space V and \sigma: \mathfrak{g} \to \mathfrak{gl}(W) on W, the representation \rho \oplus \sigma acts on V \oplus W by (\rho \oplus \sigma)(X)(v \oplus w) = \rho(X)v \oplus \sigma(X)w for all X \in \mathfrak{g}, v \in V, and w \in W. This construction preserves the module structure over the universal enveloping algebra U(\mathfrak{g}), making V \oplus W a \mathfrak{g}-module where the action is componentwise. are associative and commutative up to isomorphism, allowing arbitrary finite of representations, which decompose semisimple representations into irreducibles under suitable conditions, such as for semisimple over \mathbb{C}. The construction extends representations to the of vector spaces, leveraging the Leibniz rule inherent to actions. For the same representations \rho and \sigma, the \rho \otimes \sigma acts on V \otimes W via the derivation property: (\rho \otimes \sigma)(X)(v \otimes w) = \rho(X)v \otimes w + v \otimes \sigma(X)w for X \in \mathfrak{g}, or equivalently in operator form, (\rho \otimes \sigma)(X) = \rho(X) \otimes I_W + I_V \otimes \sigma(X), where I_V and I_W are the identity operators. This defines a representation because the Lie bracket is preserved: the action satisfies [(\rho \otimes \sigma)(X), (\rho \otimes \sigma)(Y)] = (\rho \otimes \sigma)([X,Y]), as it follows from the bilinearity and the original representations' properties. are not necessarily irreducible; their decomposition into irreducibles depends on the specific and representations involved. A concrete example arises for the Lie algebra \mathfrak{su}(2) \cong \mathfrak{sl}(2, \mathbb{C}), where irreducible representations are labeled by highest weights j \in \frac{1}{2}\mathbb{N}_0 with dimension $2j+1. The tensor product of two such irreducibles, V^{j_1} \otimes V^{j_2}, decomposes via the Clebsch-Gordan series into a direct sum of irreducibles: V^{j_1} \otimes V^{j_2} \cong \bigoplus_{j = |j_1 - j_2|}^{j_1 + j_2} V^j, where the sum is over integer or half-integer steps matching the labels. This multiplicity-free decomposition is fundamental in applications like angular momentum addition in quantum mechanics and illustrates how tensor products build higher-dimensional representations from lower ones. For instance, the product of two spin-1/2 representations yields a triplet (j=1) plus a singlet (j=0). Regarding intertwining operators, if \tau: V \otimes W \to V \otimes W is a linear map such that [\rho \otimes \sigma, \tau] = 0, meaning \tau (\rho \otimes \sigma)(X) = (\rho \otimes \sigma)(X) \tau for all X \in \mathfrak{g}, then \tau preserves the \mathfrak{g}-module structure and maps invariant subspaces to invariant subspaces. In the decomposed tensor product, such \tau must respect the direct summands, acting as intertwiners between corresponding irreducibles. This property underscores the role of tensor products in studying representation categories and their endomorphism algebras.

Dual Representations

Given a representation \rho: \mathfrak{g} \to \mathfrak{gl}(V) of a \mathfrak{g} on a finite-dimensional V over a of characteristic zero, the (or contragredient representation) \rho^*: \mathfrak{g} \to \mathfrak{gl}(V^*) is defined on the V^* by (\rho^*(X)\phi)(v) = -\phi(\rho(X)v) for all X \in \mathfrak{g}, \phi \in V^*, and v \in V. This action preserves the Lie algebra structure via the pairing equation \langle \rho^*(X)\phi, v \rangle = -\langle \phi, \rho(X)v \rangle, which ensures that \rho^* is indeed a homomorphism. In terms of matrices, if \rho(X) is represented by a matrix A with respect to a basis of V, then \rho^*(X) is represented by -A^T with respect to the dual basis of V^*. The dual representation inherits key structural properties from the original: \rho^* is irreducible if and only if \rho is irreducible. Moreover, for finite-dimensional V, there is a natural V \cong V^{**} identifying the double dual representation (\rho^*)^* with \rho. Contragredient representations are central to highest weight theory for semisimple algebras, where the highest weight of the contragredient to an irreducible highest weight of highest weight \lambda is -w_0(\lambda), with w_0 the longest element of the ; further details appear in the discussion of highest weight representations.

