Alternative algebra

Alternative algebra is a branch of abstract algebra concerned with nonassociative algebras over a field that satisfy the left alternative law x(xy) = x^2 y and the right alternative law (yx)x = y x^2 for all elements x, y in the algebra.^[1] These laws ensure a partial relaxation of the associativity condition, where the associator (x, y, z) = (xy)z - x(yz) vanishes when two of the arguments are equal, specifically (x, x, y) = 0 and (x, y, y) = 0.^[1] Key properties of alternative algebras include flexibility, given by the identity (xy)x = x(yx), which follows directly from the alternative laws, and power-associativity, meaning that the subalgebra generated by any single element is associative.^[1] A fundamental result, known as Artin's theorem, states that the subalgebra generated by any two elements is associative, implying that alternative algebras are "nearly associative" in their binary-generated substructures.^[1] Finite-dimensional alternative algebras that are nil (every product of sufficiently many elements is zero) are necessarily nilpotent, providing structure theorems for their radical and decomposition.^[1] These algebras arise in the study of nonassociative structures, particularly in connection with division algebras and exceptional Lie groups. Prominent examples of alternative algebras include the octonions, an 8-dimensional real algebra that extends the quaternions via the Cayley-Dickson construction and serves as the largest normed division algebra over the reals. Unlike the reals, complexes, and quaternions, the octonions are nonassociative but remain alternative, enabling applications in physics, such as string theory^[2] and exceptional groups like G_2. By Hurwitz's theorem, the octonions are one of only four normed division algebras (alongside \mathbb{R}, \mathbb{C}, and \mathbb{H}), all of which are alternative. Other examples encompass Clifford-like constructions and free alternative algebras, which model universal nonassociative behaviors.^[1]

Definition and Fundamentals

Definition

An alternative algebra over a field F (or more generally over a commutative ring) is a vector space A equipped with a bilinear multiplication operation (x, y) \mapsto xy that satisfies the left alternative law x(xy) = (xx)y and the right alternative law (yx)x = y(xx) for all elements x, y \in A.^[3] These laws ensure a partial form of associativity focused on repeated multiplications by the same element, distinguishing alternative algebras from fully associative ones while excluding more general nonassociative structures. Associative algebras satisfy both alternative laws as a direct consequence of the associative identity (xy)z = x(yz), but alternative algebras allow nonassociativity in products involving distinct elements.^[3] In the unital case, where A possesses a two-sided unit element e such that ex = xe = x for all x \in A, the alternative laws imply that e distributes over multiplication in the sense e(xy) = (ex)y = x(ye) = (xy)e.^[3] Some key properties of alternative algebras, such as those related to power-associativity, are typically established under the assumption that the characteristic of the underlying field is not 2 or 3.^[4] The notion of alternative algebras originated in the 1930s with Ruth Moufang's foundational work on alternative rings and their structural properties.^[5]

The Associator

In non-associative algebras, the associator serves as a fundamental trilinear map that quantifies deviations from associativity. For an algebra A over a field, it is defined by

[x, y, z] = (xy)z - x(yz)

for all x, y, z \in A, mapping A^3 to A. This expression captures the extent to which the product of elements fails to satisfy the associative law.^[3] An algebra A is alternative if and only if the associator vanishes whenever any two of its arguments are equal, specifically [x, x, y] = 0 and [y, x, x] = 0 for all x, y \in A. These conditions are equivalent to the left and right alternative laws, respectively, providing a precise characterization of alternativity in terms of the associator.^[3] In alternative algebras, the associator exhibits strong symmetry properties: it is alternating, meaning it vanishes if any two arguments are identical and changes sign under odd permutations of its arguments, such as [x, y, z] = -[y, x, z] = -[x, z, y]. This alternating nature follows directly from the alternativity conditions and underscores the structured non-associativity in such algebras.^[3] The connection to alternativity is particularly evident in cases involving repeated elements. For instance, setting y = x in the left alternative condition yields [x, x, x] = 0, which implies (xx)x = x(xx); this demonstrates power-associativity for cubes, ensuring that triple products of a single element are well-defined regardless of parenthesization.^[3]

Key Properties

Power-Associativity and Artin's Theorem

In alternative algebras over fields of characteristic not 2, power-associativity holds, meaning that the subalgebra generated by any single element x is associative. This implies that powers of x satisfy x^m x^n = x^{m+n} unambiguously for all nonnegative integers m and n.^[3] To see this, consider the associator [x, y, z] = (xy)z - x(yz), which vanishes under alternativity when two arguments coincide, such as [x, x, x] = 0 by the left alternative identity x(xy) = (xx)y. Higher powers associate by induction: assume associativity up to degree n, then use alternativity to expand [x^k, x, x^{n-k}] and show it vanishes, ensuring the subalgebra spanned by \{x, x^2, \dots\} is associative.^[6]^[3] Artin's theorem extends this to two generators: in an alternative algebra over a field of characteristic not 2, the subalgebra generated by any two elements is associative. The result, due to Emil Artin, was first published by Max Zorn in 1930. This result characterizes alternative algebras, as the converse also holds. The proof proceeds by showing that all multilinear products in the two-generated subalgebra can be rearranged associatively using the alternative identities to control associators involving powers of the generators.^[6]^[3] A key implication is that alternative algebras exhibit local associativity, with every two-generated subalgebra being associative (and a division algebra if the parent algebra has no zero divisors). This locality underpins many structural results, such as the classification of finite-dimensional alternative division algebras. These properties rely on the characteristic not being 2, as the alternative identities involve even permutations that fail in characteristic 2.^[6]^[3]

