7.2 Alternative rings

Alternative rings expand our understanding of algebraic structures beyond associative rings. They maintain key properties like distributivity and identity elements while allowing for non-associative multiplication, opening up new avenues for mathematical exploration.

These structures play a crucial role in non-associative algebra, connecting to concepts like Moufang loops and octonions. Alternative rings provide a framework for studying algebras that don't follow the associative property, bridging the gap between associative and fully non-associative structures.

Definition of alternative rings

• Alternative rings represent a class of non-associative algebraic structures generalizing associative rings
• These structures play a crucial role in Non-associative Algebra by providing a framework for studying algebras that do not satisfy the associative property

Properties of alternative rings

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• Satisfy the left and right alternative laws: $(x \cdot x) \cdot y = x \cdot (x \cdot y)$ and $(y \cdot x) \cdot x = y \cdot (x \cdot x)$
• Contain a multiplicative identity element (1)
• Exhibit flexible property: $(x \cdot y) \cdot x = x \cdot (y \cdot x)$
• Possess a unique left and right inverse for each non-zero element

Relation to associative rings

• Form a broader class that includes all associative rings
• Retain many properties of associative rings while allowing for non-associativity
• Preserve distributivity of multiplication over addition
• Maintain commutativity and associativity of addition

Artin's theorem

• Artin's theorem establishes a fundamental connection between alternative rings and associative subrings
• This result significantly impacts the study of Non-associative Algebra by providing insights into the structure of alternative rings

Statement of Artin's theorem

• Any subalgebra generated by two elements in an alternative algebra is associative
• Formally expressed as: For any elements a and b in an alternative algebra A, the subalgebra generated by {a, b} is associative
• Applies to both finite and infinite-dimensional alternative algebras

Implications for alternative rings

• Limits the extent of non-associativity in alternative rings
• Provides a powerful tool for analyzing the structure of alternative rings
• Allows for the application of associative ring theory techniques to certain subrings
• Helps identify associative subrings within alternative rings

Moufang loops

• Moufang loops represent a class of non-associative algebraic structures closely related to alternative rings
• These structures play a significant role in Non-associative Algebra by providing examples of non-associative multiplication

Connection to alternative rings

• Alternative rings with division form Moufang loops under multiplication
• Moufang identities hold in alternative rings
• Multiplicative structure of alternative division rings can be studied through Moufang loops
• Provide a geometric interpretation of alternative ring properties

Examples of Moufang loops

• Octonions under multiplication form a Moufang loop
• Projective and affine planes over octonions yield finite Moufang loops
• Paige loops constructed from finite fields
• Chein loops obtained by doubling groups

Octonions

• Octonions represent a fundamental example of alternative rings in Non-associative Algebra
• These hypercomplex numbers extend complex numbers and quaternions, providing a rich structure for study

Octonions as alternative rings

• Form an 8-dimensional alternative division algebra over the real numbers
• Satisfy the alternative laws but not the associative law
• Possess a multiplicative identity and multiplicative inverses for non-zero elements
• Exhibit the Moufang property

Properties of octonions

• Non-associative multiplication: $(a \cdot b) \cdot c \neq a \cdot (b \cdot c)$ in general
• Normed division algebra: $\|xy\| = \|x\| \cdot \|y\|$ for all octonions x and y
• Composition algebra: Satisfy the composition identity $N(xy) = N(x)N(y)$
• Contain subalgebras isomorphic to real numbers, complex numbers, and quaternions

Cayley-Dickson construction

• Cayley-Dickson construction provides a systematic method for generating higher-dimensional algebras
• This process plays a crucial role in Non-associative Algebra by creating a sequence of algebras with decreasing algebraic properties

Process of Cayley-Dickson construction

• Start with a base algebra A (typically the real numbers)
• Define new elements of the form (a, b) where a and b are elements of A
• Introduce a new multiplication rule: $(a, b)(c, d) = (ac - d*b, da + bc*)$
• Define conjugation as $(a, b)* = (a*, -b)$
• Iterate the process to obtain higher-dimensional algebras

Application to alternative rings

• Generates the sequence: real numbers → complex numbers → quaternions → octonions
• Produces alternative rings (octonions) from associative rings (quaternions)
• Creates sedenions and higher-dimensional algebras beyond octonions
• Illustrates the progressive loss of algebraic properties (commutativity, associativity, alternativity)