Representations on Homomorphisms

Given representations \rho: \mathfrak{g} \to \mathfrak{gl}(V) and \sigma: \mathfrak{g} \to \mathfrak{gl}(W) of a Lie algebra \mathfrak{g} on finite-dimensional vector spaces V and W over \mathbb{C}, the space \mathrm{Hom}(V, W) of linear maps \phi: V \to W admits a natural structure as a \mathfrak{g}-module. This construction equips \mathrm{Hom}(V, W) with an action \pi: \mathfrak{g} \to \mathfrak{gl}(\mathrm{Hom}(V, W)) that preserves the Lie algebra homomorphism property, making it a representation of \mathfrak{g}. The action is equivalent to the tensor product representation \sigma \otimes \rho^* on the isomorphic space W \otimes V^*, where \rho^* denotes the dual representation on V^*. The explicit formula for the action is given by (\pi(X) \phi)(v) = \sigma(X) (\phi(v)) - \phi(\rho(X) v) for all X \in \mathfrak{g}, \phi \in \mathrm{Hom}(V, W), and v \in V. This definition arises as the infinitesimal counterpart to the induced action on linear maps under a corresponding Lie group representation, ensuring compatibility with the bracket [\cdot, \cdot] in \mathfrak{g}. It transforms \mathrm{Hom}(V, W) into a \mathfrak{g}-module where the operators \pi(X) are derivations with respect to the compositions of linear maps. Within this representation, the of \mathfrak{g}- elements consists of the intertwiners, defined as \mathrm{Hom}_\mathfrak{g}(V, W) = \{ \phi \in \mathrm{Hom}(V, W) \mid \pi(X) \phi = 0 \ \forall X \in \mathfrak{g} \}. These are precisely the linear maps satisfying the equivariance condition \sigma(X) \phi = \phi \rho(X) for all X \in \mathfrak{g}, forming the fixed points under the \mathfrak{g}-action. This captures the \mathfrak{g}-equivariant morphisms between V and W, central to classifying representations up to . For irreducible representations V and W, provides a precise for the intertwiners: \dim \mathrm{Hom}_\mathfrak{g}(V, W) = 1 if V \cong W as \mathfrak{g}-modules (in which case the intertwiners are scalar multiples of the ), and \dim \mathrm{Hom}_\mathfrak{g}(V, W) = 0 otherwise. This result holds over \mathbb{C} and underscores the orthogonality of distinct irreducibles, facilitating multiplicity computations in decompositions.

Induced Representations

In the context of Lie algebra representations, induced representations extend modules from a subalgebra to the full using the universal enveloping algebra, analogous to the induction process for group representations. Given a \mathfrak{g} over a k (typically \mathbb{C} or \mathbb{R}) and a \mathfrak{h} \subset \mathfrak{g}, let W be a representation of \mathfrak{h}, meaning W is a over the universal enveloping algebra U(\mathfrak{h}). The induced representation is then defined as the U(\mathfrak{g})- \Ind_{\mathfrak{h}}^{\mathfrak{g}} W = U(\mathfrak{g}) \otimes_{U(\mathfrak{h})} W, where the tensor product identifies elements of the form u \cdot a \otimes w = u \otimes a \cdot w for u \in U(\mathfrak{g}), a \in U(\mathfrak{h}), and w \in W. The action of \mathfrak{g} on \Ind_{\mathfrak{h}}^{\mathfrak{g}} W is induced from the natural left U(\mathfrak{g})-module structure on U(\mathfrak{g}) itself, extended to the tensor product. Specifically, for X \in \mathfrak{g} and u \otimes w \in U(\mathfrak{g}) \otimes_{U(\mathfrak{h})} W, X \cdot (u \otimes w) = Xu \otimes w. For X \in \mathfrak{h}, this action is consistent with the original \mathfrak{h}-action on W, since Xu \otimes w = uX \otimes w = u \otimes (X \cdot w) in the quotient tensor product. This construction ensures that \Ind_{\mathfrak{h}}^{\mathfrak{g}} W is indeed a representation of \mathfrak{g}. A key property of this induction functor is its adjointness to the restriction functor \Res_{\mathfrak{g}}^{\mathfrak{h}}, which forgets the \mathfrak{g}-action to yield an \mathfrak{h}-module. By Frobenius reciprocity, \Hom_{\mathfrak{g}}(\Ind_{\mathfrak{h}}^{\mathfrak{g}} W, V) \cong \Hom_{\mathfrak{h}}(W, \Res_{\mathfrak{g}}^{\mathfrak{h}} V) for any \mathfrak{g}-module V, where the isomorphism is natural in both arguments. This duality facilitates the study of representation categories and decomposition patterns. Induced representations play a role in broader frameworks, such as constructing modules in category \mathcal{O}.

