🔁Lie Algebras and Lie Groups Unit 4 – Structure Theory of Lie Algebras

Key Concepts and Definitions

• Lie algebra defined as a vector space $V$ over a field $F$ with a bilinear operation called the Lie bracket $[x,y]$ satisfying antisymmetry and the Jacobi identity
• Lie subalgebra consists of a subspace of a Lie algebra that is closed under the Lie bracket operation
• Ideal of a Lie algebra analogous to a normal subgroup in group theory, where $[x,y]$ is in the ideal for all $x$ in the Lie algebra and $y$ in the ideal
• Derived series of a Lie algebra defined recursively as $L^{(0)}=L$, $L^{(i+1)}=[L^{(i)},L^{(i)}]$, measuring the solvability of the Lie algebra
  • Lie algebra is solvable if the derived series terminates at zero after finitely many steps
• Lower central series of a Lie algebra defined recursively as $L_0=L$, $L_{i+1}=[L,L_i]$, measuring the nilpotency of the Lie algebra
  • Lie algebra is nilpotent if the lower central series terminates at zero after finitely many steps
• Killing form defined as a symmetric bilinear form $B(x,y)=\text{tr}(\text{ad}_x \text{ad}_y)$, where $\text{ad}_x$ is the adjoint representation of $x$
  • Plays a crucial role in the classification of Lie algebras and the study of their structure
• Cartan subalgebra defined as a maximal abelian subalgebra of a Lie algebra consisting of semisimple elements
  • Semisimple elements are diagonalizable under the adjoint representation

Historical Context and Development

• Lie algebras introduced by Sophus Lie in the late 19th century as a tool to study continuous transformation groups, now known as Lie groups
• Élie Cartan made significant contributions to the structure theory of Lie algebras in the early 20th century, including the classification of simple Lie algebras
• Hermann Weyl further developed the representation theory of Lie algebras and introduced the concept of weights and root systems
• Claude Chevalley provided a unified approach to the classification of simple Lie algebras over arbitrary fields using root systems
• Harish-Chandra made fundamental contributions to the representation theory of Lie algebras and Lie groups, particularly in the context of infinite-dimensional representations
• Kac-Moody algebras introduced as a generalization of finite-dimensional simple Lie algebras, allowing for infinite-dimensional representations
• Quantum groups emerged as a deformation of the universal enveloping algebra of Lie algebras, leading to new developments in mathematical physics and representation theory

Fundamental Theorems and Properties

• Engel's theorem states that a Lie algebra is nilpotent if and only if the adjoint representation $\text{ad}_x$ is nilpotent for all $x$ in the Lie algebra
  • Provides a characterization of nilpotent Lie algebras in terms of the nilpotency of the adjoint representation
• Lie's theorem asserts that every finite-dimensional solvable Lie algebra over an algebraically closed field has a common eigenvector for all elements in the Lie algebra acting via the adjoint representation
  • Implies that every finite-dimensional representation of a solvable Lie algebra over an algebraically closed field is upper-triangularizable
• Weyl's complete reducibility theorem states that every finite-dimensional representation of a semisimple Lie algebra is completely reducible
  • Consequence of the existence of a non-degenerate invariant bilinear form on the Lie algebra, such as the Killing form
• Schur's lemma asserts that any linear map between two irreducible representations of a Lie algebra that commutes with the action of the Lie algebra is either zero or an isomorphism
  • Plays a crucial role in the study of irreducible representations and their decompositions
• Universal enveloping algebra of a Lie algebra is an associative algebra that captures the algebraic structure of the Lie algebra
  • Poincaré-Birkhoff-Witt theorem provides a basis for the universal enveloping algebra in terms of ordered monomials of the Lie algebra elements
• Levi decomposition states that every finite-dimensional Lie algebra over a field of characteristic zero can be written as a semidirect product of a solvable ideal and a semisimple subalgebra
  • Semisimple part is unique up to isomorphism and is called the Levi factor

Classification of Lie Algebras

• Simple Lie algebras are the building blocks of semisimple Lie algebras and are classified into four infinite families ($A_n$, $B_n$, $C_n$, $D_n$) and five exceptional cases ($G_2$, $F_4$, $E_6$, $E_7$, $E_8$)
  • Classification is based on the Dynkin diagrams associated with the root systems of the Lie algebras
• Classical Lie algebras correspond to the infinite families in the classification and are related to the classical matrix groups (special linear, orthogonal, and symplectic groups)
  • $A_n$ corresponds to the special linear Lie algebra $\mathfrak{sl}_{n+1}$
  • $B_n$ corresponds to the odd orthogonal Lie algebra $\mathfrak{so}_{2n+1}$
  • $C_n$ corresponds to the symplectic Lie algebra $\mathfrak{sp}_{2n}$
  • $D_n$ corresponds to the even orthogonal Lie algebra $\mathfrak{so}_{2n}$
• Exceptional Lie algebras are the five special cases that do not fit into the infinite families and have unique properties
  • $G_2$ is the smallest exceptional Lie algebra and is related to the octonions
  • $F_4$, $E_6$, $E_7$, and $E_8$ are larger exceptional Lie algebras with intricate structures and representations
• Solvable and nilpotent Lie algebras play a crucial role in the study of general Lie algebras, as they appear in the Levi decomposition and the structure of Borel subalgebras
  • Borel subalgebras are maximal solvable subalgebras and are used in the study of flag varieties and representation theory

