4.2 Alternative algebras
3 min read•august 21, 2024
Alternative algebras generalize associative algebras while maintaining key properties. They satisfy , exhibit , and possess the , bridging the gap between associative and fully non-associative structures.
establishes that subalgebras generated by two elements in an are associative. This allows for the application of techniques to certain subalgebras, simplifying computations and providing structural insights.
Definition of alternative algebras
- Alternative algebras form a crucial subset of non-associative algebras in the broader field of Non-associative Algebra
- These algebras generalize associative algebras while maintaining certain key properties
- Alternative algebras bridge the gap between associative and fully non-associative structures
Properties of alternative algebras
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- Satisfy the alternative laws: for all elements x and y
- Exhibit flexibility: for all elements x and y
- Possess the Moufang identity: for all elements x, y, and z
- Contain subalgebras generated by any two elements (associative)
- Satisfy the : powers of a single element associate
Examples of alternative algebras
- (non-associative division algebra over real numbers)
- (non-division alternative algebra)
- (finite-dimensional alternative algebras)
- of dimension 8 (generalization of octonions)
Artin's theorem
- Establishes a fundamental connection between alternative algebras and associativity
- Provides a powerful tool for analyzing the structure of alternative algebras
- Highlights the unique position of alternative algebras within non-associative algebra theory
Statement of Artin's theorem
- Any generated by two elements in an alternative algebra is associative
- Equivalently, for any three elements a, b, c in an alternative algebra:
- If two of them are equal, then their associator vanishes
- Generalizes to: the subalgebra generated by any two elements and their inverses is associative
Implications for alternative algebras
- Allows application of associative algebra techniques to certain subalgebras
- Simplifies computations involving pairs of elements
- Provides a basis for understanding the structure of alternative algebras
- Facilitates the study of representations and modules over alternative algebras
Moufang loops
- Non-associative algebraic structures closely related to alternative algebras
- Generalize groups by relaxing the associativity condition
- Play a crucial role in the theory of alternative algebras and non-associative structures
Connection to alternative algebras
- Multiplicative structure of an alternative division algebra forms a Moufang loop
- Alternative loop algebras over a field are alternative algebras
- Moufang identities in alternative algebras correspond to Moufang loop properties
- Study of provides insights into the structure of alternative algebras
Properties of Moufang loops
- Satisfy the Moufang identities: , ,
- Possess the : each element has a unique two-sided inverse
- Exhibit the : and $y(xx) = (yx)x$$ for all elements x and y
- Contain associative subloops generated by any two elements (Moufang theorem)
Octonions
- Non-associative division algebra over the real numbers
- Extend the quaternions and form the largest normed division algebra
- Serve as a fundamental example of an alternative algebra
Structure of octonions
- 8-dimensional algebra over the real numbers
- Basis elements: 1, e1, e2, e3, e4, e5, e6, e7
- Non-commutative and non-associative multiplication
- Possess a conjugation operation similar to complex numbers and quaternions
- Satisfy the alternative laws and Moufang identities
Multiplication table for octonions
- Compact representation using the Fano plane
- Multiplication rules:
- for i = 1, 2, ..., 7
- for i ≠ j
- where (i,j,k) is a triple in the Fano plane
- Multiplication is non-associative:
Key Terms to Review (22)
Alternative Algebra: Alternative algebra refers to a type of non-associative algebra where the product of any two elements is associative when either element is repeated. This means that in an alternative algebra, the identity \(x \cdot (x \cdot y) = (x \cdot x) \cdot y\) holds for all elements \(x\) and \(y\). This property creates a unique structure that connects to various mathematical concepts, showcasing its importance in areas like Lie algebras, composition algebras, and Jordan algebras.
Alternative laws: Alternative laws are specific properties that govern alternative algebras, which are algebraic structures where certain products of elements obey alternative identities. In these algebras, any element squared results in an expression that remains invariant under the permutation of the other elements involved in the operation. These laws create a framework for understanding how elements interact, particularly in systems that do not conform to conventional associative laws.
Alternative Property: The alternative property is a characteristic of certain algebraic structures where the operation satisfies specific conditions, allowing certain elements to behave in a predictable manner under multiplication. This property is crucial for understanding alternative rings and alternative algebras, which maintain a balance between associative and non-associative behaviors, ensuring certain expressions yield consistent results, even when the full associative law does not apply.
Artin's Theorem: Artin's Theorem states that in an alternative ring, every element can be expressed as a sum of an idempotent and a nilpotent element. This theorem is significant because it provides insight into the structure of alternative rings and highlights the relationship between idempotents and nilpotents within these algebraic systems. Understanding this theorem helps in studying properties of non-associative rings, alternative algebras, and their representations.
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.
Cayley-Dickson Algebras: Cayley-Dickson algebras are a class of non-associative algebras constructed by recursively doubling the dimension of algebras while modifying the multiplication operation. This process creates new algebras that may possess various algebraic properties, including being alternative or even non-commutative. The construction of Cayley-Dickson algebras connects deeply with concepts like non-associative rings and alternative algebras, showcasing their rich structure and applications in mathematical frameworks.
