1.4 Power-associative algebras

Power-associative algebras relax associativity while maintaining certain power-related properties. They bridge the gap between associative and fully non-associative structures, allowing for well-defined powers without full associativity.

These algebras include octonions, Jordan algebras, and flexible algebras. They satisfy key identities like the fourth-degree condition and have important relationships with other algebraic structures, subalgebras, and ideals.

Definition of power-associative algebras

• Non-associative algebras form the broader context for power-associative algebras in algebraic structures
• Power-associative algebras relax the associativity condition while maintaining certain power-related properties
• Crucial concept in Non-associative Algebra bridges the gap between associative and fully non-associative structures

Key properties

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• Powers of elements associate regardless of parenthesization
• Satisfies the identity $(x^m)[x^n](https://www.fiveableKeyTerm:x^n) = x^{m+n}$ for all positive integers m and n
• Allows for well-defined notion of powers without full associativity
• Generalizes associative algebras while maintaining some computational convenience
• Includes both associative and some non-associative algebras as special cases

Algebraic structure

• Defined over a field or ring, typically denoted as F
• Vector space structure with an additional bilinear multiplication operation
• Multiplication not necessarily associative or commutative
• Closure property ensures product of any two elements remains in the algebra
• Identity element may or may not exist, depending on the specific algebra

Examples of power-associative algebras

• Power-associative algebras encompass a wide range of algebraic structures
• Studying examples helps illuminate the diversity and applications of these algebras
• Crucial for understanding the broader landscape of Non-associative Algebra

Octonions

• Non-associative division algebra over the real numbers
• 8-dimensional algebra with a basis of 8 unit octonions
• Satisfies power-associativity but not full associativity
• Generalizes complex numbers and quaternions
• Applications in theoretical physics (string theory, M-theory)
• Multiplication table defines the octonion product

Jordan algebras

• Commutative power-associative algebras
• Satisfy the Jordan identity: $(x^2y)x = x^2(yx)$
• Originally developed to formalize quantum mechanics observables
• Special Jordan algebras derived from associative algebras
• Exceptional Jordan algebras not derived from associative algebras (Albert algebras)
• Connection to projective geometry and quantum information theory

Flexible algebras

• Satisfy the flexibility condition: $(xy)x = x(yx)$ for all elements x and y
• Generalize associative and alternative algebras
• Include Lie-admissible algebras as a subclass
• Power-associativity follows from flexibility in characteristic ≠ 2, 3
• Examples include Jordan algebras and some genetic algebras

Power-associativity conditions

• Conditions defining power-associative algebras form the foundation of their theory
• Understanding these conditions essential for analyzing and classifying power-associative structures
• Crucial concept in Non-associative Algebra for generalizing associativity

Degree conditions

• Third-degree condition: $(xx)x = x(xx)$ for all elements x
• Fourth-degree condition: $(xx)(xx) = (x(xx))x$ for all elements x
• Higher-degree conditions ensure consistency of powers beyond the fourth power
• Sufficient conditions for power-associativity in characteristic 0 or sufficiently large
• Interplay between degree conditions and characteristic of the base field

Linearization techniques

• Method to derive equivalent conditions by replacing variables with linear combinations
• Allows for the study of power-associativity using multilinear identities
• Partial linearization replaces some occurrences of a variable
• Full linearization replaces all occurrences of a variable
• Useful for proving theorems and deriving new identities in power-associative algebras

Relation to other algebras

• Power-associative algebras occupy a unique position in the hierarchy of algebraic structures
• Understanding their relationships to other algebra types crucial for broader context in Non-associative Algebra
• Comparisons highlight the distinctive features and generalizations of power-associative algebras

vs Associative algebras

• Associative algebras form a proper subclass of power-associative algebras
• Power-associative algebras relax the full associativity condition
• Associative algebras satisfy $(ab)c = a(bc)$ for all elements a, b, c
• Power-associative algebras only require associativity for powers of the same element
• Examples of power-associative algebras that are not associative (octonions, Jordan algebras)

vs Alternative algebras

• Alternative algebras satisfy $(xx)y = x(xy)$ and $(yx)x = y(xx)$ for all elements x, y
• All alternative algebras are power-associative, but not vice versa
• Alternative algebras include associative algebras and octonions
• Power-associative algebras allow for more general structures than alternative algebras
• Flexibility holds in alternative algebras but not necessarily in all power-associative algebras

Power-associative identities

• Identities characterize and define the behavior of power-associative algebras
• Understanding these identities crucial for proving theorems and analyzing structures
• Form the foundation for studying power-associative algebras in Non-associative Algebra

Fourth-degree identity

• Fundamental identity for power-associative algebras: $(xx)(xx) = (x(xx))x$
• Ensures consistency of the fourth power regardless of parenthesization
• Equivalent to the condition that all polynomials of degree 4 in one variable associate
• Can be derived from third-degree associativity in characteristic ≠ 2, 3
• Plays a crucial role in the classification of power-associative algebras

Higher-degree identities

• Generalize the fourth-degree identity to higher powers
• Ensure consistency of all powers beyond the fourth power
• Can be expressed as multilinear identities through linearization techniques
• Examples include fifth-degree identity: $((xx)x)(xx) = (xx)((xx)x)$
• Higher-degree identities become increasingly complex and challenging to verify

Subalgebras and ideals

• Subalgebras and ideals form essential building blocks for understanding algebraic structures
• Studying these substructures crucial for analyzing and classifying power-associative algebras
• Important concepts in Non-associative Algebra for decomposing and constructing algebras

Power-associative subalgebras

• Subalgebras of power-associative algebras that maintain power-associativity
• Closed under addition, scalar multiplication, and algebra multiplication
• Generated by a single element (one-generated subalgebras always power-associative)
• May possess additional properties not present in the parent algebra
• Examples include the real subalgebra of octonions and special Jordan subalgebras

Ideals in power-associative algebras

• Subspaces closed under multiplication by arbitrary elements from either side
• Play a crucial role in the structure theory of power-associative algebras
• Allow for the construction of quotient algebras
• Nil ideals consist of nilpotent elements and are important for classification
• Proper ideals in simple power-associative algebras must be zero

Representation theory

• Representation theory studies algebras through their actions on vector spaces
• Crucial for understanding the structure and properties of power-associative algebras
• Bridges Non-associative Algebra with linear algebra and group theory

Linear representations

• Homomorphisms from power-associative algebras to endomorphism algebras of vector spaces
• Preserve power-associativity but not necessarily other algebraic properties
• Regular representation uses the algebra itself as the representation space
• Adjoint representation based on the left multiplication operators
• Challenges in developing a comprehensive representation theory for non-associative algebras

Modules over power-associative algebras

• Vector spaces equipped with an action of a power-associative algebra
• Generalize the notion of modules over associative algebras
• Right modules, left modules, and bimodules defined based on the action
• Homomorphisms between modules preserve the algebra action
• Challenges in defining tensor products and developing a full module theory