8.4 Character theory for non-associative algebras

Character theory for non-associative algebras extends classical representation theory to broader algebraic structures. It provides powerful tools for analyzing properties and symmetries in non-associative settings, bridging abstract algebra with practical applications in physics and computer science.

This topic explores how characters, functions mapping algebra elements to complex numbers, capture essential information about representations. It covers character properties, tables, and formulas, highlighting unique challenges and adaptations required for non-associative algebras like Lie and Jordan algebras.

Fundamentals of character theory

• Character theory in non-associative algebras extends classical representation theory to broader algebraic structures
• Provides powerful tools for analyzing algebraic properties and symmetries in non-associative settings
• Bridges abstract algebra with practical applications in physics and computer science

Definition of characters

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• Functions mapping algebra elements to complex numbers preserving algebraic structure
• Trace functions of matrix representations generalized to non-associative contexts
• Capture essential information about algebra representations in a concise form
• Defined as $\chi(a) = \text{tr}(ρ(a))$ where ρ is a representation and a is an algebra element

Properties of characters

• Linearity allows character addition and scalar multiplication
• Conjugation invariance ensures $\chi(bab^{-1}) = \chi(a)$ for all elements a, b
• Character values often algebraic integers in cyclotomic field extensions
• Satisfy orthogonality relations crucial for decomposing representations

Character tables

• Organize character values for all irreducible representations of an algebra
• Rows correspond to irreducible characters, columns to conjugacy classes
• Reveal symmetries and structural properties of the algebra at a glance
• Powerful tool for classifying and distinguishing non-isomorphic algebras
• Example: Octonion algebra character table highlights its non-associativity through unique patterns

Characters in non-associative algebras

• Extend traditional character theory to algebras without associativity
• Require careful handling of multiplication order and bracketing
• Provide insights into the structure of algebras like Lie algebras and Jordan algebras

Differences from associative algebras

• Non-associativity complicates definition of representations and characters
• Require generalized notions of conjugacy and centralizers
• May involve multilinear character functions for certain algebra types
• Character values can depend on bracketing order of elements (Lie algebras)

Challenges in non-associative settings

• Lack of general associative matrix algebra embedding limits representation theory
• Irreducible representations may not decompose tensor products uniquely
• Character formulas often require algebra-specific modifications
• Schur's lemma may not apply, affecting orthogonality relations

Representation theory connection

• Characters provide a bridge between abstract algebra and concrete matrix representations
• Allow study of infinite-dimensional representations through finite-dimensional tools
• Crucial for understanding symmetry groups in physics and chemistry applications

Representations vs characters

• Representations map algebra elements to linear transformations on vector spaces
• Characters condense representation information into complex-valued functions
• Representations contain more information but characters often easier to work with
• Indecomposable representations may share characters in non-semisimple cases

Character-based approach benefits

• Simplifies calculations by working with complex numbers instead of matrices
• Allows classification of representations without explicit construction
• Reveals global algebra structure through character relations and orthogonality
• Facilitates comparison between different algebras and their representations

Character formulas

• Provide computational tools for deriving and manipulating characters
• Generalize classical results to non-associative settings with appropriate modifications
• Essential for practical applications of character theory in algebra computations

Frobenius formula

• Expresses characters of induced representations in terms of subgroup characters
• Adapted for non-associative algebras by considering appropriate subalgebras
• General form: $\chi_G^H(g) = \frac{1}{|H|} \sum_{x \in G} \chi_H(xgx^{-1})$ where G is the full algebra and H is a subalgebra
• Requires careful interpretation of conjugation in non-associative contexts

Schur orthogonality relations

• Fundamental identities relating different irreducible characters
• For non-associative algebras, may involve modified inner products
• First relation: $\sum_{g \in G} \chi_i(g)\overline{\chi_j(g)} = |G|\delta_{ij}$ where χi and χj are irreducible characters
• Second relation: $\sum_i \chi_i(g)\overline{\chi_i(h)} = |C_G(g)|\delta_{gh}$ where C_G(g) is the centralizer of g

Character degrees

• Provide important numerical invariants for representations and algebras
• Reflect dimensionality and complexity of representations
• Play crucial role in various character-theoretic results and applications

Degree definition

• Character degree defined as value of character on algebra identity element
• Represents dimension of corresponding representation vector space
• For irreducible characters, degrees are positive integers
• Sum of squares of degrees equals order of the algebra (in finite case)

Degree properties

• Divisibility properties often reveal structural information about algebra
• Degrees of irreducible characters divide order of algebra in certain cases
• Degree 1 characters correspond to one-dimensional representations (linear characters)
• Frobenius-Schur indicator relates degrees to real, complex, or quaternionic nature of representations

Irreducible characters

• Correspond to fundamental building blocks of representations
• Essential for decomposing arbitrary representations into simpler components
• Provide complete set of orthogonal functions for character space

Irreducibility criteria

• Character is irreducible if and only if its inner product with itself equals 1
• Irreducible characters cannot be written as sum of other characters
• Number of irreducible characters equals number of conjugacy classes in finite case
• Burnside's theorem relates irreducible character degrees to algebra structure

Orthogonality of irreducible characters

• Irreducible characters form orthonormal basis for character space
• Inner product of distinct irreducible characters always zero
• Generalizes to modified inner products for certain non-associative algebras
• Crucial for decomposing arbitrary characters into irreducible components