Moufang Identities

Alternative algebras satisfy the Moufang identities, which are a set of identities that generalize aspects of associativity. Specifically, for all elements x, y, z in the algebra, the identities

(xyx)z = x(y(xz)), \quad z(xyx) = ((zx)y)x, \quad (xy)(zx) = x(yz)x

hold. These identities ensure that products involving repeated elements behave in a controlled manner, facilitating computations in nonassociative settings.^[7]^[3] Alternative algebras are flexible, satisfying the identity (xy)x = x(yx) for all x, y. This flexibility follows directly from the alternative laws, as the associator (x, y, x) = 0. In the unital case, the set of invertible elements under multiplication forms a Moufang loop, inheriting these identities and providing a loop-theoretic structure embedded within the algebra. The Moufang identities in alternative algebras derive from the core alternative laws and the structure's properties, as established in foundational treatments of nonassociative algebras. Historically, these identities emerged from Ruth Moufang's 1931 investigation into alternative division rings, where she analyzed the multiplicative structure of non-Desarguesian projective planes and derived key identities for their coordinate systems.^[8] This work laid the groundwork for understanding alternative algebras beyond associative cases.^[8]

Examples

Alternative Algebras

Alternative algebras include several prominent examples arising from classical constructions in nonassociative algebra. The quaternion algebra \mathbb{H} over the real numbers is a 4-dimensional division algebra that is associative and thus alternative by definition, serving as a foundational case where alternativity holds trivially due to full associativity. The octonions \mathbb{O} provide a key nonassociative example, forming an 8-dimensional division algebra over the reals that satisfies the alternative property despite failing general associativity. Constructed via the Cayley-Dickson process applied to the quaternions, the octonions feature a multiplication table derived from this doubling procedure, ensuring left and right alternativity for all elements.^[9] The Cayley-Dickson construction extends beyond the octonions to produce higher-dimensional algebras, yielding alternative structures up to dimension 8; however, the subsequent 16-dimensional sedenions, while power-associative, introduce zero divisors and fail alternativity. These sedenions retain some structural similarities to their predecessors but lose the division property and the alternative identities. Matrix algebras over alternative division rings offer further examples through Zorn's vector-matrix construction, which defines a nonstandard multiplication on 2×2 "matrices" with entries from the octonions or related rings, resulting in an alternative algebra of dimension 32 that preserves alternativity despite the underlying nonassociativity. This construction, originally used to realize the split octonions, generalizes to produce alternative structures mimicking matrix multiplication over nonassociative bases.^[9]^[10] Certain polynomial algebras over alternative bases also yield alternative structures, where nonassociative multiplication is defined on polynomials with coefficients in an alternative algebra like the octonions, ensuring the overall algebra satisfies left and right alternative laws while allowing for noncommutative and nonassociative terms.^[11]

Non-Alternative Algebras

Lie algebras provide a prominent class of nonassociative algebras that generally fail to satisfy the alternative laws. In a Lie algebra, the multiplication is given by the Lie bracket, which is bilinear and skew-symmetric, ensuring that [x, x] = 0 for all x, but the right alternative identity [x, y, y] = 0 does not hold in general. For instance, consider the special linear Lie algebra \mathfrak{sl}(2, \mathbb{R}) with standard basis elements H = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, X = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, and Y = \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, where the bracket is the commutator. Here, [X, Y] = H and [H, Y] = -2Y, so the associator [X, Y, Y] = [[X, Y], Y] - X [Y, Y] = [H, Y] = -2Y \neq 0.^[12] Jordan algebras offer another category where alternativity does not always hold, despite all Jordan algebras being power-associative. While certain subclasses, such as spin factor algebras, are alternative, the general case fails. The exceptional Jordan algebra, also known as the Albert algebra and realized as the 27-dimensional algebra of 3×3 Hermitian matrices over the octonions with symmetrized multiplication, is not alternative. This follows from the fact that it is exceptional and non-special: special Jordan algebras arise as subspaces of associative algebras under the Jordan product, which would imply alternativity for commutative cases, but the Albert algebra cannot be embedded in this way.^[13] A concrete low-dimensional example of a non-alternative algebra is the 3-dimensional Lie algebra corresponding to the cross product on \mathbb{R}^3, with basis e_1, e_2, e_3 and multiplication defined by e_1 e_2 = e_3, e_2 e_1 = -e_3, e_2 e_3 = e_1, e_3 e_2 = -e_1, e_3 e_1 = e_2, e_1 e_3 = -e_2, and all squares zero. This structure satisfies the Jacobi identity but fails alternativity, as [e_1, e_2, e_2] = (e_1 e_2) e_2 - e_1 (e_2 e_2) = e_3 e_2 = -e_1 \neq 0. Power-associative algebras, which ensure that powers of any element associate, do not necessarily satisfy alternativity. The exceptional Jordan algebra serves as such an example, being power-associative by the Jordan identity but failing the alternative laws, as detailed above. This distinction highlights that alternativity imposes stricter conditions beyond mere power-associativity.^[13]

