8.3 Representations of alternative algebras

Alternative algebras bridge the gap between associative and non-associative structures. They maintain certain associativity properties while generalizing associative algebras, providing insights into complex non-associative structures in advanced math and physics.

Representations of alternative algebras extend concepts from associative algebra representation theory. They offer tools for studying alternative algebras' structure and properties, connecting them to other areas of mathematics and physics.

Definition of alternative algebras

• Alternative algebras generalize associative algebras while maintaining certain associativity properties
• These algebras play a crucial role in non-associative algebra, bridging the gap between associative and non-associative structures
• Understanding alternative algebras provides insights into more complex non-associative structures encountered in advanced mathematics and physics

Properties of alternative algebras

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• Satisfy the alternative laws: $(x,x,y) = (y,x,x) = 0$ for all elements x and y
• Flexible identity holds: $(xy)x = x(yx)$ for all elements x and y
• Every subalgebra generated by two elements is associative
• Artin's theorem states that any three elements of an alternative algebra generate an associative subalgebra
• Alternative algebras are power-associative, meaning all powers of a single element associate

Examples of alternative algebras

• Octonions form the most well-known non-associative alternative algebra
• Cayley algebras over fields of characteristic not 2
• Quaternion algebras provide examples of associative alternative algebras
• Zorn's vector matrix algebras serve as another class of alternative algebras
• Real alternative algebras of dimension ≤ 4 are always associative

Representations of alternative algebras

• Representations of alternative algebras extend concepts from associative algebra representation theory
• These representations provide tools for studying the structure and properties of alternative algebras
• Understanding representations aids in connecting alternative algebras to other areas of mathematics and physics

Linear representations

• Map elements of an alternative algebra A to linear transformations on a vector space V
• Preserve the algebraic structure: $ρ(xy) = ρ(x)ρ(y)$ for all x, y in A
• Can be finite-dimensional or infinite-dimensional depending on the vector space V
• Regular representations form an important class of linear representations
• Faithful representations preserve the structure of the algebra without loss of information

Bimodule representations

• Consist of a vector space M with left and right actions of an alternative algebra A
• Satisfy compatibility conditions: $(a,m,b) = (m,a,b) = (a,b,m) = 0$ for all a, b in A and m in M
• Generalize the notion of modules over associative algebras
• Allow for the study of alternative algebras through their action on other structures
• Can be used to construct new alternative algebras through extensions and products

Cayley-Dickson process

• Fundamental construction method in non-associative algebra for generating new algebras
• Produces a sequence of algebras with doubling dimension at each step
• Plays a crucial role in understanding the structure of alternative and other non-associative algebras

Construction of octonions

• Start with the real numbers and apply the Cayley-Dickson process three times
• First step produces complex numbers: $(a,b) = a + bi$
• Second step yields quaternions: $(a,b) = a + bj$ where a and b are complex numbers
• Final step creates octonions: $(a,b) = a + be$ where a and b are quaternions
• Multiplication rule for octonions: $(a,b)(c,d) = (ac - d*b, da + bc*)$
• Octonions form an 8-dimensional alternative algebra over the real numbers

Generalized Cayley-Dickson algebras

• Continue the Cayley-Dickson process beyond octonions to create higher-dimensional algebras
• Sedenions (16-dimensional) result from applying the process to octonions
• Each iteration doubles the dimension of the previous algebra
• Algebras beyond octonions lose the alternative property
• Generalized Cayley-Dickson algebras exhibit progressively weaker algebraic properties
• Study of these algebras provides insights into the limits of algebraic structures

Regular representations

• Fundamental class of representations for alternative algebras
• Allow the study of an algebra through its action on itself
• Play a crucial role in understanding the structure and properties of alternative algebras

Left regular representations

• Represent elements of an algebra A as left multiplication operators on A itself
• For each a in A, define L_a(x) = ax for all x in A
• Preserve the algebraic structure: L_ab = L_a L_b
• Provide a concrete realization of the algebra as linear transformations
• Can be used to study the left ideals and left multiplication properties of the algebra

