8.1 Representations of Lie algebras

Lie algebras are a key part of non-associative algebra, providing a framework for studying continuous symmetries in math and physics. They capture the infinitesimal properties of Lie groups, enabling powerful analysis techniques across various fields.

Representations of Lie algebras extend this study by showing how Lie algebras act on vector spaces. This allows Lie algebraic techniques to be applied to a wide range of mathematical and physical problems, bridging abstract algebra and practical applications.

Definition of Lie algebras

• Lie algebras form a crucial part of non-associative algebra, providing a framework for studying continuous symmetries in mathematical and physical systems
• These algebraic structures capture the infinitesimal properties of Lie groups, allowing for powerful analysis techniques in various fields of mathematics and physics

Axioms of Lie algebras

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• Vector space $L$ over a field $F$ equipped with a binary operation $[,]: L \times L \rightarrow L$ called the Lie bracket
• Bilinearity requires $[ax + by, z] = a[x,z] + b[y,z]$ and $[z, ax + by] = a[z,x] + b[z,y]$ for all $x,y,z \in L$ and $a,b \in F$
• Alternating property dictates $[x,x] = 0$ for all $x \in L$
• Jacobi identity states $[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0$ for all $x,y,z \in L$
• Anticommutativity follows from bilinearity and alternating property, resulting in $[x,y] = -[y,x]$

Examples of Lie algebras

• Three-dimensional real Lie algebra $\mathfrak{so}(3)$ represents infinitesimal rotations in 3D space
  • Basis elements correspond to rotations around x, y, and z axes
  • Lie bracket given by $[e_i, e_j] = \epsilon_{ijk}e_k$, where $\epsilon_{ijk}$ is the Levi-Civita symbol
• General linear Lie algebra $\mathfrak{gl}(n,F)$ consists of all $n \times n$ matrices over field $F$
  • Lie bracket defined as the commutator $[A,B] = AB - BA$
  • Special linear Lie algebra $\mathfrak{sl}(n,F)$ is a subalgebra of traceless matrices

Relationship to Lie groups

• Lie algebras serve as tangent spaces to Lie groups at the identity element
• Exponential map connects Lie algebra elements to Lie group elements
• Baker-Campbell-Hausdorff formula relates group multiplication to Lie bracket operations
• Adjoint representation of a Lie group on its Lie algebra encodes the group's structure

Representations of Lie algebras

• Representations of Lie algebras extend the study of non-associative algebra by providing a way to understand how Lie algebras act on vector spaces
• These representations allow for the application of Lie algebraic techniques to a wide range of mathematical and physical problems, bridging abstract algebra and practical applications

Definition of representation

• Linear map $\rho: L \rightarrow \text{End}(V)$ from a Lie algebra $L$ to the endomorphisms of a vector space $V$
• Preserves the Lie bracket structure $\rho([x,y]) = \rho(x)\rho(y) - \rho(y)\rho(x)$ for all $x,y \in L$
• Representation space $V$ becomes an $L$-module under the action $x \cdot v = \rho(x)(v)$ for $x \in L$ and $v \in V$
• Dimension of the representation refers to the dimension of the vector space $V$

Homomorphisms and isomorphisms

• Lie algebra homomorphism $\phi: L_1 \rightarrow L_2$ preserves the Lie bracket $\phi([x,y]) = [\phi(x),\phi(y)]$
• Isomorphism between Lie algebras establishes a bijective homomorphism
• Representation homomorphism $\psi: V_1 \rightarrow V_2$ satisfies $\psi(\rho_1(x)v) = \rho_2(x)\psi(v)$ for all $x \in L$ and $v \in V_1$
• Equivalent representations connected by an invertible intertwining operator

Adjoint representation

• Canonical representation of a Lie algebra on itself
• Defined by $\text{ad}_x(y) = [x,y]$ for all $x,y \in L$
• Jacobi identity ensures $\text{ad}_{[x,y]} = [\text{ad}_x, \text{ad}_y]$
• Kernel of the adjoint representation forms the center of the Lie algebra
• Killing form $K(x,y) = \text{Tr}(\text{ad}_x \circ \text{ad}_y)$ provides an invariant bilinear form

Types of representations

• Understanding different types of representations enhances the study of non-associative algebra by revealing the structure and properties of Lie algebras through their actions on vector spaces
• This classification allows for a systematic approach to analyzing and applying Lie algebraic concepts in various mathematical and physical contexts

Irreducible representations

• Cannot be decomposed into smaller subrepresentations
• Every non-zero vector generates the entire representation space
• Schur's lemma states that any intertwining operator is a scalar multiple of the identity
• Finite-dimensional irreducible representations of semisimple Lie algebras classified by highest weights
• Casimir operators act as scalar multiples of the identity on irreducible representations

Reducible representations

• Can be decomposed into a direct sum of subrepresentations
• Completely reducible representations decompose into a direct sum of irreducible representations
• Weyl's theorem states that all finite-dimensional representations of semisimple Lie algebras are completely reducible
• Decomposition process involves finding invariant subspaces and quotient representations
• Multiplicity of an irreducible component refers to the number of times it appears in the decomposition