Root Systems and Weight Spaces

• Root system of a semisimple Lie algebra is a finite set of vectors in a Euclidean space that satisfy certain axioms and encode the structure of the Lie algebra
  • Roots are the non-zero eigenvalues of the adjoint representation of a Cartan subalgebra on the Lie algebra
  • Simple roots are a subset of the roots that form a basis for the root system and determine the Dynkin diagram
• Weight space decomposition expresses a representation of a Lie algebra as a direct sum of weight spaces, which are eigenspaces of the Cartan subalgebra
  • Weights are the eigenvalues of the Cartan subalgebra acting on the representation
  • Highest weight of a representation is the weight that is maximal with respect to the partial order induced by the simple roots
• Weyl group is a finite reflection group associated with the root system of a semisimple Lie algebra
  • Acts on the root system and the weight lattice, permuting the roots and weights while preserving the structure of the Lie algebra
  • Length function on the Weyl group measures the minimal number of simple reflections needed to express an element and plays a role in representation theory
• Weyl character formula expresses the character of an irreducible representation in terms of the Weyl group and the highest weight of the representation
  • Provides a powerful tool for computing characters and multiplicities of weights in representations

Representation Theory

• Representation of a Lie algebra is a linear action of the Lie algebra on a vector space that preserves the Lie bracket
  • Finite-dimensional representations are characterized by their dimension and can be studied using linear algebra techniques
  • Infinite-dimensional representations, such as Verma modules and highest weight representations, require more advanced tools from functional analysis and algebraic geometry
• Irreducible representations are the building blocks of representation theory and cannot be decomposed into non-trivial subrepresentations
  • Classification of irreducible representations for semisimple Lie algebras is given by the highest weight theorem, which associates an irreducible representation to each dominant integral weight
• Tensor products of representations can be decomposed into irreducible components using the Littlewood-Richardson rule or the Klimyk formula
  • Decomposition of tensor products is essential for studying the representation theory of Lie groups and their applications in physics
• Characters of representations are functions on the Lie algebra that encode the trace of the representation and satisfy certain invariance properties
  • Characters determine the isomorphism class of a representation and can be used to compute multiplicities and decompositions
• Schur-Weyl duality relates the representation theory of the general linear group and the symmetric group, providing a powerful tool for studying the representations of both groups
  • Generalizations of Schur-Weyl duality, such as Howe duality and the double commutant theorem, have applications in various areas of mathematics

Applications in Physics and Mathematics

• Lie algebras and their representations play a fundamental role in quantum mechanics, where observables are represented by self-adjoint operators on a Hilbert space
  • Commutation relations between observables are encoded by the Lie bracket of the corresponding Lie algebra elements
  • Unitary representations of Lie groups describe the symmetries of quantum systems and give rise to conservation laws
• Gauge theories in particle physics rely heavily on the structure of Lie algebras and their representations
  • Gauge fields are described by connections on principal bundles with structure group given by a Lie group
  • Lie algebra of the gauge group determines the interactions between particles and the symmetries of the theory
• Conformal field theories (CFTs) have a rich algebraic structure governed by the Virasoro algebra, which is an infinite-dimensional Lie algebra
  • Representations of the Virasoro algebra, known as primary fields, determine the spectrum and correlation functions of the CFT
  • Kac-Moody algebras and their representations also appear in the study of CFTs and integrable models
• Lie algebras and their cohomology play a significant role in deformation theory and the study of moduli spaces in algebraic geometry
  • Deformations of algebraic structures, such as complex manifolds or vector bundles, are controlled by the cohomology of certain Lie algebras
  • Moduli spaces of geometric objects often have a natural Lie algebraic description, which allows for the application of representation-theoretic techniques

Advanced Topics and Current Research

• Kac-Moody algebras are infinite-dimensional generalizations of semisimple Lie algebras that have found applications in various areas of mathematics and physics
  • Affine Kac-Moody algebras are central to the study of conformal field theories and integrable models in statistical mechanics
  • Hyperbolic Kac-Moody algebras, such as the $E_{10}$ algebra, have been proposed as a possible framework for unifying gravity and other forces in physics
• Quantum groups are deformations of the universal enveloping algebra of a Lie algebra that have a rich algebraic structure and representation theory
  • Yangians and quantized enveloping algebras are examples of quantum groups that have been studied extensively in the context of integrable systems and conformal field theory
  • Representations of quantum groups have led to new developments in knot theory, topological quantum field theory, and the study of invariants of 3-manifolds
• Categorification is a process of lifting algebraic structures to a higher categorical level, where vector spaces are replaced by categories and linear maps by functors
  • Categorification of Lie algebras and their representations has led to the development of Khovanov homology and other invariants in low-dimensional topology
  • Higher representation theory aims to study the categorified analogues of classical representation-theoretic objects and their applications
• Infinite-dimensional Lie algebras, such as the Virasoro algebra and Kac-Moody algebras, continue to be an active area of research with connections to various areas of mathematics and physics
  • Representation theory of infinite-dimensional Lie algebras poses new challenges and requires the development of novel techniques and tools
  • Connections between infinite-dimensional Lie algebras, integrable systems, and conformal field theories are being explored in current research
• Geometric representation theory seeks to understand the representations of Lie algebras and Lie groups using tools from algebraic geometry and topology
  • Techniques such as localization, equivariant cohomology, and the Riemann-Hilbert correspondence have been used to study the geometry of representation spaces and flag varieties
  • Langlands program, which aims to unify various areas of mathematics, has deep connections to the geometric representation theory of Lie groups and their Lie algebras