Occurrences and Applications

In Composition Algebras

Composition algebras are unital algebras equipped with a nondegenerate quadratic form N: A \to K, where K is the base field, satisfying the composition property N(xy) = N(x)N(y) for all x, y \in A.^[14] This structure ensures that the algebra preserves the norm under multiplication, making it a central object in the study of normed algebras. All finite-dimensional composition algebras are alternative, meaning they satisfy the alternative laws x(xy) = (xx)y and (yx)x = y(xx) for all elements.^[14] This property follows from the norm's multiplicativity and the nondegeneracy of the quadratic form, as established in standard classifications.^[15] Over the real numbers, Hurwitz's theorem provides a complete classification: the only finite-dimensional composition algebras exist in dimensions 1, 2, 4, and 8.^[16] These are, up to isomorphism, the real numbers \mathbb{R} (dimension 1, associative), the complex numbers \mathbb{C} (dimension 2, associative), the quaternions \mathbb{H} (dimension 4, associative), and the octonions \mathbb{O} (dimension 8, alternative but nonassociative).^[14] The octonions serve as the prime example of a nonassociative alternative composition algebra, equipped with the Euclidean norm N(x) = \sum x_i^2, which satisfies the required multiplicativity.^[14] In all these cases, the algebras are division algebras, meaning every nonzero element has a multiplicative inverse. Over the complex numbers, finite-dimensional composition algebras are also alternative, but the classification differs due to the algebraic closure of \mathbb{C}. They include structures like \mathbb{C} \oplus \mathbb{C} (dimension 2, split form), the matrix algebra M_2(\mathbb{C}) (dimension 4, associative), and split octonion algebras (dimension 8, alternative), all preserving the composition property for suitably defined norms.^[14] However, unlike the real case, higher-dimensional examples beyond dimension 8 do not arise as composition algebras, consistent with generalizations of Hurwitz's theorem. Higher-dimensional algebras constructed via the Cayley-Dickson process, such as the sedenions (dimension 16), introduce zero divisors and fail to be composition algebras, as the norm no longer satisfies N(xy) = N(x)N(y) for all elements. Moreover, unlike the octonions, the sedenions are not alternative, violating the left and right alternative laws.^[18] This breakdown highlights the exceptional nature of dimensions up to 8 in the theory of composition algebras.

In Geometry and Loops

Alternative algebras play a pivotal role in the coordinatization of certain projective planes, particularly Moufang planes, which are translation planes satisfying specific geometric axioms derived from the Moufang identities. A projective plane is a Moufang plane if and only if it can be coordinatized by an alternative division ring, where the ring's non-associative multiplication enables the plane's structure without requiring full associativity.^[19] For instance, projective planes over the octonions form non-Desarguesian Moufang planes, as the octonions' alternative but non-associative nature violates Desargues' theorem, leading to geometries that differ fundamentally from those over associative division rings like fields or quaternions.^[20] A key theorem establishes that any alternative division ring yields a Moufang plane through standard coordinatization procedures, where points and lines are defined via the ring's elements and operations, preserving the plane's translation properties and Moufang axioms.^[21] This construction highlights how the alternative property suffices for the geometric incidences and collinearity relations in Moufang planes, bridging algebraic non-associativity with synthetic geometry. In loop theory, the multiplicative structure of an alternative division algebra—excluding zero—forms a Moufang loop, a quasigroup with the Moufang identities that supports unique solvability of equations and facilitates geometric constructions like those in projective planes.^[22] These loops enable the definition of coordinates and transformations in Moufang geometries, allowing for the embedding of algebraic operations into spatial configurations. An exceptional case arises with the octonion plane, known as the Cayley plane, which is the unique non-Desarguesian Moufang plane up to isomorphism, as all finite Moufang planes are Desarguesian by the Artin-Zorn theorem, leaving the infinite octonion example as the sole non-associative counterpart.^[23] Beyond projective geometry, alternative algebras underpin structures in exceptional Lie groups, such as G₂, F₄, and E₈, where octonionic constructions yield their defining representations and spinor varieties.^[24] Historically, these algebras have informed physics applications, including octonionic formulations in string theory for modeling higher-dimensional symmetries.^[24]