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Encyclopedia > Algebra over a field

In mathematics, an algebra over a field K, or a K-algebra, is a vector space A over K equipped with a compatible notion of multiplication of elements of A. A straightforward generalisation allows K to be any commutative ring. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ... The fundamental concept in linear algebra is that of a vector space or linear space. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...


(Some authors use the term "algebra" synonymously with "associative algebra", but Wikipedia does not. Note also the other uses of the word listed in the algebra article.) In mathematics, an associative algebra is a vector space (or more generally module) which also allows the multiplication of vectors in a distributive and associative manner. ... Algebra is a branch of mathematics which may be roughly characterized as a generalization and extension of arithmetic, in which symbols are employed to denote operations, and letters to represent number and quantity; it also refers to a particular kind of abstract algebra structure, the algebra over a field. ...

Contents

Definitions

To be precise, let K be a field, and let A be a vector space over K. Suppose we are given a binary operation A×AA, with the result of this operation applied to the vectors x and y in A written as xy. Suppose further that the operation is bilinear, i.e.: In mathematics, a binary operation, or binary operator, is a calculation involving two input quantities and one kind of a specific operation. ... In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ...

  • (x + y)z = xz + yz;
  • x(y + z) = xy + xz;
  • (ax)y = a(xy); and
  • x(by) = b(xy);

for all scalars a and b in K and all vectors x, y, and z. Then with this operation, A becomes an algebra over K, and K is the base field of A. The operation is called "multiplication". In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ...


In general, xy is the product of x and y, and the operation is called multiplication. However, the operation in several special kinds of algebras goes by different names.


Algebras can also more generally be defined over any commutative ring K: we need a module A over K and a bilinear multiplication operation which satisfies the same identities as above; then A is a K-algebra, and K is the base ring of A. In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In abstract algebra, a module is a generalization of a vector space. ...


Two algebras A and B over K are isomorphic if there exists a bijective K-linear map f : AB such that f(xy) = f(x) f(y) for all x,y in A. For all practical purposes, isomorphic algebras are identical; they just differ in the notation of their elements. In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...


Properties

For algebras over a field, the bilinear multiplication from A × A to A is completely determined by the multiplication of basis elements of A. Conversely, once a basis for A has been chosen, the products of basis elements can be set arbitrarily, and then extended in a unique way to a bilinear operator on A, i.e. so that the resulting multiplication will satisfy the algebra laws. In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V...


Thus, given the field K, any algebra can be specified up to isomorphism by giving its dimension (say n), and specifying n3 structure coefficients ci,j,k, which are scalars. These structure coefficients determine the multiplication in A via the following rule: In mathematics, the term up to xxxx is used to describe a situation in which members of an equivalence class can be regarded as a single entity for some purpose. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In mathematics, the dimension of a vector space V is the cardinality (i. ... The concept of a scalar is used in mathematics, physics, and computing. ...

where e1,...,en form a basis of A. The only requirement on the structure coefficients is that, if the dimension n is an infinite number, then this sum must always converge (in whatever sense is appropriate for the situation). Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... In mathematics, a series is a sum of a sequence of terms. ...


Note however that several different sets of structure coefficients can give rise to isomorphic algebras.


When the algebra can be endowed with a metric, then the structure coefficients are written with upper and lower indices, so as to distinguish their transformation properties under coordinate transformations. Specifically, lower indices are covariant indices, and transform via pullbacks, while upper indices are contravariant, transforming under pushforwards. Thus, in mathematical physics, the structure coefficients are often written ci,jk, and their defining rule is written using the Einstein notation as In mathematics, a metric space is a set (or space) where a distance between points is defined. ... In category theory, see covariant functor. ... This article discusses the pullback in differential geometry. ... Contravariant is a mathematical term with a precise definition in tensor analysis. ... In mathematics, the push forward (or pushforward) of a smooth map F : M → N between smooth manifolds at a point p is, in some sense, the best linear approximation of F near p. ... Mathematical physics is the scientific field in between mathematics and physics; it studies the problems inspired by physics within a mathematically rigorous framework. ... For other topics related to Einstein see Einstein (disambig) In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate equations or formulas. ...

eiej = ci,jke k.