Semisimple Lie Algebras

Structure Theory Overview

A \mathfrak{g} over an of characteristic zero is defined as a of Lie algebras, or equivalently, as a Lie algebra satisfying [\mathfrak{g}, \mathfrak{g}] = \mathfrak{g} with no nonzero abelian ideals. This structure ensures that \mathfrak{g} has no nontrivial solvable ideals, distinguishing it from solvable or nilpotent algebras. Central to the Cartan-Weyl theory is the existence of a \mathfrak{h}, which is a maximal toral subalgebra—meaning it is abelian and consists entirely of ad-diagonalizable elements—and all are conjugate under the adjoint action of \mathfrak{g}. The root system \Phi of \mathfrak{g} relative to \mathfrak{h} consists of the nonzero linear functionals \alpha \in \mathfrak{h}^* such that the root space \mathfrak{g}_\alpha = \{ X \in \mathfrak{g} \mid [H, X] = \alpha(H) X \ \forall H \in \mathfrak{h} \} is nonzero. These root spaces decompose \mathfrak{g} as \mathfrak{g} = \mathfrak{h} \oplus \bigoplus_{\alpha \in \Phi} \mathfrak{g}_\alpha, providing a basis of eigenvectors for the of \mathfrak{h}. The of \mathfrak{g} decomposes into the trivial representation on \mathfrak{h} and one-dimensional representations on each \mathfrak{g}_\alpha. The Killing form B(X, Y) = \operatorname{Tr}(\operatorname{ad}_X \operatorname{ad}_Y) on \mathfrak{g} is a nondegenerate for semisimple \mathfrak{g}, which restricts to a nondegenerate form on \mathfrak{h} and induces a scalar product on \mathfrak{h}^*. This form plays a key role in defining the W, the generated by the reflections s_\alpha: \mathfrak{h}^* \to \mathfrak{h}^* given by s_\alpha(\lambda) = \lambda - \frac{2 B(\lambda, \alpha)}{B(\alpha, \alpha)} \alpha for \alpha \in \Phi. The acts on the by permuting and preserves the set of positive roots once a choice of positive system is made.

Highest Weight Representations

In the representation theory of complex semisimple Lie algebras, consider a finite-dimensional representation \rho: \mathfrak{g} \to \mathfrak{gl}(V) on a vector space V, where \mathfrak{g} has Cartan subalgebra \mathfrak{h}. The \mathfrak{h}-weight spaces are defined as V_\lambda = \{ v \in V \mid \rho(H)v = \lambda(H) v \ \forall H \in \mathfrak{h} \} for \lambda \in \mathfrak{h}^*, and \lambda is a weight of the representation if V_\lambda \neq 0. The set of weights P(V) \subset \mathfrak{h}^* forms a finite subset, partially ordered by \mu \leq \lambda if \lambda - \mu is a non-negative integer linear combination of positive roots from the root system of \mathfrak{g}. A \lambda is a highest weight of V if V_\lambda \neq 0 and there exists no \mu > \lambda. A nonzero v \in V_\lambda is a highest weight if \rho(X)v = 0 for all X \in \mathfrak{g}_\alpha with \alpha a positive . Such a generates the entire under the action of the universal enveloping algebra U(\mathfrak{g}), and in an irreducible , the highest is unique up to scalar multiple. A highest \lambda is dominant integral if \langle \lambda, \alpha \rangle \in \mathbb{Z}_{\geq 0} for all simple positive \alpha, ensuring it lies in the and the dominant Weyl chamber. The fundamental classification result is the highest weight theorem: every finite-dimensional irreducible representation of \mathfrak{g} admits a unique highest weight \lambda, which is dominant integral, and possesses a unique (up to scalars) highest weight vector v annihilated by the positive root spaces \mathfrak{g}_\alpha for \alpha > 0. Moreover, V is generated as U(\mathfrak{g})v. This parametrizes all such representations uniquely by their highest weights. The Borel subalgebra \mathfrak{b} = \mathfrak{h} \oplus \bigoplus_{\alpha > 0} \mathfrak{g}_\alpha plays a central role, as the highest weight vector v spans a one-dimensional irreducible \mathfrak{b}- on which \mathfrak{h} acts by \lambda and the nilradical \mathfrak{n}^+ = \bigoplus_{\alpha > 0} \mathfrak{g}_\alpha acts trivially. Finite-dimensional irreducible \mathfrak{g}-representations arise as extensions of such \mathfrak{b}-modules to \mathfrak{g}, specifically the unique irreducible quotient of the induced module from \mathfrak{b} to \mathfrak{g}.