Right regular representations

• Represent elements of an algebra A as right multiplication operators on A
• For each a in A, define R_a(x) = xa for all x in A
• Preserve the algebraic structure: R_ab = R_b R_a (note the reversal of order)
• Allow for the investigation of right ideals and right multiplication properties
• Comparison with left regular representations reveals information about the commutativity of the algebra

Irreducible representations

• Fundamental building blocks in representation theory of alternative algebras
• Cannot be decomposed into smaller subrepresentations
• Play a crucial role in understanding the structure of more complex representations

Classification of irreducible representations

• Finite-dimensional irreducible representations often classified by highest weight vectors
• For alternative algebras, classification may involve additional parameters or constraints
• Wedderburn-Artin theorem for associative algebras does not directly apply to alternative algebras
• Classification often requires case-by-case analysis depending on the specific alternative algebra
• Representation theory of octonions involves unique challenges due to their non-associativity

Schur's lemma for alternative algebras

• Extends the classical Schur's lemma from associative algebra representation theory
• States that any endomorphism of an irreducible representation is a scalar multiple of the identity
• Proof requires careful consideration of the alternative laws
• Applies to both finite and infinite-dimensional representations
• Plays a crucial role in the development of character theory for alternative algebras

Representation theory vs associative algebras

• Compares and contrasts representation theory of alternative algebras with that of associative algebras
• Highlights the unique challenges and features of studying representations in the non-associative setting
• Provides insights into the broader landscape of algebraic structures and their representations

Similarities and differences

• Both theories study homomorphisms from algebras to endomorphism rings of vector spaces
• Alternative algebra representations must preserve the alternative laws, not just multiplication
• Tensor products of representations behave differently in the alternative setting
• Wedderburn-Artin structure theory for associative algebras has limited applicability to alternative algebras
• Character theory for alternative algebras requires modifications to account for non-associativity

Challenges in alternative algebra representations

• Non-associativity complicates the construction and analysis of representations
• Lack of a general structure theory analogous to the Wedderburn-Artin theorem for associative algebras
• Limited applicability of classical techniques from Lie algebra and associative algebra representation theory
• Need for new tools and approaches tailored to the specific properties of alternative algebras
• Computational complexity often increases due to the need to verify alternative laws

Applications of representations

• Representations of alternative algebras find applications in various fields of mathematics and science
• These applications demonstrate the practical importance of studying alternative algebra representations
• Understanding these applications motivates further research in the field of non-associative algebra

Physics and quantum mechanics

• Octonions and their representations appear in certain models of particle physics
• Alternative algebras play a role in the study of exceptional Lie groups and their applications in physics
• Representations of alternative algebras contribute to the development of generalized quantum mechanics
• Non-associative structures arise in the study of magnetic monopoles and string theory
• Alternative algebra representations provide tools for exploring higher-dimensional spacetime models

Computer science and coding theory

• Alternative algebras and their representations contribute to the development of error-correcting codes
• Octonion-based algorithms find applications in computer graphics and signal processing
• Non-associative structures play a role in certain cryptographic protocols
• Alternative algebra representations inspire new approaches to quantum computing algorithms
• Coding theory benefits from the unique properties of alternative algebras in constructing efficient codes

Modules over alternative algebras

• Generalize the concept of vector spaces to the setting of alternative algebras
• Provide a framework for studying representations and the action of alternative algebras on other structures
• Play a crucial role in developing a structural theory for alternative algebras

Definition and properties

• An A-module M over an alternative algebra A consists of an abelian group M and bilinear maps A × M → M and M × A → M
• Satisfy the alternative laws: $(a,b,m) = (a,m,b) = (m,a,b) = 0$ for all a, b in A and m in M
• Left A-modules, right A-modules, and bimodules are distinguished based on the defined actions
• Submodules, quotient modules, and direct sums of modules can be defined analogously to the associative case
• Faithful modules provide representations of the alternative algebra without loss of information

Homomorphisms and isomorphisms

• Module homomorphisms preserve the action of the alternative algebra
• For A-modules M and N, a homomorphism f: M → N satisfies f(am) = af(m) and f(ma) = f(m)a for all a in A and m in M
• Isomorphisms are bijective homomorphisms with homomorphic inverses
• Kernel and image of module homomorphisms are defined similarly to the vector space case
• The first isomorphism theorem holds: M/ker(f) ≅ im(f) for any homomorphism f: M → N

