Alternative algebra

In abstract algebra, an alternative algebra is a non-associative algebra in which the multiplication operation is not assumed to be associative, but is required to satisfy the alternative properties, meaning that the associator of any three elements is an alternating multilinear map.

Definition and Basic Properties

An algebra $A$ over a field is called alternative if it satisfies the left and right alternative identities for all elements $x$ and $y$ in $A$:

• Left alternative identity: $x(xy) = (xx)y$
• Right alternative identity: $(yx)x = y(xx)$

These conditions can be elegantly expressed using the associator, a trilinear map defined as $[x, y, z] = (xy)z - x(yz)$. An algebra is alternative if and only if its associator is an alternating function, meaning that it vanishes whenever any two of its arguments are equal. Consequently, swapping any two arguments changes the sign of the associator:
$[x, y, z] = -[y, x, z] = -[x, z, y] = -[z, y, x]$

Because the associator is alternating, it immediately follows that $[x, x, y] = 0$ (left alternative) and $[y, x, x] = 0$ (right alternative). Furthermore, the alternating property implies that the associator is totally antisymmetric, which provides a powerful tool for manipulating expressions in alternative algebras.

Examples

The most trivial examples of alternative algebras are associative algebras, where the associator is identically zero for all elements. In this case, both the left and right alternative identities are naturally satisfied.

The most prominent and historically significant example of a strictly non-associative alternative algebra is the algebra of octonions (also known as Cayley numbers). The octonions form an 8-dimensional normed division algebra over the real numbers. While they lack full associativity, their multiplication is alternative.

Another important class of examples is the Zorn vector-matrix algebras, which provide a construction for split-octonions and other alternative algebras using a matrix-like structure with vector entries and a modified multiplication rule involving cross products.

It is worth noting that not all algebras constructed via the Cayley-Dickson process are alternative. While the real numbers, complex numbers, quaternions, and octonions are alternative (the first three being fully associative), the 16-dimensional sedenions fail to satisfy the alternative identities and are therefore not alternative algebras.

Key Theorems and Identities

Alternative algebras possess several remarkable properties that make them highly structured despite the lack of full associativity.

Artin's Theorem: Formulated by Emil Artin, this fundamental theorem states that in any alternative algebra, the subalgebra generated by any two elements is associative. This implies that any expression involving only two variables can be evaluated without worrying about the placement of parentheses, greatly simplifying algebraic manipulations.

Power Associativity: As a direct consequence of Artin's theorem, alternative algebras are power-associative. This means that the subalgebra generated by a single element is associative, allowing for the unambiguous definition of integer powers $x^n$ for any element $x$.

Moufang Identities: Alternative algebras satisfy the Moufang identities, which are weak forms of associativity that hold for any elements $x, y,$ and $z$:

1. $z(x(zy)) = ((zx)z)y$
2. $x(z(yz)) = ((xz)y)z$
3. $(zx)(yz) = z((xy)z)$
4. $(zx)(yz) = (z(xy))z$

These identities are crucial in the study of Moufang loops, which are the algebraic structures formed by the invertible elements of an alternative algebra.

Structure Theory

The structure of alternative algebras is heavily constrained by their defining identities, leading to profound classification theorems.

Kleinfeld's Theorem: Erwin Kleinfeld proved that any simple alternative algebra over a field is either a simple associative algebra or is isomorphic to the algebra of octonions over some extension field. This result highlights the exceptional nature of the octonions, showing that they are the only strictly non-associative building blocks for simple alternative algebras.

Composition Algebras: Alternative algebras are closely related to composition algebras. By Hurwitz's theorem, the only real normed division algebras are the real numbers, complex numbers, quaternions, and octonions. The octonions represent the maximum dimension for a composition algebra, and their alternative nature is intrinsically linked to the properties of the norm.

Relationship to Other Algebras

Alternative algebras serve as a bridge between associative algebras and other classes of non-associative algebras, particularly Jordan and Lie algebras.

Jordan Algebras: If $A$ is an alternative algebra, one can define a new commutative product, the Jordan product, as $x \circ y = \frac{1}{2}(xy + yx)$. The algebra $A$ equipped with this new product forms a Jordan algebra. This connection is vital in quantum mechanics and the study of symmetric spaces, as it allows the extraction of commutative, non-associative structures from alternative ones.

Lie Algebras: Similarly, defining the commutator bracket as $[x, y] = xy - yx$ turns any alternative algebra into a Lie algebra. The Jacobi identity, which is required for a Lie algebra, is satisfied because the associator of an alternative algebra is alternating. This provides a natural way to construct Lie algebras from the automorphisms and derivations of alternative algebras, such as the exceptional Lie algebra $G_2$, which is the derivation algebra of the octonions.