Weyl's Theorems

Weyl's theorem on complete reducibility asserts that every finite-dimensional representation of a semisimple Lie algebra over the complex numbers is completely reducible. Specifically, for a semisimple Lie algebra \mathfrak{g} and a finite-dimensional \mathfrak{g}-module V, any \mathfrak{g}-submodule W \subseteq V admits a complementary submodule U such that V = W \oplus U, allowing V to decompose as a direct sum of irreducible representations. This result holds over algebraically closed fields of characteristic zero and underpins the semisimple nature of the representation category for such algebras. The character of a representation provides a key invariant for studying these modules. For a finite-dimensional representation \rho: \mathfrak{g} \to \mathfrak{gl}(V) of a semisimple Lie algebra \mathfrak{g}, the character \chi_V is defined by \chi_V(X) = \operatorname{Tr}(\rho(X)) for X \in \mathfrak{g}, or more precisely in terms of the Cartan subalgebra \mathfrak{h} \subseteq \mathfrak{g}, \chi_V(e^h) = \sum_{\mu} \dim V_\mu e^\mu(h), where the sum is over weights \mu and V = \bigoplus V_\mu is the weight space decomposition. For an irreducible representation L_\lambda with highest weight \lambda, the character is \chi_\lambda(X) = \operatorname{Tr}(\rho_\lambda(X)). Irreducible finite-dimensional representations of semisimple Lie algebras are uniquely determined up to isomorphism by their highest weights \lambda, which are dominant integral weights. The Weyl character formula gives an explicit expression for the character of an irreducible highest weight module. For a dominant weight \lambda, the character is \chi_\lambda = \frac{\sum_{w \in W} \varepsilon(w) e^{w(\lambda + \rho)}}{\sum_{w \in W} \varepsilon(w) e^{w(\rho)}}, where W is the , \varepsilon(w) = (-1)^{\ell(w)} is the sign of w with \ell(w) its , and \rho is the half-sum of the positive . This formula, derived using the denominator \Delta = \sum_{w \in W} \varepsilon(w) e^{w(\rho)} = \prod_{\alpha \in R^+} (e^{\alpha/2} - e^{-\alpha/2}), encodes the weight multiplicities and symmetries of the representation. To compute individual weight multiplicities from the character formula, Kostant's multiplicity formula provides a combinatorial tool. In the irreducible representation L_\lambda with highest weight \lambda, the multiplicity m_\mu^\lambda of a weight \mu is m_\mu^\lambda = \sum_{w \in W} \varepsilon(w) \, P\bigl( w(\lambda + \rho) - (\mu + \rho) \bigr), where P(\nu) is the Kostant partition function, counting the number of ways to write the positive root combination \nu as a non-negative integer sum of positive roots. This formula applies to integral weights \mu in the weight lattice and facilitates explicit calculations for finite-dimensional representations of semisimple Lie algebras.

Enveloping Algebras

Universal Enveloping Algebra

The U(\mathfrak{g}) of a \mathfrak{g} over a k with unity is constructed as the quotient of the T(\mathfrak{g}) by the two-sided ideal I generated by all elements of the form X \otimes Y - Y \otimes X - [X, Y] for X, Y \in \mathfrak{g}. This construction ensures that U(\mathfrak{g}) is an associative unital containing \mathfrak{g} as a Lie subalgebra via the canonical inclusion map i: \mathfrak{g} \hookrightarrow U(\mathfrak{g}), where the Lie bracket in U(\mathfrak{g}) coincides with the commutator induced from the associative product. The U(\mathfrak{g}) satisfies a universal property: for any associative unital A and Lie algebra homomorphism \phi: \mathfrak{g} \to A (where A is viewed as a Lie algebra under the commutator), there exists a unique algebra homomorphism \tilde{\phi}: U(\mathfrak{g}) \to A extending \phi \circ i^{-1}. A fundamental result concerning the structure of U(\mathfrak{g}) is the Poincaré–Birkhoff–Witt (PBW) theorem, which provides an explicit basis when \mathfrak{g} is of finite over k. If \{X_1, \dots, X_r\} is a basis for \mathfrak{g}, then the set of all ordered monomials X_1^{n_1} X_2^{n_2} \cdots X_r^{n_r} with n_i \in \mathbb{N}_0 forms a k-basis for U(\mathfrak{g}). This implies that U(\mathfrak{g}) is over k with equal to the cardinality of the basis monomials, and it establishes a monomial basis independent of the choice of ordered basis for \mathfrak{g}, facilitating computations in the algebra. Representations of \mathfrak{g} are equivalently modules over U(\mathfrak{g}) via the action through the inclusion i. The center Z(U(\mathfrak{g})) = \{ z \in U(\mathfrak{g}) \mid [z, u] = 0 \ \forall u \in U(\mathfrak{g}) \} consists of elements that commute with every generator of U(\mathfrak{g}). For semisimple Lie algebras over fields of characteristic zero, Z(U(\mathfrak{g})) is generated by the operators, which are specific central elements constructed from invariant bilinear forms on \mathfrak{g}. By , these Casimir operators act as scalar multiples of the identity on any of U(\mathfrak{g}). An important algebra homomorphism associated with U(\mathfrak{g}) is the augmentation map \epsilon: U(\mathfrak{g}) \to k, defined uniquely by the requirements that it is an algebra and \epsilon(X) = 0 for all X \in \mathfrak{g}. The of \epsilon, known as the augmentation ideal, is the two-sided ideal generated by the image of \mathfrak{g} in U(\mathfrak{g}).

Representations via U(g)