Faithful representations

• Injective representations where distinct Lie algebra elements act differently on the representation space
• Provide isomorphisms between the Lie algebra and a subalgebra of $\mathfrak{gl}(V)$
• Adjoint representation is faithful if and only if the Lie algebra has a trivial center
• Existence of faithful representations allows for concrete matrix realizations of abstract Lie algebras
• Ado's theorem guarantees the existence of finite-dimensional faithful representations for finite-dimensional Lie algebras

Weight theory

• Weight theory forms a fundamental part of the representation theory of Lie algebras in non-associative algebra
• This framework provides powerful tools for analyzing the structure of representations and connecting algebraic properties to geometric concepts

Weight spaces and vectors

• Weight space $V_\lambda$ consists of vectors $v \in V$ such that $h \cdot v = \lambda(h)v$ for all $h$ in the Cartan subalgebra
• Weight $\lambda$ is a linear functional on the Cartan subalgebra
• Representation decomposes as a direct sum of weight spaces $V = \bigoplus_\lambda V_\lambda$
• Weight diagram visually represents the weights of a representation in the dual space of the Cartan subalgebra
• Multiplicity of a weight refers to the dimension of its corresponding weight space

Root systems

• Roots are non-zero weights of the adjoint representation
• Root system $\Phi$ consists of all roots and satisfies specific axioms (closure under reflection, integrality)
• Simple roots form a basis for the root system, allowing for classification of roots as positive or negative
• Weyl group generated by reflections with respect to simple roots acts on the weight lattice
• Dynkin diagrams provide a graphical representation of simple root systems, classifying simple Lie algebras

Cartan subalgebra

• Maximal abelian subalgebra $\mathfrak{h}$ of a Lie algebra $L$
• Consists of semisimple elements that are diagonalizable in the adjoint representation
• Dimension of the Cartan subalgebra defines the rank of the Lie algebra
• Roots and weights are elements of the dual space $\mathfrak{h}^*$
• Cartan-Killing form restricted to the Cartan subalgebra provides a non-degenerate bilinear form

Highest weight theory

• Highest weight theory plays a crucial role in the classification and construction of representations in non-associative algebra
• This approach provides a systematic way to understand and generate representations of Lie algebras, particularly for semisimple Lie algebras

Highest weight modules

• Representation containing a highest weight vector $v_\lambda$ annihilated by all positive root vectors
• Highest weight $\lambda$ determines the entire structure of the module
• Finite-dimensional irreducible representations of semisimple Lie algebras are highest weight modules
• Weight space decomposition of highest weight modules follows a specific pattern determined by the root system
• Contravariant form on highest weight modules provides a tool for studying their structure

Verma modules

• Universal highest weight modules $M(\lambda)$ generated by a highest weight vector
• Infinite-dimensional for non-integral dominant weights
• Unique maximal proper submodule leads to the construction of irreducible highest weight modules
• BGG resolution expresses finite-dimensional modules in terms of Verma modules
• Category $\mathcal{O}$ provides a framework for studying Verma modules and their quotients

Character formulas

• Formal sum $\text{ch}(V) = \sum_\lambda \dim(V_\lambda)e^\lambda$ encoding weight space dimensions
• Weyl character formula expresses characters of finite-dimensional irreducible representations
• Freudenthal's formula allows for recursive computation of weight multiplicities
• Kostant's multiplicity formula provides an alternative approach using partition functions
• Characters form a ring under tensor product operations, reflecting the decomposition of tensor products

Classification of representations

• Classification of representations is a fundamental aspect of non-associative algebra, particularly in the study of Lie algebras
• This systematic categorization allows for a deeper understanding of the structure and properties of different types of representations

Finite-dimensional representations

• Completely classified for semisimple Lie algebras using highest weight theory
• Dimension formula expresses the dimension of irreducible representations in terms of roots and weights
• Tensor product decomposition rules determined by the Littlewood-Richardson coefficients
• Branching rules describe the decomposition of representations under restriction to subalgebras
• Weyl's dimension formula provides a closed-form expression for dimensions of irreducible representations

Infinite-dimensional representations

• Include Verma modules, generalized Verma modules, and their quotients
• Harish-Chandra modules for semisimple Lie groups with finite-dimensional weight spaces
• Principal series representations arising from parabolic induction
• Discrete series representations for semisimple Lie groups of Hermitian type
• Complementary series representations occurring in specific ranges of parameters

Unitary representations

• Preserve an inner product on the representation space
• Crucial in quantum mechanics and harmonic analysis on Lie groups
• Unitary dual problem seeks to classify all irreducible unitary representations
• Bargmann's classification of unitary representations of $SL(2,\mathbb{R})$
• Restriction of unitary representations to compact subgroups yields discrete decompositions

Representation theory applications

• Applications of representation theory in non-associative algebra extend far beyond pure mathematics
• This powerful framework provides essential tools for understanding and solving problems in various fields of physics and geometry

