🔁Lie Algebras and Lie Groups Unit 3 – Lie Algebra Representations

Key Concepts and Definitions

• Lie algebra $\mathfrak{g}$ is a vector space over a field $\mathbb{F}$ equipped with a bilinear operation called the Lie bracket $[\cdot,\cdot]: \mathfrak{g} \times \mathfrak{g} \rightarrow \mathfrak{g}$ satisfying antisymmetry and the Jacobi identity
  • Antisymmetry: $[x,y] = -[y,x]$ for all $x,y \in \mathfrak{g}$
  • Jacobi identity: $[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0$ for all $x,y,z \in \mathfrak{g}$
• Lie group $G$ is a smooth manifold that is also a group, where the group operations (multiplication and inversion) are smooth maps
• Representation of a Lie algebra $\mathfrak{g}$ on a vector space $V$ is a Lie algebra homomorphism $\rho: \mathfrak{g} \rightarrow \mathfrak{gl}(V)$, where $\mathfrak{gl}(V)$ is the Lie algebra of endomorphisms of $V$
• Adjoint representation $\text{ad}: \mathfrak{g} \rightarrow \mathfrak{gl}(\mathfrak{g})$ is a representation of a Lie algebra $\mathfrak{g}$ on itself defined by $\text{ad}_x(y) = [x,y]$ for all $x,y \in \mathfrak{g}$
• Irreducible representation is a representation that has no proper, non-trivial subrepresentations
• Weight of a representation $\rho: \mathfrak{g} \rightarrow \mathfrak{gl}(V)$ is a linear functional $\lambda \in \mathfrak{h}^*$ such that there exists a non-zero vector $v \in V$ with $\rho(h)(v) = \lambda(h)v$ for all $h \in \mathfrak{h}$, where $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$
• Character of a representation $\rho: \mathfrak{g} \rightarrow \mathfrak{gl}(V)$ is the trace function $\chi_\rho: \mathfrak{g} \rightarrow \mathbb{F}$ defined by $\chi_\rho(x) = \text{tr}(\rho(x))$ for all $x \in \mathfrak{g}$

Fundamental Theorems

• Lie's Theorem establishes a correspondence between Lie groups and Lie algebras, stating that every finite-dimensional Lie algebra over $\mathbb{R}$ or $\mathbb{C}$ is the Lie algebra of a unique connected, simply connected Lie group
• Ado's Theorem states that every finite-dimensional Lie algebra over a field of characteristic zero has a faithful, finite-dimensional representation
  • Consequently, every finite-dimensional Lie algebra can be viewed as a matrix Lie algebra
• Weyl's Theorem on complete reducibility asserts that every finite-dimensional representation of a semisimple Lie algebra is completely reducible (can be decomposed into a direct sum of irreducible representations)
• Schur's Lemma describes the structure of homomorphisms between irreducible representations
  • If $\rho_1: \mathfrak{g} \rightarrow \mathfrak{gl}(V_1)$ and $\rho_2: \mathfrak{g} \rightarrow \mathfrak{gl}(V_2)$ are irreducible representations, then any homomorphism $\phi: V_1 \rightarrow V_2$ is either zero or an isomorphism
• Weyl Character Formula expresses the character of an irreducible representation of a semisimple Lie algebra in terms of its highest weight
• Tensor Product Decomposition Theorem states that the tensor product of two irreducible representations of a semisimple Lie algebra can be decomposed into a direct sum of irreducible representations
• Weyl's Dimension Formula gives the dimension of an irreducible representation of a semisimple Lie algebra in terms of its highest weight