Lie algebra representations are intimately connected to modules over the universal enveloping algebra U(\mathfrak{g}). Specifically, every representation of a \mathfrak{g} on a V, which is a Lie algebra \mathfrak{g} \to \mathfrak{gl}(V), extends uniquely to a left U(\mathfrak{g})-module structure on V. This extension is defined by the universal property of U(\mathfrak{g}), where elements u \in U(\mathfrak{g}) act on v \in V via the algebra multiplication, preserving the Lie bracket action of \mathfrak{g} on V. Conversely, every U(\mathfrak{g})-module restricts to a \mathfrak{g}-module by the natural embedding \mathfrak{g} \hookrightarrow U(\mathfrak{g}), establishing an equivalence of categories between representations of \mathfrak{g} and left U(\mathfrak{g})-modules. A key feature of this perspective is the role of the center Z(U(\mathfrak{g})) of the enveloping algebra, which consists of elements that commute with all of U(\mathfrak{g}) and thus act by scalars on irreducible representations. The quadratic Casimir operator provides a fundamental example: for a semisimple Lie algebra \mathfrak{g} over \mathbb{C} with nondegenerate invariant bilinear form B, choose dual bases \{e_i\} and \{f_i\} such that B(e_i, f_i) = \delta_{ij}. The Casimir element is then \Omega = \sum_i e_i f_i \in Z(U(\mathfrak{g})). On an irreducible representation of highest weight \lambda, \Omega acts as the scalar c_\lambda I, where c_\lambda = (\lambda, \lambda + 2\rho) and \rho is half the sum of the positive roots. This eigenvalue formula distinguishes irreducible representations and plays a central role in character computations. The structure of Z(U(\mathfrak{g})) is fully elucidated by the Harish-Chandra , which for a \mathfrak{g} with \mathfrak{h} identifies the center with the ring of invariants in the algebra on \mathfrak{h}^*: Z(U(\mathfrak{g})) \cong \mathbb{C}[\mathfrak{h}^*]^W, where W is the acting on \mathfrak{h}^*. This maps central elements to symmetric , providing a description of how the center acts on representations via highest weights. While all representations of \mathfrak{g} yield U(\mathfrak{g})-modules, possibly infinite-dimensional, the finite-dimensional ones are precisely those annihilated by some power of the augmentation ideal \mathfrak{a} of U(\mathfrak{g}), the kernel of the counit map U(\mathfrak{g}) \to \mathbb{C} sending \mathfrak{g} to zero. This condition ensures the module is supported in finite degrees of the associated graded algebra \mathrm{Sym}(\mathfrak{g}), aligning with the finite-dimensionality of the original \mathfrak{g}-representation.

Infinite-Dimensional Representations

Category O

Category O is a fundamental abelian category in the representation theory of complex semisimple Lie algebras, introduced by Bernstein, Gelfand, and Gelfand to study infinite-dimensional modules with controlled growth. For a semisimple Lie algebra \mathfrak{g} with Cartan subalgebra \mathfrak{h} and Borel subalgebra \mathfrak{b} = \mathfrak{h} \oplus \mathfrak{n}, where \mathfrak{n} is the nilpotent radical, the objects of Category O are the finitely generated \mathfrak{g}-modules M that are \mathfrak{h}-semisimple—meaning M decomposes as a direct sum of finite-dimensional weight spaces M = \bigoplus_{\lambda \in \mathfrak{h}^*} M_\lambda—and locally finite-dimensional over \mathfrak{U}(\mathfrak{b}), so that for every v \in M, the \mathfrak{U}(\mathfrak{b})-submodule generated by v is finite-dimensional. This structure ensures that modules in O have weights in a finite union of cosets of the root lattice and exhibit locally nilpotent action by \mathfrak{n}, making O suitable for highest weight theory. The category O decomposes into a of blocks O = \bigoplus_\chi O_\chi, where each block O_\chi consists of modules on which Z(\mathfrak{U}(\mathfrak{g})) of the universal enveloping acts via a fixed central character \chi: Z(\mathfrak{U}(\mathfrak{g})) \to \mathbb{C}. These blocks are indecomposable in the sense that simple modules with different central characters cannot appear together in extensions, and each O_\chi is equivalent to the principal block up to linkage via the Weyl group action on weights. Verma modules, which are indecomposable highest weight modules induced from one-dimensional \mathfrak{b}-modules, belong to Category O and form a key class of projective generators within their blocks. Projective objects in O play a central role in homological algebra, with each block O_\chi possessing enough projectives that resolve the simple highest weight modules L(\lambda). The Bernstein–Gelfand–Gelfand (BGG) resolution provides an explicit projective resolution for every simple module L(\lambda) in O, constructed as a complex of projective modules P(\mu) indexed by Weyl group elements, yielding \Ext^i_O(L(\lambda), L(\nu)) \cong \mathbb{C} for i=0 and specific \nu = w \cdot \lambda, and zero otherwise. This resolution, built using translation functors and Verma module embeddings, enables computations of extension groups and characters, underpinning much of the homological structure of O.