Character calculations

• Involve various techniques for deriving and manipulating character values
• Combine algebraic properties with number-theoretic insights
• Essential for practical applications of character theory in algebra research

Methods for computing characters

• Trace method uses matrix representations when available
• Induction calculates characters of larger algebras from smaller subalgebras
• Restriction derives subalgebra characters from full algebra characters
• Tensor product method combines known characters to produce new ones
• Clifford theory relates characters of algebra to those of normal subalgebra

Character-based proofs

• Leverage character properties to prove structural results about algebras
• Often provide elegant alternatives to direct algebraic manipulation
• Character sum formulas yield insights into conjugacy class sizes
• Orthogonality relations used to count fixed points of group actions
• Burnside's lemma proven using character-theoretic techniques

Applications of characters

• Extend beyond pure mathematics to physics, chemistry, and computer science
• Provide powerful tools for analyzing symmetries in various scientific contexts
• Enable efficient algorithms for algebraic computations and classifications

Structure determination

• Character tables reveal central series and derived series of algebras
• Degrees of irreducible characters constrain possible algebra orders
• Induced character formulas help identify normal substructures
• Brauer's permutation lemma uses characters to analyze automorphism groups

Isomorphism testing

• Character tables serve as invariants for non-isomorphic algebras
• Comparison of character values can quickly rule out isomorphism
• Power maps of characters reveal cyclicity of algebra elements
• Character degrees and their multiplicities provide isomorphism invariants

Character extensions

• Techniques for relating characters of algebras to those of sub- or superalgebras
• Essential for building up character theory of complex algebras from simpler ones
• Provide insights into relationships between different algebraic structures

Induced characters

• Construct characters of larger algebra from characters of subalgebra
• Frobenius reciprocity relates induction to restriction process
• Mackey decomposition formula describes interaction of induction with conjugation
• Artin's induction theorem expresses any character as rational combination of induced characters

Restricted characters

• Derive characters of subalgebra by restricting characters of full algebra
• Often decompose into sum of irreducible characters of subalgebra
• Branching rules describe decomposition patterns for specific algebra types
• Clifford's theorem relates irreducible characters to those of normal subalgebras

Advanced character theory

• Extends classical character theory to more general settings
• Addresses challenges posed by modular representations and non-semisimple algebras
• Provides tools for analyzing algebras over fields of positive characteristic

Brauer characters

• Generalize ordinary characters to modular representations
• Defined using eigenvalues of representing matrices in algebraic number fields
• Satisfy modified orthogonality relations and character formulas
• Brauer-Nesbitt theorem relates Brauer characters to ordinary characters

Modular characters

• Arise in representations over fields of positive characteristic
• Capture information lost in reduction modulo prime characteristic
• Decomposition matrices relate ordinary characters to modular characters
• Block theory organizes modular characters into p-blocks with common defect groups

Characters in specific algebras

• Apply general character theory to important classes of non-associative algebras
• Reveal unique features and challenges posed by different algebraic structures
• Provide concrete examples illustrating abstract character-theoretic concepts

Lie algebras

• Characters defined via formal exponential series due to non-associativity
• Weyl character formula expresses irreducible characters in terms of roots
• Tensor product decomposition governed by Littlewood-Richardson rule
• Harish-Chandra isomorphism relates characters to invariant polynomials

Jordan algebras

• Characters reflect graded structure of Jordan algebras
• Peirce decomposition plays crucial role in character computations
• Reduced trace form used to define inner product on character space
• McCrimmon-Zelmanov classification theorem utilizes character theory

Octonion algebras

• Non-associativity and non-commutativity pose unique challenges for character theory
• G2 exceptional Lie group appears as automorphism group, influencing character structure
• Cayley-Dickson construction reflected in character values and degrees
• Connections to exceptional Jordan algebras through Freudenthal-Tits magic square

Computational aspects

• Leverage computer algebra systems for complex character calculations
• Develop efficient algorithms for generating and manipulating character data
• Address computational challenges posed by high-dimensional and infinite algebras

Software for character computations

• GAP (Groups, Algorithms, Programming) system includes extensive character theory functionality
• Magma computer algebra system offers tools for non-associative algebra characters
• SageMath provides open-source implementations of character-theoretic algorithms
• Custom packages developed for specific algebra types (LiE for Lie algebras)

Algorithmic challenges

• Efficient computation of character tables for large algebras
• Decomposition of tensor products into irreducible components
• Isomorphism testing using character-based invariants
• Generation of all irreducible characters for infinite-dimensional algebras
• Numerical approximation techniques for continuous character theory

Open problems

• Highlight active areas of research in character theory for non-associative algebras
• Identify connections between character theory and other branches of mathematics
• Motivate future directions for theoretical and computational investigations

Current research directions

• Extending modular character theory to broader classes of non-associative algebras
• Developing character theories for quantum groups and Hopf algebras
• Investigating connections between character theory and geometric representation theory
• Applying character-theoretic methods to problems in algebraic combinatorics
• Exploring character varieties and their role in geometric invariant theory

Unsolved questions in character theory

• McKay conjecture relating character degrees to normalizers of Sylow subgroups
• Generalization of Artin's conjecture on induced characters to non-associative settings
• Classification of all simple modules for exceptional Lie algebras in positive characteristic
• Development of a comprehensive character theory for Malcev algebras
• Characterization of algebras with identical character tables but non-isomorphic structures