Representation Theory Basics

• Representation theory studies abstract algebraic structures (Lie algebras) by representing their elements as linear transformations of vector spaces, providing a concrete way to analyze their properties
• Homomorphism between representations $\rho_1: \mathfrak{g} \rightarrow \mathfrak{gl}(V_1)$ and $\rho_2: \mathfrak{g} \rightarrow \mathfrak{gl}(V_2)$ is a linear map $\phi: V_1 \rightarrow V_2$ such that $\phi(\rho_1(x)(v)) = \rho_2(x)(\phi(v))$ for all $x \in \mathfrak{g}$ and $v \in V_1$
• Isomorphic representations are representations that are related by an invertible homomorphism
• Subrepresentation of a representation $\rho: \mathfrak{g} \rightarrow \mathfrak{gl}(V)$ is a subspace $W \subseteq V$ that is invariant under the action of $\rho$, i.e., $\rho(x)(w) \in W$ for all $x \in \mathfrak{g}$ and $w \in W$
• Direct sum of representations $\rho_1: \mathfrak{g} \rightarrow \mathfrak{gl}(V_1)$ and $\rho_2: \mathfrak{g} \rightarrow \mathfrak{gl}(V_2)$ is a representation $\rho: \mathfrak{g} \rightarrow \mathfrak{gl}(V_1 \oplus V_2)$ defined by $\rho(x)(v_1, v_2) = (\rho_1(x)(v_1), \rho_2(x)(v_2))$ for all $x \in \mathfrak{g}$, $v_1 \in V_1$, and $v_2 \in V_2$
• Tensor product of representations $\rho_1: \mathfrak{g} \rightarrow \mathfrak{gl}(V_1)$ and $\rho_2: \mathfrak{g} \rightarrow \mathfrak{gl}(V_2)$ is a representation $\rho: \mathfrak{g} \rightarrow \mathfrak{gl}(V_1 \otimes V_2)$ defined by $\rho(x)(v_1 \otimes v_2) = \rho_1(x)(v_1) \otimes v_2 + v_1 \otimes \rho_2(x)(v_2)$ for all $x \in \mathfrak{g}$, $v_1 \in V_1$, and $v_2 \in V_2$
• Dual representation of a representation $\rho: \mathfrak{g} \rightarrow \mathfrak{gl}(V)$ is a representation $\rho^*: \mathfrak{g} \rightarrow \mathfrak{gl}(V^*)$ defined by $(\rho^*(x)(f))(v) = -f(\rho(x)(v))$ for all $x \in \mathfrak{g}$, $f \in V^*$, and $v \in V$

Types of Representations

• Trivial representation is a representation $\rho: \mathfrak{g} \rightarrow \mathfrak{gl}(V)$ where $\rho(x) = 0$ for all $x \in \mathfrak{g}$
• Faithful representation is a representation $\rho: \mathfrak{g} \rightarrow \mathfrak{gl}(V)$ that is injective, i.e., $\rho(x) = 0$ implies $x = 0$ for all $x \in \mathfrak{g}$
• Irreducible representation is a representation that has no proper, non-trivial subrepresentations
  • Every non-zero vector in an irreducible representation generates the entire representation space under the action of the Lie algebra
• Completely reducible representation (semisimple representation) is a representation that can be decomposed into a direct sum of irreducible representations
• Indecomposable representation is a representation that cannot be expressed as a direct sum of proper subrepresentations
• Unitary representation is a representation $\rho: \mathfrak{g} \rightarrow \mathfrak{gl}(V)$ on a Hilbert space $V$ that preserves the inner product, i.e., $\langle \rho(x)(v_1), v_2 \rangle + \langle v_1, \rho(x)(v_2) \rangle = 0$ for all $x \in \mathfrak{g}$ and $v_1, v_2 \in V$
• Highest weight representation is an irreducible representation that is uniquely determined by its highest weight (a dominant integral weight)
  • Highest weight representations play a crucial role in the classification of irreducible representations of semisimple Lie algebras