Verma Modules

Verma modules form a central class of infinite-dimensional highest weight representations for semisimple Lie algebras, playing the role of universal indecomposable objects within category O. They were introduced by D. N. Verma to study the structure of representations via induction from Borel subalgebras. Given a complex semisimple Lie algebra \mathfrak{g} with Cartan subalgebra \mathfrak{h}, fixed Borel subalgebra \mathfrak{b} = \mathfrak{h} \oplus \mathfrak{n}, and \lambda \in \mathfrak{h}^*, the Verma module M_\lambda is constructed as the induced module M_\lambda = U(\mathfrak{g}) \otimes_{U(\mathfrak{b})} k_\lambda, where k_\lambda denotes the one-dimensional \mathfrak{b}-module on which \mathfrak{h} acts via the character \lambda (i.e., h \cdot 1 = \lambda(h) \cdot 1 for h \in \mathfrak{h}) and \mathfrak{n} acts trivially (i.e., n \cdot 1 = 0 for n \in \mathfrak{n}). This yields a left U(\mathfrak{g})-module freely generated over U(\mathfrak{n}^-) by a highest weight vector v_\lambda = 1 \otimes 1 of weight \lambda, annihilated by \mathfrak{n}, with weights of M_\lambda given by \lambda - \mathbb{N} \Delta^+ (where \Delta^+ is the set of positive roots). The Verma module M_\lambda satisfies a universal property: it is the initial object among all highest weight \mathfrak{g}-modules of highest weight \lambda. Specifically, for any such module V with highest weight vector v \in V, there exists a unique \mathfrak{g}-module homomorphism \phi: M_\lambda \to V such that \phi(v_\lambda) = v. This follows directly from the universal property of induced modules and the definition of highest weight modules. In general, M_\lambda is indecomposable but not necessarily irreducible; it possesses a unique maximal proper submodule, often denoted \mathrm{rad} M_\lambda, whose quotient is the simple highest weight module L_\lambda. When \lambda(h_\alpha) is a non-negative integer for each simple root \alpha (with h_\alpha \in \mathfrak{h} the coroot), this maximal submodule is generated by the vectors f_\alpha^{\lambda(h_\alpha) + 1} v_\lambda over the simple roots \alpha, where f_\alpha is a root vector lowering the weight by \alpha. For more general \lambda, the structure is determined by embeddings of other Verma modules, with \mathrm{rad} M_\lambda generated by singular vectors arising from these embeddings. A key structural result is the linkage principle, which asserts that the highest weights of all irreducible subquotients (composition factors) of M_\lambda lie within the dotted orbit W \cdot \lambda = \{ w(\lambda + \rho) - \rho \mid w \in W \}, where W is the and \rho is the half-sum of the positive roots. This confines the in each block of category O to weights linked by the action.

Harish-Chandra Modules

In the context of representations of real semisimple Lie algebras, consider a connected real semisimple G with finite center and \mathfrak{g}, together with a maximal compact K \subset G and its \mathfrak{k}. A (\mathfrak{g}, K)-module is defined as a complex vector space V equipped with compatible actions of \mathfrak{g} (via a ) and of K (via a representation), such that V is \mathfrak{h}-semisimple for a \mathfrak{h} of \mathfrak{g}, the action of K on V is locally finite (every vector generates a finite-dimensional K-), and V is finitely generated as a over the universal enveloping U(\mathfrak{g}). Harish-Chandra modules form a fundamental subcategory of (\mathfrak{g}, K)-modules, characterized by admissibility: each irreducible representation of K (known as a K-type) appears with finite multiplicity in V. This finite multiplicity condition ensures that the K-isotypic components of V are finite-dimensional, distinguishing Harish-Chandra modules from more general infinite-dimensional representations. The category of Harish-Chandra modules is abelian and artinian, with irreducible objects corresponding to the algebraic components of irreducible smooth representations of G. A result is Harish-Chandra's admissibility , which states that for any irreducible unitary \pi of G on a , the subspace of K-finite vectors (vectors generating finite-dimensional K-subrepresentations) carries the structure of an admissible (\mathfrak{g}, K)-module, meaning it is a Harish-Chandra module. This reduces the study of infinite-dimensional unitary representations of G to algebraic questions about their underlying Harish-Chandra modules, facilitating classification via infinitesimal characters and support varieties. To construct explicit examples of modules, Zuckerman functors provide derived functors of the process from representations of K (or more generally, from Levi subgroups) to (\mathfrak{g}, K)-modules. Specifically, for a finite-dimensional representation W of K, the zeroth Zuckerman functor applies the algebraic U(\mathfrak{g}) \otimes_{U(\mathfrak{k})} W, and higher derived functors R^j \Gamma_{\mathfrak{g}, K} yield projective resolutions in the category of modules, often producing irreducible quotients under suitable cohomological vanishing conditions. These functors, initially developed for computing relative , enable the cohomological of representations and link algebraic modules to geometric constructions like varieties. Harish-Chandra modules play a pivotal role in the theory of automorphic forms on G, where the space of K-finite automorphic forms on G / \Gamma (for a discrete subgroup \Gamma) inherits a natural (\mathfrak{g}, K)-module structure, with irreducibles corresponding to cuspidal automorphic representations via the Langlands correspondence. In the context of Beilinson-Bernstein localization for real groups, every Harish-Chandra module embeds as the global sections of a coherent sheaf of twisted differential operators on the flag variety G/P, providing a geometric realization that parallels the complex case but accounts for the non-compact structure through compactly supported cohomology.

