Algebra over a field

:This article is about a particular kind of vector space. For other uses of the term "algebra" see algebra (disambiguation).

Kinds of algebras and examples

A commutative algebra is one whose multiplication is commutative; an associative algebra is one whose multiplication is associative. These include the most familiar kinds of algebras.

Related Topics:
Commutative - Associative algebra - Associative

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• Associative algebras:
• the algebra of all n-by-n matrices over the field (or commutative ring) K. Here the multiplication is ordinary matrix multiplication.
• Group algebras, where a group serves as a basis of the vector space and algebra multiplication extends group multiplication
• the commutative algebra K of all polynomials over K
• algebras of functions, such as the R-algebra of all real-valued continuous functions defined on the interval , or the C-algebra of all holomorphic functions defined on some fixed open set in the complex plane. These are also commutative.
• Incidence algebras are built on certain partially ordered sets.
• algebras of linear operators, for example on a Hilbert space. Here the algebra multiplication is given by the composition of operators. These algebras also carry a topology; many of them are defined on an underlying Banach space which turns them into Banach algebras. If an involution is given as well, we obtain B-star-algebras and C*-algebras. These are studied in functional analysis.

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

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• Lie algebras, for which we require xx = 0 and the Jacobi identity (xy)z + (yz)x + (zx)y = 0. For these algebras the product is called the Lie bracket and is conventionally written instead of xy. Examples include:
• Euclidean space R³ with multiplication given by the vector cross product (with K the field R of real numbers);
• algebras of vector fields on a differentiable manifold (if K is R or the complex numbers C) or an algebraic variety (for general K);
• every associative algebra gives rise to a Lie algebra by using the commutator 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 algebras, for which we require (xy)x² = x(yx²) and also xy = yx.
• every associative algebra over a field of characteristic 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 algebras, for which we 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 algebras are the reals, complexes, quaternions and octonions.
• Power-associative algebras, for which we require that x^mxⁿ = x^m+n, where m≥1 and n≥1. (Here we formally define xⁿ recursively as x(x^n-1).) Examples include all associative algebras, all alternative algebras, and the sedenions.

More classes of algebras:

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• Division algebras, in which multiplicative inverses exist or division can be carried out. The finite-dimensional division algebras over the field of real numbers can be classified nicely.
• Quadratic algebras, for which we 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 quaternions H (an associative algebra);
• the octonions (an alternative algebra);
• the sedenions (a power-associative algebra, like all of the Cayley-Dickson algebras).
• The Poisson algebras are considered in geometric quantization. They carry two multiplications, turning them into commutative algebras and Lie algebras in different ways.