🔁Lie Algebras and Lie Groups Unit 6 – Semisimple Lie Algebra Representations

Key Concepts and Definitions

• Lie algebra $L$ vector space over field $F$ with bilinear operation $[,]: L \times L \to L$ called the Lie bracket satisfying antisymmetry and Jacobi identity
• Lie group $G$ smooth manifold with group structure where multiplication and inversion are smooth maps
• Representation of Lie algebra $L$ linear map $\rho: L \to \mathfrak{gl}(V)$ preserving the Lie bracket structure ($\rho([x,y]) = [\rho(x),\rho(y)]$)
  • Representation of Lie group $G$ smooth homomorphism $\phi: G \to GL(V)$ where $V$ is a vector space
• Semisimple Lie algebra direct sum of simple Lie algebras (non-abelian with no non-trivial ideals)
• Irreducible representation has no proper non-trivial subrepresentations
• Weights elements of dual space of Cartan subalgebra $\mathfrak{h}^*$ occurring as eigenvalues in representations
  • Highest weight unique weight with maximal eigenvalue

Historical Context and Importance

• Sophus Lie (1842-1899) introduced concepts in late 19th century to study symmetries of differential equations
• Élie Cartan further developed theory in early 20th century, classifying simple Lie algebras and studying their representations
• Hermann Weyl connected Lie theory with quantum mechanics in 1920s, using representations to describe particle states
• Representation theory of semisimple Lie algebras plays central role in modern theoretical physics (gauge theories, string theory)
• Classification of simple Lie algebras over $\mathbb{C}$ completed by Killing and Cartan
  • Four infinite families ($A_n, B_n, C_n, D_n$) and five exceptional cases ($G_2, F_4, E_6, E_7, E_8$)
• Representations used to construct and analyze symmetries in various branches of mathematics (harmonic analysis, algebraic geometry, number theory)

Fundamental Structures

• Cartan subalgebra $\mathfrak{h}$ maximal abelian subalgebra of $L$ consisting of semisimple elements (diagonalizable in some faithful representation)
• Root system $\Phi \subset \mathfrak{h}^*$ set of non-zero weights occurring in adjoint representation of $L$
  • Simple roots linearly independent subset of $\Phi$ spanning the root space with each root expressible as linear combination with integer coefficients of the same sign
• Weyl group $W$ finite group generated by reflections through hyperplanes orthogonal to roots, acting on $\mathfrak{h}^*$
• Universal enveloping algebra $U(L)$ associative algebra containing $L$ as Lie subalgebra, with universal mapping property for Lie algebra homomorphisms into associative algebras
• Casimir elements central elements of $U(L)$ acting as scalars on irreducible representations (used to classify representations)

Classification and Types

• Lie algebras classified by root systems and Dynkin diagrams (graphs encoding simple roots and their angles)
  • Type $A_n$ corresponds to special linear Lie algebra $\mathfrak{sl}_{n+1}(\mathbb{C})$
  • Type $B_n$ corresponds to odd orthogonal Lie algebra $\mathfrak{so}_{2n+1}(\mathbb{C})$
  • Type $C_n$ corresponds to symplectic Lie algebra $\mathfrak{sp}_{2n}(\mathbb{C})$
  • Type $D_n$ corresponds to even orthogonal Lie algebra $\mathfrak{so}_{2n}(\mathbb{C})$
• Exceptional Lie algebras ($G_2, F_4, E_6, E_7, E_8$) have more intricate structures and representations
• Real forms of complex Lie algebras obtained by considering compatible anti-linear involutions (e.g., $\mathfrak{su}(n)$ compact real form of $\mathfrak{sl}_n(\mathbb{C})$)
• Affine Lie algebras infinite-dimensional extensions of semisimple Lie algebras, central in conformal field theory and integrable systems

Representation Theory Basics

• Representations of Lie algebras and Lie groups encode symmetries and provide tools for studying their structure
• Irreducible representations building blocks of representation theory, cannot be decomposed into non-trivial subrepresentations
• Characters $\chi_V(g) = \text{tr}(\phi(g))$ for $g \in G$ and representation $\phi: G \to GL(V)$, contain essential information about the representation
  • Character formula expresses characters in terms of Weyl group and highest weights
• Tensor products of representations correspond to combining symmetries, decompose into direct sum of irreducible representations (Clebsch-Gordan coefficients)
• Schur's lemma states that morphisms between irreducible representations are either zero or isomorphisms
• Weyl's character formula expresses characters of irreducible representations in terms of Weyl group and highest weights

Semisimple Lie Algebras

• Semisimple Lie algebras direct sum of simple Lie algebras, classified by Dynkin diagrams
• Killing form $B(x,y) = \text{tr}(\text{ad}_x \circ \text{ad}_y)$ non-degenerate symmetric bilinear form on semisimple Lie algebras, used to define root systems
• Weyl-Kac character formula generalizes Weyl character formula to affine Lie algebras
• Representations of semisimple Lie algebras uniquely determined by highest weight (finite-dimensional, complex)
  • Highest weight representations constructed using Verma modules and unique irreducible quotients
• Weyl's dimension formula expresses dimensions of irreducible representations in terms of highest weights and Weyl group
• Tensor product decomposition rules (e.g., Littlewood-Richardson rule for $\mathfrak{sl}_n(\mathbb{C})$) describe the decomposition of tensor products of irreducible representations

Representation Techniques

• Verma modules universal highest weight modules, generated by highest weight vector with relations determined by root system and highest weight
  • Unique irreducible quotient is the corresponding highest weight representation
• Borel-Weil theorem realizes irreducible representations as spaces of sections of line bundles on flag varieties (homogeneous spaces for Lie groups)
• Peter-Weyl theorem decomposes regular representation of compact Lie group into direct sum of irreducible representations with multiplicities equal to their dimensions
• Schur-Weyl duality relates representations of $\mathfrak{sl}_n(\mathbb{C})$ and symmetric group $S_n$, providing combinatorial tools for studying representations
• Crystal bases combinatorial objects encoding structure of representations, useful for studying tensor product decompositions and character formulas
• Kazhdan-Lusztig theory studies representations of Lie algebras and quantum groups using geometry of Schubert varieties and intersection cohomology

Applications and Examples

• Representation theory of $\mathfrak{sl}_2(\mathbb{C})$ foundational example, irreducible representations labeled by non-negative integers (dimensions)
  • Representations of $\mathfrak{sl}_2(\mathbb{C})$ used in quantum mechanics for describing spin states
• Adjoint representation of Lie algebra $L$ acts on itself via Lie bracket, decomposes into irreducible representations corresponding to root spaces
• Fundamental representations of $\mathfrak{sl}_n(\mathbb{C})$ exterior powers of the standard representation, play important role in invariant theory and algebraic geometry
• Representations of Lorentz and Poincaré groups describe particle states in relativistic quantum field theory
  • Spinor representations correspond to fermions, tensor representations to bosons
• Virasoro algebra central extension of Witt algebra (vector fields on circle), plays fundamental role in conformal field theory and string theory
  • Representations of Virasoro algebra describe symmetries and states in these theories
• Representations of Lie groups and Lie algebras used in geometric quantization to construct quantum systems from classical systems with symmetries