Flexibility: Flexibility in algebra refers to a specific property of non-associative algebras where the relation $a(bc) = (ab)c$ holds for all elements $a$, $b$, and $c$ in the algebra. This property allows for the manipulation of expressions without losing structure and is crucial in understanding how different types of algebras interact and behave, particularly in alternative algebras and Jordan algebras, where flexibility plays a significant role in their definitions and applications.
Idempotent Element: An idempotent element in a non-associative algebraic structure is an element 'e' such that when it is operated on by itself, it yields itself again, meaning that $$ e ullet e = e $$ for a given operation 'bullet'. This concept plays a vital role in understanding the structure and behavior of various algebraic systems, revealing important properties about their elements, such as how they interact and behave under certain operations.
Inverse Property: The inverse property refers to the principle that states for every element in a set, there exists another element that, when combined with it under a specific operation, results in the identity element of that operation. This concept is crucial in understanding how certain algebraic structures maintain balance and identity, particularly in alternative algebras where non-standard operations may apply.
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.
L. E. Dickson: L. E. Dickson was a prominent mathematician known for his contributions to the field of algebra, particularly in the study of alternative algebras and finite groups. His work established foundational concepts in the classification and theory of alternative algebras, which are structures where the associativity condition is relaxed, and he made significant strides in connecting these structures to other areas of mathematics.
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.
Moufang Identity: The Moufang identity refers to a specific type of algebraic identity that is satisfied by certain algebraic structures, particularly in the context of loops and alternative algebras. This identity has a crucial role in defining Moufang loops, which are a subclass of loops where certain conditions hold, providing a framework for understanding the relationship between non-associative operations. The importance of the Moufang identity extends to various areas, including alternative algebras and octonions, influencing their properties and applications, particularly in advanced mathematical theories like string theory.
Moufang Loops: Moufang loops are a special type of non-associative algebraic structure characterized by a loop (a set with a binary operation that has an identity element and where every element has an inverse) satisfying the Moufang identities. These identities ensure certain conditions about how elements combine, making them a bridge between groups and more complex algebraic structures. Understanding moufang loops involves exploring their properties and relationships to alternative rings and algebras, as well as their historical significance and applications in various mathematical areas, including differential geometry.
Multiplicative identity: The multiplicative identity is a special element in a set with a binary operation that, when multiplied by any element in that set, leaves the original element unchanged. In most number systems, this identity is represented by the number 1, meaning for any element 'a', the equation 'a * 1 = a' holds true. Understanding the multiplicative identity is crucial for grasping the underlying structure and properties of alternative algebras.
N. jacobson: N. Jacobson is a prominent mathematician known for his significant contributions to non-associative algebra, particularly in the classification of simple Malcev algebras and alternative algebras. His work has provided crucial insights into the structure and behavior of these algebras, which are essential in understanding broader algebraic systems and their applications, such as in coding theory. Jacobson's influence extends to the study of derivations and automorphisms, where he introduced various concepts that have shaped modern algebraic theory.
Nilpotent Element: A nilpotent element is an element 'a' in a ring such that there exists a positive integer 'n' where the nth power of 'a' equals zero, represented mathematically as $$a^n = 0$$. This concept plays a crucial role in understanding the structure of non-associative rings, alternative algebras, and special Jordan algebras, highlighting how certain elements behave under multiplication and their implications on ring properties and algebraic identities.
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.
Power-associative property: The power-associative property states that in a power-associative algebra, the result of taking products of elements and then raising them to a power is independent of how the products are grouped. This means that for any element 'a' in the algebra and any positive integer 'n', it holds that $(a^n)^m = a^{n imes m}$. This property allows for a clearer understanding of the structure and behavior of alternative algebras, emphasizing how associative behavior emerges at higher powers.
Split-octonions: Split-octonions are a type of non-associative algebra that extends the concept of octonions, characterized by their distinct multiplication properties and a non-positive signature. They are notable for having an alternative algebra structure, meaning that any two elements can be multiplied in a way that satisfies certain associative-like properties, albeit not for all three elements. This structure provides a fascinating insight into higher-dimensional algebra and its applications in various mathematical fields.
Subalgebra: A subalgebra is a subset of an algebraic structure that is closed under the operations defined in that structure and itself forms an algebraic structure. This means that a subalgebra retains the same operations and properties of the larger algebra while being contained within it, allowing for the study of smaller, manageable sections of complex algebras.
Zorn's Vector Matrix Algebras: Zorn's Vector Matrix Algebras refer to a class of algebras that combine vector spaces and matrix operations while adhering to the principles of alternative algebras. These structures allow for the manipulation of vectors through matrix representations, enabling the development of linear transformations and other algebraic operations within a defined framework. They are particularly significant in exploring various algebraic properties and theorems associated with non-associative systems.