(g,K)-Modules

Definition and Properties

A (g, K)-module, where g is the Lie algebra of a real reductive Lie group G and K is a maximal compact subgroup with Lie algebra k, is a complex vector space V that carries both a representation \pi_g: g \to \mathrm{End}(V) and a continuous representation \pi_K: K \to \mathrm{GL}(V) satisfying the following compatibility conditions: V decomposes algebraically as a direct sum V = \bigoplus_{\sigma} V_\sigma over all irreducible finite-dimensional representations \sigma of K, where each V_\sigma is the \sigma-isotypic component; the infinitesimal action of k via \pi_g|_{k} coincides with the differential of \pi_K; and for all X \in g, k \in K, v \in V, \pi_K(k) \pi_g(X) \pi_K(k^{-1}) v = \pi_g(\mathrm{Ad}(k) X) v. The irreducible constituents \sigma of the K-action on V are termed K-types, and the multiplicity m_\sigma = \dim \mathrm{Hom}_K(\sigma, V) measures their occurrence in V. Since K is compact, the possible K-types correspond to discrete elements in the weight lattice of k, ensuring that the weights of V form a set. A (g, K)-module V is admissible if every K-type appears with finite multiplicity, i.e., m_\sigma < \infty for all \sigma, which implies that V is a of finitely many copies of each \sigma in its decomposition. The center Z(\mathfrak{U}(\mathfrak{g})) of the universal enveloping algebra \mathfrak{U}(\mathfrak{g}) acts on V by commuting endomorphisms; in an irreducible admissible (g, K)-module, this action is scalar, given by a central character \chi: Z(\mathfrak{U}(\mathfrak{g})) \to \mathbb{C}. By the Harish-Chandra isomorphism Z(\mathfrak{U}(\mathfrak{g})) \cong \mathrm{Sym}(\mathfrak{h}^*)^W, where \mathfrak{h} is a Cartan subalgebra of \mathfrak{g} and W is the Weyl group, the central character corresponds to an infinitesimal character, which is a W-orbit in \mathfrak{h}^*. Admissible (g, K)-modules arise as the K-finite vectors in admissible unitary representations of G, and such representations decompose as direct integrals into irreducible unitary components, each of whose K-finite part is an irreducible admissible (g, K)-module appearing with finite multiplicity.

Applications to Symmetric Spaces

Symmetric spaces of the form G/K, where G is a semisimple Lie group and K is a maximal compact subgroup, provide a geometric setting for studying representations via (\mathfrak{g}, K)-modules. The Hilbert space L^2(G/K) of square-integrable functions on the symmetric space decomposes as a direct integral of irreducible unitary (\mathfrak{g}, K)-modules, reflecting the unitary dual of G restricted to the geometry of G/K. This decomposition arises from the Peter-Weyl theorem applied to the compact group K and extends to the non-compact case through Harish-Chandra's theory, where the K-finite vectors in L^2(G/K) are dense and form an admissible (\mathfrak{g}, K)-module. A key family of representations appearing in this decomposition is the principal series, constructed as induced representations from a minimal parabolic P of G. These modules are realized on sections of line bundles over the flag variety G/B, where B is a containing P, and they capture the continuous spectrum of L^2(G/K). For Hermitian symmetric spaces, where G/K admits a G-invariant complex structure, discrete series representations also contribute to the discrete spectrum; these are irreducible unitary (\mathfrak{g}, K)-modules with square-integrable matrix coefficients, existing for groups G satisfying the condition that the real rank equals the compact rank, i.e., \mathrm{rank}(G) = \mathrm{rank}(K), such as when G has a compact Cartan subgroup. Examples include the discrete series of \mathrm{SL}(2, \mathbb{R}) on the hyperbolic plane \mathbb{H}^2 \cong \mathrm{SL}(2, \mathbb{R})/\mathrm{SO}(2). Knapp-Stein intertwining operators play a crucial role in relating different principal series modules, particularly by intertwining the left action of G with actions induced from parabolics \overline{P}. These operators, defined via meromorphic continuation of integrals over unipotent subgroups, establish isomorphisms between principal series at complementary parameters (e.g., \pi_\nu \cong \pi_{-\overline{\nu}}) and facilitate the unitarization process essential for the on G/K. In the symmetric space setting, analogs of these operators extend to spherical principal series for pairs (G, H), enabling the study of and branching rules. Beyond classical real groups, (\mathfrak{g}, K)-modules find modern applications in the representation theory of p-adic groups, where analogous constructions decompose spaces of automorphic forms on adelic quotients. The principal series and intertwining operators inform the local Langlands correspondence, parametrizing irreducible representations of p-adic reductive groups via tempered (\mathfrak{g}, K)-modules, with direct implications for global automorphic representations and the trace formula. This framework bridges on symmetric spaces to number-theoretic problems, such as functoriality in the .