If you apply this to vectors written in index notation, then this becomes Index notation is used in mathematics to refer to the elements of matrices or the components of a vector. ...

(xy)k = ci,jkx iyj.

If K is only a commutative ring and not a field, then the same process works if A is a free module over K. If it isn't, then the multiplication is still completely determined by its action on a generating set of A; however, the structure constants can't be specified arbitrarily in this case, and knowing only the structure constants does not specify the algebra up to isomorphism. In mathematics, a free module is a module having a free basis. ... In mathematics, a subset S of a algebraic structure G is a generating set of G (or G is generated by S) if the smallest subset of G that includes S and is closed under the algebraic operations on G is G itself. ...


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. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, an associative algebra is a vector space (or more generally module) which also allows the multiplication of vectors in a distributive and associative manner. ... In mathematics, associativity is a property that a binary operation can have. ...

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: For the square matrix section, see square matrix. ... This article gives an overview of the various ways to multiply matrices. ... In the theory of group representations, the group algebra is any of various constructions to assign to a group (either a locally compact topological group, or a group without a topology, i. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In elementary algebra, an interval is a set that contains every real number between two indicated numbers, and possibly the two numbers themselves. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In order theory, a field of mathematics, a locally finite partially ordered set is one for which every closed interval [a, b] = {x : a ≤ x ≤ b} within it is finite. ... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ... In functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space. ... B*-algebras are mathematical structures studied in functional analysis. ... C*-algebras are an important area of research in functional analysis. ... Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ...

  • 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 xmxn = xm+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 sedenions.

More classes of algebras: In mathematics, a Lie algebra (named after Sophus Lie, pronounced lee) is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... The Jacobi identity is the name for the following equation: [X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0 for all X,Y,Z. Lie algebras are the primary example of an algebra which satisfies the Jacobi identity. ... In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, the cross product is a binary operation on vectors in three dimensions. ... Please refer to Real vs. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in Euclidean space. ... In mathematics, a manifold M is a type of space, characterised in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ... For an electrical switch that periodically reverses the current see commutator (electric) In mathematics the commutator of two elements g and h of a group G is the element g −1 h −1 gh, often denoted by [ g, h ]. It is equal to the groups identity if and only... In mathematics, a Jordan algebra is defined in abstract algebra as an algebra over a field with multiplication satisfying the following axioms: (commutative law) (Jordan identity) Jordan algebras were first introduced by Pascual Jordan in quantum mechanics. ... The word characteristic has several meanings: In mathematics, see characteristic (algebra) characteristic function characteristic subgroup Euler characteristic method of characteristics In genetics, see characteristic (genetics). ... In abstract algebra, an algebra is called alternative if (xx)y=x(xy) and y(xx)=(yx)x for all x and y in the algebra, that is, if the multiplication is alternative. ... In mathematics, the octonions are a nonassociative extension of the quaternions. ... In abstract algebra, power associativity is a weak form of associativity. ... The sedenions form a 16-dimensional algebra over the reals obtained by applying the Cayley-Dickson construction to the octonions. ...

  • 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:

In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division is possible. ... Please refer to Real vs. ... In mathematics, the Cayley-Dickson construction produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In mathematics, the octonions are a nonassociative extension of the quaternions. ... In abstract algebra, an algebra is called alternative if (xx)y=x(xy) and y(xx)=(yx)x for all x and y in the algebra, that is, if the multiplication is alternative. ... The sedenions form a 16-dimensional algebra over the reals obtained by applying the Cayley-Dickson construction to the octonions. ... In abstract algebra, power associativity is a weak form of associativity. ... A Poisson algebra is an associative algebra together with a Lie bracket, satisfying Leibniz law. ... In mathematical physics, geometric quantization is a mathematical approach to define a quantum theory corresponding to a given classical theory in such a way that certain analogies between the classical theory and the quantum theory remain manifest, for example the similarity between the Heisenberg equation in the Heisenberg picture of...

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