🧮Non-associative Algebra Unit 8 – Non-associative Structures: Representation

Key Concepts and Definitions

• Non-associative algebra generalizes associative algebra by removing the requirement that multiplication is associative, meaning $(ab)c$ does not necessarily equal $a(bc)$
• Representation theory studies abstract algebraic structures by representing their elements as linear transformations of vector spaces
• Lie algebras are a key example of non-associative algebras, consisting of a vector space equipped with a bilinear operation called the Lie bracket, denoted $[x, y]$, satisfying anticommutativity and the Jacobi identity
  • Anticommutativity: $[x, y] = -[y, x]$
  • Jacobi identity: $[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0$
• Jordan algebras are another important class of non-associative algebras, characterized by the Jordan identity $(x^2y)x = x^2(yx)$
• Representation dimension refers to the dimension of the vector space on which the non-associative algebra is represented
• Faithful representation implies that distinct elements of the algebra are represented by distinct linear transformations
• Irreducible representation cannot be decomposed into smaller subrepresentations

Historical Context and Development

• Non-associative algebras emerged in the early 20th century as a generalization of associative algebras
• Lie algebras were introduced by Sophus Lie in the late 19th century in the context of studying continuous transformation groups
  • Lie's work laid the foundation for the development of representation theory
• Jordan algebras were introduced by Pascual Jordan in the 1930s in the context of quantum mechanics
• The study of non-associative algebras gained momentum in the mid-20th century with the work of mathematicians such as Nathan Jacobson and Irving Kaplansky
• Representation theory of non-associative algebras has been influenced by developments in physics, particularly quantum mechanics and particle physics
• The classification of simple Lie algebras and simple Jordan algebras over algebraically closed fields was a major milestone in the theory
• Recent decades have seen the application of non-associative algebras and their representations in various areas of mathematics and physics, such as differential geometry, combinatorics, and string theory

Types of Non-associative Structures

• Lie algebras are the most extensively studied class of non-associative algebras
  • Examples include the vector space of square matrices with the commutator $[A, B] = AB - BA$ as the Lie bracket
• Jordan algebras are another important class, satisfying the Jordan identity and power-associativity $(x^n)x = x(x^n)$
  • Examples include the space of self-adjoint matrices with the product $A \circ B = \frac{1}{2}(AB + BA)$
• Malcev algebras generalize Lie algebras by satisfying a weaker form of the Jacobi identity
• Quasi-associative algebras satisfy a generalized associativity condition involving a bilinear form
• Noncommutative Jordan algebras are a generalization of Jordan algebras that do not require commutativity
• Lie superalgebras and Jordan superalgebras incorporate a $\mathbb{Z}_2$-grading into the algebraic structure
• Vertex algebras combine aspects of Lie algebras and commutative algebras, with applications in conformal field theory

Representation Theory Basics

• Representation theory aims to understand abstract algebraic structures by representing their elements as linear transformations on vector spaces
• A representation of an algebra $A$ is a homomorphism $\rho: A \to \text{End}(V)$, where $V$ is a vector space and $\text{End}(V)$ is the space of linear transformations on $V$
  • The homomorphism property requires $\rho(ab) = \rho(a)\rho(b)$ for all $a, b \in A$
• The dimension of the representation is the dimension of the vector space $V$
• Subrepresentations are subspaces of $V$ that are invariant under the action of $A$ via $\rho$
• Irreducible representations cannot be decomposed into non-trivial subrepresentations
• Schur's lemma states that any homomorphism between irreducible representations is either zero or an isomorphism
• Characters are traces of representation matrices, providing a way to study representations up to isomorphism
• Tensor products and direct sums allow the construction of new representations from existing ones

