🧩Representation Theory Unit 11 – Representations of Lie Algebras

Key Concepts and Definitions

• Lie algebras are vector spaces equipped with a bilinear operation called the Lie bracket, denoted as $[x, y]$, satisfying anticommutativity and the Jacobi identity
• Representations of Lie algebras map elements of the algebra to linear transformations on a vector space while preserving the Lie bracket structure
• The dimension of a representation refers to the dimension of the vector space on which the Lie algebra acts
• Irreducible representations cannot be decomposed into smaller subrepresentations and serve as building blocks for constructing more complex representations
• The adjoint representation is a special representation where the Lie algebra acts on itself via the Lie bracket operation
  • It plays a crucial role in understanding the structure and properties of the Lie algebra
• Weights are eigenvalues of the Cartan subalgebra elements in a representation and provide a way to classify and study representations
• The highest weight of a representation is the weight that is dominant with respect to a chosen ordering and uniquely determines the irreducible representation

Fundamentals of Lie Algebras

• Lie algebras are characterized by their dimension, which is the dimension of the underlying vector space, and their structure constants, which determine the Lie bracket operation
• Simple Lie algebras are non-abelian Lie algebras that have no non-trivial ideals and play a central role in the classification of Lie algebras
• The Cartan subalgebra is a maximal abelian subalgebra of a Lie algebra consisting of semisimple elements and is essential for studying representations
  • Its dimension is called the rank of the Lie algebra
• Root systems are sets of vectors in the dual space of the Cartan subalgebra that arise from the adjoint representation and encode the structure of the Lie algebra
  • They are used to classify simple Lie algebras and construct their representations
• The Weyl group is a finite reflection group associated with a root system that acts on the Cartan subalgebra and plays a crucial role in the representation theory of Lie algebras
• The universal enveloping algebra of a Lie algebra is an associative algebra that contains the Lie algebra as a subspace and allows for the construction of representations

Representation Basics

• A 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 linear transformations on $V$
• The action of a Lie algebra element $x \in \mathfrak{g}$ on a vector $v \in V$ in a representation is given by $\rho(x)(v)$
• Subrepresentations are subspaces of a representation that are invariant under the action of the Lie algebra and form representations themselves
• Morphisms between representations are linear maps that commute with the action of the Lie algebra and preserve the representation structure
• The direct sum of representations is a new representation formed by taking the direct sum of the underlying vector spaces and defining the Lie algebra action componentwise
• The tensor product of representations is a new representation formed by taking the tensor product of the underlying vector spaces and extending the Lie algebra action using the Leibniz rule
  • It allows for the construction of new representations from existing ones

Types of Representations

• Finite-dimensional representations are representations where the underlying vector space has finite dimension and are the most well-studied type of representations
• Infinite-dimensional representations, such as Verma modules and highest weight representations, arise naturally in the study of Lie algebras and have a rich structure
• Unitary representations are representations on Hilbert spaces where the Lie algebra elements act as skew-Hermitian operators and are important in mathematical physics
  • They ensure that the representation preserves the inner product structure of the Hilbert space
• Projective representations are generalizations of ordinary representations where the Lie algebra elements act up to a scalar multiple and arise in the study of covering groups
• Spinor representations are special representations of Lie algebras that are constructed using Clifford algebras and are essential in physics, particularly in the description of fermions
• Adjoint representations, as mentioned earlier, are representations where the Lie algebra acts on itself via the Lie bracket operation and provide insight into the structure of the Lie algebra

Construction Techniques

• The highest weight method is a powerful technique for constructing irreducible representations of Lie algebras using highest weight vectors and Verma modules
  • It relies on the existence of a highest weight vector, which is annihilated by the positive root spaces of the Lie algebra
• Induced representations are constructed by starting with a representation of a subalgebra and extending it to a representation of the entire Lie algebra using a process called induction
• The Borel-Weil theorem provides a geometric realization of irreducible representations of complex semisimple Lie algebras as spaces of sections of line bundles over flag varieties
• The Peter-Weyl theorem decomposes the regular representation of a compact Lie group into a direct sum of irreducible representations, establishing a connection between representation theory and harmonic analysis
• The Weyl character formula is an explicit formula for computing the characters of irreducible representations of semisimple Lie algebras using data from the root system and Weyl group
• Tensor product decomposition formulas, such as the Clebsch-Gordan coefficients for $\mathfrak{sl}_2(\mathbb{C})$, describe how tensor products of irreducible representations decompose into direct sums of irreducible representations

Important Theorems and Results

• Weyl's theorem asserts that every finite-dimensional representation of a semisimple Lie algebra is completely reducible, meaning it can be decomposed into a direct sum of irreducible representations
• The Poincaré-Birkhoff-Witt theorem provides a basis for the universal enveloping algebra of a Lie algebra, allowing for the construction of representations and the study of their properties
• The Weyl character formula, as mentioned earlier, gives an explicit expression for the characters of irreducible representations of semisimple Lie algebras
• The Kostant multiplicity formula computes the multiplicities of weight spaces in irreducible representations using data from the root system and Weyl group
• The Borel-Weil-Bott theorem is a generalization of the Borel-Weil theorem that relates the cohomology of line bundles over flag varieties to the representation theory of Lie algebras
  • It provides a geometric approach to studying representations
• The Kazhdan-Lusztig conjecture, now a theorem, describes the characters of irreducible highest weight representations of semisimple Lie algebras in terms of certain polynomials arising from the Weyl group
• The Kirillov-Kostant orbit method relates the unitary representations of a Lie group to the orbits of its coadjoint action on the dual of its Lie algebra, providing a geometric perspective on representation theory

Applications in Physics and Mathematics

• Representation theory of Lie algebras plays a fundamental role in quantum mechanics, where physical observables are represented by self-adjoint operators on a Hilbert space and symmetries are described by Lie groups and their representations
• In particle physics, Lie algebras and their representations are used to classify elementary particles and understand their interactions through gauge theories like the Standard Model
• Conformal field theories, which are important in string theory and statistical mechanics, heavily rely on the representation theory of infinite-dimensional Lie algebras such as the Virasoro algebra and affine Kac-Moody algebras
• Representation theory is a key tool in the study of symmetric spaces, which are manifolds with a high degree of symmetry and have applications in geometry, topology, and mathematical physics
• The Langlands program, a far-reaching network of conjectures connecting representation theory, number theory, and geometry, heavily relies on the representation theory of Lie groups and their algebras
• Representation theory has applications in combinatorics, such as the study of symmetric functions and the representation theory of the symmetric group, which is related to the representation theory of $\mathfrak{gl}_n(\mathbb{C})$

Challenges and Advanced Topics

• Infinite-dimensional representations, such as those arising from Kac-Moody algebras and the Virasoro algebra, pose significant challenges and require the development of new techniques and ideas
• The representation theory of Lie superalgebras, which are generalizations of Lie algebras that include both even and odd elements, is an active area of research with connections to supersymmetry in physics
• Quantum groups, which are deformations of universal enveloping algebras of Lie algebras, have a rich representation theory that generalizes and extends classical results
• The geometric Langlands program aims to relate the representation theory of Lie groups and Lie algebras to the geometry of certain moduli spaces and has deep connections to mathematical physics and number theory
• Categorification is a process of lifting algebraic structures, such as Lie algebras and their representations, to categorical level, leading to new insights and connections to other areas of mathematics
• The study of unitary representations of real reductive groups is a challenging and active area of research with applications in automorphic forms and the Langlands program
• The representation theory of affine Lie algebras and their connections to vertex operator algebras and conformal field theory is a rich and highly active area of research at the intersection of mathematics and physics