Representations on Algebras

General Setup

In the context of Lie algebra representations, a representation on an associative algebra A over a k equips A with a \mathfrak{g}-module structure that is compatible with its multiplication. Specifically, this is a Lie algebra homomorphism \rho: \mathfrak{g} \to \Der_k(A), where \Der_k(A) denotes the Lie algebra of k-linear derivations of A, consisting of k-linear endomorphisms D: A \to A satisfying the Leibniz rule D(ab) = D(a)b + a D(b) for all a, b \in A, with the Lie bracket given by the commutator [D, D'] = D \circ D' - D' \circ D'. Equivalently, the action \cdot: \mathfrak{g} \times A \to A satisfies X \cdot (ab) = (X \cdot a)b + a(X \cdot b) for all X \in \mathfrak{g} and a, b \in A, making A a \mathfrak{g}-module algebra. The action by derivations ensures that the representation respects the associative structure of A, allowing \mathfrak{g} to "differentiate" elements while preserving products, analogous to infinitesimal symmetries in differential geometry. This compatibility extends naturally to the universal enveloping algebra U(\mathfrak{g}), where A becomes a left U(\mathfrak{g})-module algebra via the unique algebra homomorphism induced by \rho. A fundamental example is the adjoint action of \mathfrak{g} on its universal enveloping algebra U(\mathfrak{g}), where elements of \mathfrak{g} act as inner derivations: for X \in \mathfrak{g} and u \in U(\mathfrak{g}), X \cdot u = [X, u], satisfying the Leibniz rule due to the defining relations in U(\mathfrak{g}). Another example arises in algebraic geometry, where the Lie algebra \mathfrak{g} of an algebraic group G acts by derivations on the coordinate ring k[V] of an affine variety V on which G acts rationally; for X \in \mathfrak{g} and f \in k[V], the action is X \cdot f = \frac{d}{dt}\big|_{t=0} (f \circ \exp(tX)), yielding a \mathfrak{g}-module algebra structure. The universal enveloping algebra U(\mathfrak{g}) itself embodies a universal property for such representations: given any associative algebra A and a Lie algebra homomorphism \phi: \mathfrak{g} \to \Der_k(A), there exists a unique algebra homomorphism \psi: U(\mathfrak{g}) \to \End_k(A) such that \psi|_{\mathfrak{g}} = \phi, thereby extending the action to an U(\mathfrak{g})-module algebra structure on A. This property underscores U(\mathfrak{g}) as the "free" associative algebra incorporating the Lie structure of \mathfrak{g} via derivations.

Examples in Deformation Theory

In deformation theory, representations of a \mathfrak{g} on polynomial algebras, such as the \mathrm{Sym}(\mathfrak{g}^*), undergo deformation to modules over the quantum enveloping algebra U_h(\mathfrak{g}), where the parameter h (or q = e^h) governs the deformation, and the at h=0 recovers the original Lie algebra representation. This framework, introduced by Drinfeld and , allows for the quantization of classical structures on algebraic varieties associated with \mathfrak{g}, preserving key representation-theoretic properties like highest weight modules while introducing q-deformations that alter commutation relations. Such deformations are central to understanding quantum symmetries in integrable systems and have been extensively studied for finite-dimensional semisimple \mathfrak{g}. For Poisson-Lie groups, the \mathfrak{[g](/page/G)} of the group G carries a Lie bialgebra structure (\mathfrak{[g](/page/G)}, \delta), where the cobracket \delta: \mathfrak{[g](/page/G)} \to \wedge^2 \mathfrak{[g](/page/G)} induces a Lie bracket on the dual \mathfrak{[g](/page/G)}^*, equipping \mathfrak{[g](/page/G)}^* with a Poisson structure compatible with the group structure on the dual Poisson-Lie group. Representations of \mathfrak{[g](/page/G)} on this dual Poisson structure arise naturally via the coadjoint action, deforming classical Poisson manifolds into quantum analogs and facilitating the study of dressing transformations and Bruhat decompositions in the deformed setting. Drinfeld's seminal construction highlights how these representations encode Hamiltonian dynamics on groups, with the dual structure providing a canonical example of \mathfrak{[g](/page/G)}-action on a deformed Poisson algebra. Computational aspects of these deformations benefit from crystal bases, which offer a combinatorial model for the irreducible representations of quantum groups U_q(\mathfrak{g}), deforming the classical weight lattices and branching rules of \mathfrak{g}-representations as q \to 1. Developed by Kashiwara, these bases enable explicit calculations of tensor product decompositions and highest weights in the deformed category, bridging with models like the Heisenberg chain. For affine \mathfrak{g}, crystal bases extend to perfect crystals, facilitating algorithmic verification of deformation consistency in infinite-dimensional contexts. The Drinfeld-Sokolov reduction provides a key example of infinite-dimensional representations arising from affine Lie algebras \hat{\mathfrak{g}} at the critical level, mapping modules over \hat{\mathfrak{g}} to representations on W-algebras via Hamiltonian reduction. This process, originally formulated for constructing extended Virasoro algebras, yields vertex operator algebras where the W-algebra acts as a central extension, with representations classified by minimal series modules deforming classical Feigin-Frenkel isomorphisms. In the quantum setting, the reduction preserves conformal weights and fusion rules, linking affine representations to chiral algebras in two-dimensional conformal field theory.