1.4 Power-associative algebras
5 min read•august 21, 2024
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 , 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- 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 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:
- 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: 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: for all elements x
- Fourth-degree condition: 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 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 and 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:
- 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:
- 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
- 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
- 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
Key Terms to Review (18)
[x, y]: [x, y] denotes the commutator of two elements x and y in a non-associative algebra, representing the element formed by the difference of the product of these elements in different orders: $$[x, y] = xy - yx$$. This concept is important as it illustrates how non-associativity manifests through the interaction of elements and provides insight into the structure of power-associative algebras.
Associative algebra: An associative algebra is a type of algebraic structure that combines elements of both algebra and linearity, where the operations are both associative and compatible with a vector space structure. In an associative algebra, the multiplication operation satisfies the property that for any elements a, b, and c, the equation (a * b) * c = a * (b * c) holds true. This structure serves as a foundation for various algebraic theories, connecting to different types of algebras like alternative and power-associative algebras.
Enveloping Algebra: Enveloping algebra is an associative algebra that provides a way to represent a Lie algebra in a more manageable form, allowing for the study of its representations and module theory. This structure plays a crucial role in connecting Lie algebras to more familiar associative algebras, making it easier to analyze their properties and relationships through various mathematical frameworks.
Jacobson's Theorem: Jacobson's Theorem states that every finite-dimensional Jordan algebra can be represented as a subalgebra of a certain type of algebra known as a special Jordan algebra. This theorem provides insight into the structure of Jordan algebras and links them to other algebraic frameworks, particularly in understanding the classification and representation of these algebras.
Jordan Algebra: A Jordan algebra is a non-associative algebraic structure characterized by a bilinear product that satisfies the Jordan identity, which states that the product of an element with itself followed by the product of this element with any other element behaves in a specific way. This type of algebra plays a significant role in various mathematical fields, including radical theory, representation theory, and its connections to Lie algebras and alternative algebras.
K. A. McCrimmon: K. A. McCrimmon is a mathematician known for his work in the field of non-associative algebra, particularly power-associative algebras. He contributed significantly to the understanding of structures like alternative algebras and their properties, establishing foundational concepts that connect these algebras to various mathematical frameworks. His research has influenced how algebraists study and classify different algebraic systems.
Lie algebra: A Lie algebra is a vector space equipped with a binary operation called the Lie bracket, which satisfies bilinearity, antisymmetry, and the Jacobi identity. This structure is essential for studying algebraic properties and symmetries in various mathematical contexts, connecting to both associative and non-associative algebra frameworks.
Matrix algebra: Matrix algebra is a branch of mathematics that deals with the manipulation and analysis of matrices, which are rectangular arrays of numbers or functions. This area of study is foundational for understanding more complex structures in algebra, including special types of algebras where operations can be defined in terms of matrix representations. Key operations in matrix algebra include addition, multiplication, and finding determinants and inverses, which are vital for studying linear transformations and systems of equations.
Module theory: Module theory is the study of modules, which are generalizations of vector spaces where the scalars come from a ring instead of a field. This concept connects algebraic structures and has important implications in various areas of mathematics, especially in understanding power-associative algebras and their properties. Module theory also provides a framework to investigate the role of non-associative algebras in applications such as coding theory, where structures can be utilized for error correction and data transmission.
N. Bourbaki: N. Bourbaki is a pseudonym for a group of mainly French mathematicians who came together in the 1930s with the goal of reformulating mathematics on an extremely rigorous and formal basis. Their work has significantly influenced many areas of mathematics, particularly in the context of abstract algebra, including structures like non-associative rings and algebras.
Octonions: Octonions are a number system that extends the quaternions, forming an 8-dimensional non-associative algebra over the real numbers. They play a significant role in various areas of mathematics and physics, especially due to their unique properties such as being alternative but not associative, which allows for interesting applications in geometry and theoretical physics.
Peirce Decomposition: Peirce decomposition is a method used to break down Jordan algebras into simpler components based on their structure and properties. This decomposition reveals how these algebras can be understood in terms of simpler subalgebras, which is essential for studying the behavior of Jordan algebras in various mathematical contexts.
Power-associative Algebra: Power-associative algebra is an algebraic structure where every element can be expressed in terms of its powers alone, meaning that the subalgebra generated by any single element is associative. This property ensures that the operation defined on the algebra behaves consistently when applied to powers of the same element, allowing for simplified expressions and calculations in algebraic operations. In such algebras, concepts like exponentiation and polynomial identities play a significant role.
Power-associativity property: The power-associativity property states that for any element in a non-associative algebra, the repeated application of the operation to that element behaves in a consistent manner. In other words, if you take an element and repeatedly combine it with itself, the way you group the combinations does not affect the outcome. This property is crucial in understanding how operations work within power-associative algebras, where certain expressions can be simplified based on this consistent behavior.
Representation Theory: Representation theory studies how algebraic structures, like groups or algebras, can be represented through linear transformations of vector spaces. This theory provides a bridge between abstract algebra and linear algebra, revealing how these structures can act on spaces and enabling the application of linear methods to problems in abstract algebra.
Tits' Theorem: Tits' Theorem states that every finite-dimensional Jordan algebra is isomorphic to a subalgebra of an associative algebra. This theorem reveals deep connections between Jordan algebras and associative algebras, emphasizing the structural richness of Jordan algebras. Understanding Tits' Theorem is essential in studying various algebraic structures, as it bridges the concepts of power-associative algebras and computational methods within Jordan algebras.
Universal Enveloping Algebra: The universal enveloping algebra is a construction that associates a certain associative algebra to a given Lie algebra, allowing one to study representations of the Lie algebra through more manageable algebraic structures. It plays a crucial role in linking the properties of Lie algebras with those of associative algebras, providing a means to explore how Lie algebras can be represented in a broader algebraic context, especially in relation to power-associative algebras and various representation theories.
X^n: The expression $x^n$ represents a variable $x$ raised to the power of an integer $n$, where $n$ indicates the number of times $x$ is multiplied by itself. This concept is crucial in various algebraic structures, particularly in power-associative algebras, where the multiplication operation behaves in a way that ensures all elements can be expressed as powers of any given element. Understanding $x^n$ allows for deeper exploration of the properties and behaviors of elements within these algebras.