A non-associative algebra

Algebra over a field

In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...

(or distributive algebra) over a field (or a ring) K is a K-vector space (or more generally a module

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

) A equipped with a K-bilinear

Bilinear

Bilinear may refer to:* Bilinear sampling, a method in computer graphics for choosing the color of a texture* Bilinear form* Bilinear interpolation* Bilinear map, a type of mathematical function between vector spaces...

 map A × A → A. There are left and right multiplication maps
and . The enveloping algebra of A is the subalgebra of all K-endomorphisms of A generated by the multiplication maps.

An algebra is unital or unitary if it has a unit or identity element I with Ix = x = xI for all x in the algebra.

Examples

The best-known kinds of non-associative algebras are those that are nearly associative—that is, in which some simple equation constrains the differences between different ways of associating multiplication of elements. These include:

• Lie algebra
  Lie algebra
  In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
  
  s that are anticommutative
  Anticommutativity
  In mathematics, anticommutativity is the property of an operation that swapping the position of any two arguments negates the result. Anticommutative operations are widely used in algebra, geometry, mathematical analysis and, as a consequence, in physics: they are often called antisymmetric...
  
   , which require xx = 0 and the Jacobi identity
  Jacobi identity
  In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. Unlike for associative operations, order of evaluation is significant for operations satisfying Jacobi identity...
  
   (xy)z + (yz)x + (zx)y = 0. For these algebras the product is called the Lie bracket and is conventionally written [x,y] instead of xy. Examples include:
  • Euclidean space
    Euclidean space
    In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
    
     R³ with multiplication given by the vector cross product (with K the field R of real number
    Real number
    In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
    
    s)
  • Algebras of vector field
    Vector field
    In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
    
    s on a differentiable manifold
    Differentiable manifold
    A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
    
     (if K is R or the complex number
    Complex number
    A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
    
    s C) or an algebraic variety
    Algebraic variety
    In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
    
     (for general K);
  • Every associative algebra gives rise to a Lie algebra by using the commutator
    Commutator
    In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.-Group theory:...
    
     as Lie bracket. In fact every Lie algebra can either be constructed this way, or is a subalgebra of a Lie algebra so constructed.

• Jordan algebra
  Jordan algebra
  In abstract algebra, a Jordan algebra is an algebra over a field whose multiplication satisfies the following axioms:# xy = yx # = x ....
  
  s which are commutative
  Commutativity
  In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...
  
   and satisfy the Jordan property (xy)x² = x(yx²) and also xy = yx
  • every associative algebra over a field of characteristic
    Characteristic (algebra)
    In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
    
     other than 2 gives rise to a Jordan algebra by defining a new multiplication x*y = (1/2)(xy + yx). In contrast to the Lie algebra case, not every Jordan algebra can be constructed this way. Those that can are called special.

• Alternative algebra
  Alternative algebra
  In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have*x = y*x = y...
  
  s, which require that (xx)y = x(xy) and (yx)x = y(xx). The most important examples are the octonions (an algebra over the reals), and generalizations of the octonions over other fields. (Obviously all associative algebras are alternative.) Up to isomorphism the only finite-dimensional real alternative, division algebras (see below) are the reals, complexes, quaternions and octonions.

• Power-associative algebras, which require that x^mxⁿ = x^m+n, where m ≥ 1 and n ≥ 1. (Here we formally define xⁿ recursively as x(xⁿ⁻¹).) Examples include all associative algebras, all alternative algebras, and the sedenion
  Sedenion
  In abstract algebra, sedenions form a 16-dimensional non-associative algebra over the reals obtained by applying the Cayley–Dickson construction to the octonions...
  
  s.

• The hyperbolic quaternion
  Hyperbolic quaternion
  In the abstract algebra of algebras over a field, the hyperbolic quaternionq = a + bi + cj + dk, \quad a,b,c,d \in R \!is a mutated quaternion wherei^2 = j^2 = k^2 = +1 \! instead of the usual −1....
  
   algebra over R, which was an experimental algebra before the adoption of Minkowski space
  Minkowski space
  In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
  
   for special relativity
  Special relativity
  Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
  
  .

These properties are related by
1) associative implies alternative implies power associative;
2) commutative and associative implies Jordan implies power associative.
None of the converse implications hold.

More classes of algebras:

• Graded algebra
  Graded algebra
  In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a gradation ....
  
  s. These include most of the algebras of interest to multilinear algebra
  Multilinear algebra
  In mathematics, multilinear algebra extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of p-vectors and multivectors with Grassmann algebra.-Origin:In a vector space...
  
  , such as the tensor algebra
  Tensor algebra
  In mathematics, the tensor algebra of a vector space V, denoted T or T•, is the algebra of tensors on V with multiplication being the tensor product...
  
  , symmetric algebra
  Symmetric algebra
  In mathematics, the symmetric algebra S on a vector space V over a field K is the free commutative unital associative algebra over K containing V....
  
  , and exterior algebra
  Exterior algebra
  In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...
  
   over a given vector space
  Vector space
  A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
  
  . Graded algebras can be generalized to filtered algebra
  Filtered algebra
  In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory....
  
  s.

• Division algebra
  Division algebra
  In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field, in which division is possible.- Definitions :...
  
  s, in which multiplicative inverses exist or division can be carried out. The finite-dimensional alternative division algebras over the field of real numbers can be classified nicely. They are the real number
  Real number
  In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
  
  s (dimension 1), the complex number
  Complex number
  A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
  
  s (dimension 2), the quaternion
  Quaternion
  In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
  
  s (dimension 4), and the octonion
  Octonion
  In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H...
  
  s (dimension 8).

• Quadratic algebra
  Quadratic algebra
  In mathematics, a quadratic algebra is a filtered algebra generated by degree one elements, with defining relations of degree 2. It was pointed out by Yuri Manin that such algebras play an important role in the theory of quantum groups...
  
  s, which require that xx = re + sx, for some elements r and s in the ground field, and e a unit for the algebra. Examples include all finite-dimensional alternative algebras, and the algebra of real 2-by-2 matrices. Up to isomorphism the only alternative, quadratic real algebras without divisors of zero are the reals, complexes, quaternions, and octonions.

• The Cayley–Dickson algebras (where K is R), which begin with:
  • C (a commutative and associative algebra);
  • the quaternion
    Quaternion
    In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
    
    s H (an associative algebra);
  • the octonion
    Octonion
    In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold \mathbb O. There are only four such algebras, the other three being the real numbers R, the complex numbers C, and the quaternions H...
    
    s (an alternative algebra
    Alternative algebra
    In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have*x = y*x = y...
    
    );
  • the sedenion
    Sedenion
    In abstract algebra, sedenions form a 16-dimensional non-associative algebra over the reals obtained by applying the Cayley–Dickson construction to the octonions...
    
    s (a power-associative algebra, like all of the Cayley-Dickson algebras).

• The Poisson algebra
  Poisson algebra
  In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz' law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central in the study of quantum groups...
  
  s are considered in geometric quantization
  Geometric quantization
  In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory...
  
  . They carry two multiplications, turning them into commutative algebras and Lie algebras in different ways.

• Genetic algebra
  Genetic algebra
  In mathematical genetics, a genetic algebra is a algebra used to model inheritance in genetics. Some variations of these algebras are called train algebras, special train algebras, gametic algebras, Bernstein algebras, copular algebras, zygotic algebras, and baric algebras...
  
  s are non-associative algebras used in mathematical genetics.