Character theory

• Extends the concept of characters from group representation theory to alternative algebras
• Provides powerful tools for analyzing and classifying representations of alternative algebras
• Plays a crucial role in understanding the structure of alternative algebras through their representations

Characters of representations

• For a finite-dimensional representation ρ: A → End(V), the character χ_ρ is defined as χ_ρ(a) = Tr(ρ(a))
• Characters are linear: χ_ρ⊕σ = χ_ρ + χ_σ for direct sums of representations
• For alternative algebras, characters may not be multiplicative due to non-associativity
• Irreducible representations often have distinct characters, aiding in their classification
• Character values provide information about the dimension and structure of representations

Character tables for alternative algebras

• Organize character values of irreducible representations for finite-dimensional alternative algebras
• Rows correspond to irreducible representations, columns to conjugacy classes or suitable elements
• Orthogonality relations for characters may need modification compared to the group case
• Character tables aid in decomposing representations into irreducible components
• Construction and analysis of character tables often require specialized techniques for alternative algebras

Tensor products of representations

• Extend the concept of tensor products from vector spaces to representations of alternative algebras
• Provide a way to construct new representations from existing ones
• Play a crucial role in understanding the structure of representations and their decompositions

Definition and properties

• For representations ρ: A → End(V) and σ: A → End(W), define ρ ⊗ σ: A → End(V ⊗ W)
• The action on simple tensors: (ρ ⊗ σ)(a)(v ⊗ w) = ρ(a)v ⊗ σ(a)w
• Tensor products of alternative algebra representations may not preserve the alternative property
• Associativity of tensor products holds: (ρ ⊗ σ) ⊗ τ ≅ ρ ⊗ (σ ⊗ τ)
• Dimension of tensor product representations: dim(V ⊗ W) = dim(V) × dim(W)

Decomposition of tensor products

• Tensor products of irreducible representations may be reducible
• Clebsch-Gordan coefficients describe the decomposition of tensor products into irreducible components
• For alternative algebras, decomposition rules may differ from those of associative algebras
• Plethysm generalizes tensor products to more complex combinations of representations
• Understanding tensor product decompositions aids in constructing new representations and studying symmetries

Representation growth

• Studies how the number of irreducible representations of an alternative algebra grows with dimension
• Provides insights into the structure and complexity of the algebra and its representations
• Connects representation theory to other areas of mathematics, such as analytic number theory and geometry

Asymptotic behavior of representations

• Examines the growth rate of the number of irreducible representations as dimension increases
• For finite-dimensional algebras, focuses on representations of increasing dimension
• Infinite-dimensional algebras may consider growth with respect to other parameters
• Zeta functions encode information about representation growth in a generating function
• Asymptotic behavior often reveals fundamental properties of the underlying algebraic structure

Growth rates for alternative algebras

• May differ significantly from growth rates of associative algebras
• Octonions and other non-associative alternative algebras often exhibit unique growth patterns
• Polynomial growth indicates a relatively simple representation theory
• Exponential growth suggests a rich and complex structure of representations
• Intermediate growth rates may indicate interesting algebraic properties or symmetries

Computational aspects

• Addresses the practical challenges of working with representations of alternative algebras
• Develops algorithms and software tools for studying and manipulating these representations
• Bridges the gap between theoretical results and practical applications of alternative algebra representations

Algorithms for representation computations

• Develop methods for constructing and decomposing representations of alternative algebras
• Implement character calculations and tensor product decompositions
• Adapt linear algebra algorithms to handle non-associative structures
• Optimize computations for specific classes of alternative algebras (octonions)
• Develop algorithms for testing irreducibility and computing centralizers in representations

Software tools for alternative algebras

• Computer algebra systems (GAP, SageMath) with extensions for non-associative algebras
• Specialized libraries for octonion and other alternative algebra computations
• Visualization tools for understanding the structure of alternative algebra representations
• Databases of known representations and character tables for important alternative algebras
• Interfaces between theoretical results and computational implementations to facilitate research