Particle physics

• Classification of elementary particles using representations of the Poincaré group
• Standard Model based on representations of $SU(3) \times SU(2) \times U(1)$ gauge group
• Quark model utilizes $SU(3)$ flavor symmetry to organize hadrons
• Supersymmetry employs representations of super Lie algebras to relate bosons and fermions
• Grand Unified Theories explore larger Lie groups (SU(5), SO(10)) to unify fundamental forces

Quantum mechanics

• Angular momentum operators form representations of $\mathfrak{su}(2)$
• Hydrogen atom energy levels explained using $SO(4)$ symmetry
• Harmonic oscillator states organized by representations of the Heisenberg algebra
• Clebsch-Gordan coefficients describe coupling of angular momenta
• Wigner-Eckart theorem relates matrix elements of tensor operators to Clebsch-Gordan coefficients

Differential geometry

• Killing vector fields on manifolds form finite-dimensional Lie algebras
• Representation theory of $SO(n)$ classifies tensors on Riemannian manifolds
• Hodge theory employs representations of $SO(n)$ to study differential forms
• Atiyah-Singer index theorem relates analytical and topological invariants via representation theory
• Representation theory of loop groups connects to the theory of affine Lie algebras

Computational methods

• Computational methods play an increasingly important role in the study of non-associative algebra, particularly in representation theory of Lie algebras
• These tools allow for efficient calculation, exploration, and verification of theoretical results in complex algebraic structures

Lie algebra software packages

• GAP (Groups, Algorithms, Programming) provides extensive functionality for Lie algebras and their representations
• LiE specializes in computations with simple Lie algebras and their representations
• SageMath incorporates various Lie algebraic computations within a broader mathematical framework
• Mathematica's GroupMath package offers tools for Lie algebra and group theory calculations
• FORM focuses on symbolic manipulation of algebraic expressions in high-energy physics applications

Algorithms for representation theory

• LLL algorithm for finding short vectors in lattices applies to weight lattice computations
• Gram-Schmidt orthogonalization used in constructing bases for weight spaces
• Littlewood-Richardson rule implemented for tensor product decompositions
• Demazure character formula allows for recursive computation of characters
• Freudenthal's recursion formula calculates weight multiplicities in irreducible representations

Numerical techniques

• Numerical diagonalization of matrices for finding weights and weight vectors
• Monte Carlo methods for estimating dimensions of high-dimensional representations
• Iterative algorithms for solving systems of linear equations in weight space calculations
• Numerical integration techniques for computing characters and their inner products
• Machine learning approaches for pattern recognition in representation data and prediction of representation properties

Advanced topics

• Advanced topics in non-associative algebra extend the study of Lie algebras and their representations to more complex and generalized structures
• These areas of research connect representation theory to other branches of mathematics and physics, opening up new avenues for exploration and application

Kac-Moody algebras

• Infinite-dimensional generalizations of semisimple Lie algebras
• Classified by generalized Cartan matrices and Dynkin diagrams
• Affine Kac-Moody algebras correspond to extended Dynkin diagrams
• Hyperbolic Kac-Moody algebras have Lorentzian signature Cartan matrices
• Representation theory involves highest weight modules and integrable representations

Affine Lie algebras

• Central extensions of loop algebras of finite-dimensional simple Lie algebras
• Vertex operator representations connect to conformal field theory
• Modular forms arise from characters of integrable highest weight representations
• Macdonald identities generalize classical partition identities
• Affine Weyl groups describe the symmetries of affine root systems

Quantum groups

• Hopf algebra deformations of universal enveloping algebras of Lie algebras
• Quantum parameter $q$ interpolates between classical and quantum regimes
• R-matrix formalism encodes the quasi-triangular structure
• Crystal bases provide combinatorial models for representations at $q=0$
• Connections to knot theory through quantum invariants (Jones polynomial)

Historical development

• The historical development of representation theory in non-associative algebra reflects the evolution of mathematical thought and its interactions with physics
• This progression has led to a rich and diverse field with ongoing research and applications across multiple disciplines

Classical vs modern approaches

• Classical approach focused on matrix representations and character theory
• Modern approach emphasizes abstract algebraic structures and categorical methods
• Transition from concrete calculations to axiomatic foundations
• Increased emphasis on homological methods and functorial properties
• Integration of representation theory with other areas of mathematics (algebraic geometry, number theory)

Key contributors and theorems

• Sophus Lie introduced Lie groups and Lie algebras in the late 19th century
• Élie Cartan developed the structure theory of semisimple Lie algebras
• Hermann Weyl established the foundation of compact Lie group representation theory
• Harish-Chandra extended representation theory to non-compact semisimple Lie groups
• Israel Gelfand and Mark Naimark initiated the study of infinite-dimensional representations

Open problems in representation theory

• Langlands program seeks to connect representation theory of algebraic groups to number theory
• Invariant theory questions related to geometric complexity theory
• Classification of unitary representations for exceptional Lie groups
• Representation stability in sequences of groups (FI-modules)
• Categorification of quantum groups and their representations