Identities in alternative rings

• Identities in alternative rings form a crucial aspect of their study in Non-associative Algebra
• These identities characterize the algebraic properties of alternative rings and distinguish them from other non-associative structures

Moufang identities

• Set of three equivalent identities that hold in all alternative rings
• Left Moufang identity: $(xy)(zx) = x((yz)x)$
• Right Moufang identity: $(xy)(zx) = ((xy)z)x$
• Middle Moufang identity: $(xz)(yx) = x(zy)x$
• Characterize alternative rings among more general non-associative rings

Bol identities

• Generalize Moufang identities to a broader class of non-associative structures
• Left Bol identity: $x(y(xz)) = (x(yx))z$
• Right Bol identity: $((zx)y)x = z((xy)x)$
• Hold in alternative rings but also in some non-alternative structures
• Provide a connection between alternative rings and Bol loops

Nucleus and center

• Nucleus and center represent important substructures within alternative rings
• These concepts play a significant role in Non-associative Algebra by revealing the associative and commutative parts of alternative rings

Nucleus of alternative rings

• Consists of elements that associate with all other elements in the ring
• Defined as: $N(R) = \{a \in R | (a,x,y) = (x,a,y) = (x,y,a) = 0 \text{ for all } x,y \in R\}$
• Forms an associative subring of the alternative ring
• Contains all elements that make the ring "locally" associative

Center of alternative rings

• Comprises elements that commute and associate with all other elements
• Defined as: $Z(R) = \{a \in N(R) | ax = xa \text{ for all } x \in R\}$
• Forms a commutative and associative subring
• Provides insight into the structure of the alternative ring

Representation theory

• Representation theory for alternative rings extends classical representation theory to non-associative structures
• This area of study in Non-associative Algebra aims to understand alternative rings through their actions on modules

Representations of alternative rings

• Define a homomorphism from the alternative ring to the endomorphism ring of a module
• Preserve the alternative laws and ring operations
• Can be studied through their restrictions to associative subrings
• Provide a way to analyze alternative rings using linear algebra techniques

Modules over alternative rings

• Generalize the concept of vector spaces to alternative rings
• Satisfy left and right alternative laws: $(rx)y = r(xy)$ and $(xy)r = x(yr)$ for ring elements r and module elements x, y
• Include both left and right modules, as well as bimodules
• Can be used to construct representations of alternative rings

Alternative algebras

• Alternative algebras extend the concept of alternative rings to include a scalar multiplication
• These structures form an important class of non-associative algebras in Non-associative Algebra

Definition of alternative algebras

• Consist of an alternative ring R and a field F
• Include a scalar multiplication: $F \times R \rightarrow R$
• Satisfy the alternative laws for multiplication
• Maintain distributivity and compatibility with scalar multiplication

Examples of alternative algebras

• Octonion algebra over the real numbers
• Split-octonion algebra
• Zorn's vector-matrix algebra
• Cayley-Dickson algebras of dimension 8 over various fields

Applications of alternative rings

• Alternative rings find applications in various fields beyond pure mathematics
• These applications demonstrate the relevance of Non-associative Algebra in solving real-world problems

In physics

• Describe particle interactions in some quantum mechanical models
• Model supersymmetry in string theory using octonions
• Represent internal symmetries in certain elementary particle theories
• Provide a framework for studying exceptional Lie groups

In computer science

• Optimize certain algorithms using octonion arithmetic
• Enhance error-correcting codes using properties of alternative rings
• Improve cryptographic protocols through non-associative structures
• Model certain types of neural networks using alternative algebras

Alternative vs associative rings

• Comparing alternative and associative rings reveals key differences and similarities
• This comparison is crucial in Non-associative Algebra for understanding the broader landscape of ring theory

Key differences

• Alternative rings allow for non-associative multiplication
• Associative rings satisfy the associative law for all triples of elements
• Alternative rings have a more complex subring structure due to Artin's theorem
• Representation theory for alternative rings requires additional considerations

Shared properties

• Both alternative and associative rings are distributive
• Addition remains associative and commutative in both structures
• Possess multiplicative identity elements
• Exhibit similar notions of ideals and homomorphisms