Representation Methods for Non-associative Structures

• Adjoint representation of a Lie algebra $\mathfrak{g}$ represents elements $x \in \mathfrak{g}$ as linear transformations $\text{ad}_x: \mathfrak{g} \to \mathfrak{g}$ defined by $\text{ad}_x(y) = [x, y]$
• Universal enveloping algebra of a Lie algebra $\mathfrak{g}$ is an associative algebra containing $\mathfrak{g}$ as a subspace, allowing the construction of representations of $\mathfrak{g}$ from representations of the enveloping algebra
• Verma modules are induced representations of Lie algebras constructed using the universal enveloping algebra and a one-dimensional representation of a Borel subalgebra
• Highest weight theory classifies irreducible representations of semisimple Lie algebras using highest weight vectors and the action of the Cartan subalgebra
• Jordan algebras can be represented using the regular representation, where elements act on the algebra itself by multiplication
• The Tits-Kantor-Koecher construction relates Jordan algebras to Lie algebras, allowing the study of Jordan algebra representations using Lie algebra techniques
• Quasi-associative algebras can be represented using bimodules, incorporating the bilinear form into the representation structure

Important Theorems and Proofs

• Ado's theorem states that every finite-dimensional Lie algebra over a field of characteristic zero has a faithful finite-dimensional representation
  • The proof involves constructing a representation using the adjoint representation and the universal enveloping algebra
• Weyl's complete reducibility theorem asserts that every finite-dimensional representation of a semisimple Lie algebra is completely reducible (can be decomposed into irreducible subrepresentations)
  • The proof relies on the existence of an invariant inner product and the orthogonality of irreducible subrepresentations
• The Poincaré-Birkhoff-Witt theorem describes a basis for the universal enveloping algebra of a Lie algebra, enabling the construction of representations
  • The proof involves showing that monomials in a fixed basis of the Lie algebra form a basis for the enveloping algebra
• Schur's lemma has several important consequences, such as the diagonalizability of commuting semisimple elements in a representation
• The density theorem states that any irreducible representation of a simple Lie algebra is determined up to isomorphism by its highest weight
  • The proof uses the action of the Weyl group on the weight lattice and the uniqueness of the highest weight vector
• The Kronecker product theorem decomposes tensor products of irreducible representations of the general linear group $\text{GL}(n)$ into irreducible components
  • The proof involves the combinatorics of Young tableaux and Schur-Weyl duality

Applications and Examples

• Representation theory of Lie algebras has applications in quantum mechanics, where symmetries are described by Lie groups and their associated Lie algebras
  • The Lie algebra $\mathfrak{su}(2)$ is used to model the spin of particles, with its representations corresponding to different spin states
• The Lie algebra $\mathfrak{sl}(2,\mathbb{C})$ and its representations appear in the study of angular momentum and the hydrogen atom in quantum mechanics
• Representations of the Poincaré algebra, a semidirect product of the Lorentz algebra and the translation algebra, are used to describe elementary particles in relativistic quantum field theory
• The Virasoro algebra, an infinite-dimensional Lie algebra, plays a central role in conformal field theory and string theory, with its representations corresponding to different types of strings
• Jordan algebras have applications in optimization and statistics, particularly in the study of symmetric cones and self-dual cones
  • The Jordan algebra of real symmetric matrices is used in the analysis of semidefinite programming
• Vertex algebras and their representations are used to construct conformal field theories and to study the Monster group in moonshine theory
• Representations of Kac-Moody algebras, which generalize semisimple Lie algebras, appear in the study of affine Lie algebras and quantum groups

Challenges and Open Problems

• The classification of simple non-associative algebras over arbitrary fields remains an open problem, with complete results known only for certain classes of algebras and fields
• Developing a comprehensive structure theory for non-associative algebras, analogous to the theory of associative algebras, is an ongoing challenge
• Extending representation-theoretic results from characteristic zero to positive characteristic poses difficulties due to the failure of complete reducibility in general
• Constructing explicit models for representations of exceptional Lie algebras and understanding their properties is an active area of research
• Generalizing the theory of highest weight representations to infinite-dimensional Lie algebras, such as Kac-Moody algebras, presents new challenges and opportunities
• Exploring the connections between non-associative algebras and other areas of mathematics, such as geometry, topology, and number theory, is a fertile ground for interdisciplinary research
• Applying representation theory to the study of integrable systems, quantum groups, and noncommutative geometry is an emerging field with potential for new insights and discoveries
• Developing computational tools and algorithms for working with non-associative algebras and their representations is an important practical challenge, especially for applications in physics and engineering