Constructing and Analyzing Representations

• Induced representations are constructed from representations of subalgebras using a process called induction
  • Let $\mathfrak{h} \subseteq \mathfrak{g}$ be a Lie subalgebra and $\rho: \mathfrak{h} \rightarrow \mathfrak{gl}(V)$ a representation of $\mathfrak{h}$. The induced representation $\text{Ind}_\mathfrak{h}^\mathfrak{g}(\rho)$ is a representation of $\mathfrak{g}$ on a larger vector space constructed from $V$
• Restricted representations are obtained by restricting a representation of a Lie algebra to a subalgebra
  • If $\rho: \mathfrak{g} \rightarrow \mathfrak{gl}(V)$ is a representation of $\mathfrak{g}$ and $\mathfrak{h} \subseteq \mathfrak{g}$ is a Lie subalgebra, the restricted representation $\rho|_\mathfrak{h}: \mathfrak{h} \rightarrow \mathfrak{gl}(V)$ is a representation of $\mathfrak{h}$
• Decomposing representations into irreducible components is a fundamental problem in representation theory
  • For semisimple Lie algebras, Weyl's Theorem on complete reducibility guarantees that every finite-dimensional representation can be decomposed into a direct sum of irreducible representations
• Characters of representations provide a powerful tool for analyzing representations and their decompositions
  • The character of a representation $\rho: \mathfrak{g} \rightarrow \mathfrak{gl}(V)$ is the trace function $\chi_\rho: \mathfrak{g} \rightarrow \mathbb{F}$ defined by $\chi_\rho(x) = \text{tr}(\rho(x))$ for all $x \in \mathfrak{g}$
  • Characters are invariant under conjugation and satisfy certain orthogonality relations, making them useful for distinguishing non-isomorphic irreducible representations
• Weight space decomposition is a way to decompose a representation into eigenspaces (weight spaces) of a Cartan subalgebra
  • For a representation $\rho: \mathfrak{g} \rightarrow \mathfrak{gl}(V)$ and a Cartan subalgebra $\mathfrak{h} \subseteq \mathfrak{g}$, the weight space $V_\lambda$ corresponding to a weight $\lambda \in \mathfrak{h}^*$ is the subspace $V_\lambda = \{v \in V \mid \rho(h)(v) = \lambda(h)v \text{ for all } h \in \mathfrak{h}\}$
  • The representation space $V$ decomposes into a direct sum of weight spaces: $V = \bigoplus_{\lambda \in \mathfrak{h}^*} V_\lambda$

Applications to Lie Groups

• Representations of Lie groups are closely related to representations of their associated Lie algebras
  • Every representation of a Lie group $G$ induces a representation of its Lie algebra $\mathfrak{g}$ by differentiation
  • Conversely, under certain conditions (e.g., $G$ is simply connected), representations of the Lie algebra $\mathfrak{g}$ can be integrated to obtain representations of the Lie group $G$
• Adjoint representation of a Lie group $G$ is a representation $\text{Ad}: G \rightarrow \text{GL}(\mathfrak{g})$ defined by $\text{Ad}_g = d(C_g)_e$, where $C_g: G \rightarrow G$ is the conjugation map $C_g(h) = ghg^{-1}$ and $e$ is the identity element of $G$
  • The adjoint representation of a Lie group is related to the adjoint representation of its Lie algebra by $\text{Ad}_{\exp(x)} = \exp(\text{ad}_x)$ for all $x \in \mathfrak{g}$
• Representations of Lie groups are used to study the structure and properties of the groups themselves
  • Irreducible representations of compact Lie groups are finite-dimensional and unitary, and they can be classified using highest weight theory
  • The Peter-Weyl Theorem states that the matrix coefficients of irreducible unitary representations form an orthonormal basis for the space of square-integrable functions on a compact Lie group
• Representation theory plays a crucial role in the study of homogeneous spaces and equivariant vector bundles
  • A homogeneous space is a manifold $M$ on which a Lie group $G$ acts transitively, i.e., for any $x,y \in M$, there exists $g \in G$ such that $g \cdot x = y$
  • An equivariant vector bundle is a vector bundle $\pi: E \rightarrow M$ over a homogeneous space $M$ with a $G$-action on $E$ that is compatible with the $G$-action on $M$ and the bundle projection $\pi$
  • Representations of $G$ can be used to construct and classify equivariant vector bundles over homogeneous spaces

Computational Techniques

• Weight multiplicities can be computed using Freudenthal's recursion formula or the Kostant multiplicity formula
  • Freudenthal's recursion formula expresses the multiplicities of weights in an irreducible representation in terms of the multiplicities of higher weights
  • The Kostant multiplicity formula gives the multiplicity of a weight in an irreducible representation as the number of ways the weight can be expressed as a sum of positive roots
• Weyl Character Formula provides a method to compute the character of an irreducible representation of a semisimple Lie algebra
  • The formula expresses the character as an alternating sum over the Weyl group, involving the highest weight and the Weyl denominator
• Tensor product decompositions can be computed using the Littlewood-Richardson rule or the Klimyk formula
  • The Littlewood-Richardson rule is a combinatorial algorithm for decomposing the tensor product of two irreducible representations of the special linear Lie algebra $\mathfrak