Algebra over a field

This article is about a particular kind of vector space. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added For other uses of the term "algebra" see algebra (disambiguation). Algebra is a branch of Mathematics. Algebra may also mean Elementary algebra Abstract algebra

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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added A straightforward generalisation allows K to be any commutative ring. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property

(Some authors use the term "algebra" synonymously with "associative algebra", but this article does not. In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive Note also the other uses of the word listed in the algebra article. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. )

1 Definitions
2 Kinds of algebras and examples
3 Algebras and rings
4 Structure coefficients
5 See also

Definitions

Let K be a field, and let A be a vector space over K equipped with an additional binary operation from A × A to A, denoted here by juxtaposition (i. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two e. if x and y are any two elements of A, xy is the product of x and y). Then if the binary operation is bilinear, which means that the following identities hold for any three elements x, y, and z of A, and all elements ("scalars") a and b of K:

(x + y)z = xz + yz
x(y + z) = xy + xz
(ax)(by) = (ab)(xy)

we call A an algebra over K, we say that A is a K-algebra, and K is the base field of A. In Mathematics, a bilinear map is a function of two arguments that is linear in each The binary operation is often referred to as multiplication in A. According to the convention adopted in this article (see above), multiplication of elements of A is not necessarily associative. In Mathematics, associativity is a property that a Binary operation can have

More generally, algebras can be defined over an arbitrary commutative ring K instead of a field. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In this case A forms a K-module, with bilinear multiplication again satisfying the above identities. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In this case, A is a K-algebra, and K is the base ring of A.

Given K-algebras A and B, a K-algebra homomorphism is a K-linear map f: A → B such that f(xy) = f(x) f(y) for all x,y in A. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that The space of all K-algebra morphisms is frequently written as

$\mathbf{Hom}_{K\text{-alg}} (A,B).$

A K-algebra isomorphism is a bijective K-algebra morphism. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property For all practical purposes, isomorphic algebras differ only by notation.

Kinds of algebras and examples

A commutative algebra is one whose multiplication is commutative; an associative algebra is one whose multiplication is associative. In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive In Mathematics, associativity is a property that a Binary operation can have These include the most familiar kinds of algebras.

Associative algebras:
- the algebra of all n-by-n matrices over the field (or commutative ring) K. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally Here the multiplication is ordinary matrix multiplication. In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix
- Group algebras, where a group serves as a basis of the vector space and algebra multiplication extends group multiplication. In Mathematics, the group algebra is any of various constructions to assign to a Locally compact group an Operator algebra (or more generally a Banach In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element
- the commutative algebra K[x] of all polynomials over K. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations
- algebras of functions, such as the R-algebra of all real-valued continuous functions defined on the interval [0,1], or the C-algebra of all holomorphic functions defined on some fixed open set in the complex plane. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane In Mathematics, the complex plane is a geometric representation of the Complex numbers established by the real axis and the orthogonal imaginary axis These are also commutative.
- Incidence algebras are built on certain partially ordered sets. In Order theory, a field of Mathematics, an incidence algebra is an Associative algebra, defined for any locally finite Partially ordered set In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement
- algebras of linear operators, for example on a Hilbert space. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that This article assumes some familiarity with Analytic geometry and the concept of a limit. Here the algebra multiplication is given by the composition of operators. In Mathematics, a composite function represents the application of one function to the results of another These algebras also carry a topology; many of them are defined on an underlying Banach space which turns them into Banach algebras. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis In Mathematics, especially Functional analysis, a Banach algebra, named after Stefan Banach, is an Associative algebra A over the If an involution is given as well, we obtain B*-algebras and C*-algebras. B*-algebras are mathematical structures studied in Functional analysis. C*-algebras (pronounced "C-star" are an important area of research in Functional analysis, a branch of Mathematics. These are studied in functional analysis. For functional analysis as used in psychology see the Functional analysis (psychology article

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:

Lie algebras, for which we require xx = 0 and the Jacobi identity (xy)z + (yz)x + (zx)y = 0. 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 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 For these algebras the product is called the Lie bracket and is conventionally written [x,y] 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 Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which In Mathematics, the real numbers may be described informally in several different ways In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety In Mathematics, the commutator gives an indication of the extent to which a certain Binary operation fails to be Commutative. 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. In Mathematics, a Jordan algebra is defined in Abstract algebra as a (usually nonassociative) Algebra over a field with multiplication satisfying
- 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 Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's 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). In Abstract algebra, an alternative algebra is an algebra in which multiplication need not be Associative, only alternative. The most important examples are the octonions (an algebra over the reals), and generalizations of the octonions over other fields. In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real (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, for which we require that x^mxⁿ = x^m+n, where m≥1 and n≥1. In Abstract algebra, power associativity is a weak form of Associativity. (Here we formally define xⁿ recursively as x(x^n-1). ) Examples include all associative algebras, all alternative algebras, and the sedenions. In Abstract algebra, sedenions form a 16- dimensional algebra over the reals.

More classes of algebras:

Graded algebras. In Mathematics, in particular Abstract algebra, a graded algebra is an Algebra over a field (or Commutative ring) with an extra piece of structure These include most of the algebras of interest to multilinear algebra, such as the tensor algebra, symmetric algebra, and exterior algebra over a given vector space. In Mathematics, multilinear algebra extends the methods of Linear algebra. In Mathematics, the tensor algebra of a Vector space V, denoted T ( V) or T &bull( V) is the algebra In Mathematics, the symmetric algebra S ( V) (also denoted Sym ( V) on a Vector space V over a field In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added

Division algebras, in which multiplicative inverses exist or division can be carried out. In the field of Mathematics called Abstract algebra, a division algebra is roughly speaking an Algebra over a field in which division is possible The finite-dimensional alternative division algebras over the field of real numbers can be classified nicely. They are the real numbers (dimension 1), the complex numbers (dimension 2), the quaternions (dimension 4), and the octonions (dimension 8). In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real

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). Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real In Abstract algebra, an alternative algebra is an algebra in which multiplication need not be Associative, only alternative. In Abstract algebra, sedenions form a 16- dimensional algebra over the reals. In Abstract algebra, power associativity is a weak form of Associativity.

The Poisson algebras are considered in geometric quantization. In Mathematics, a Poisson algebra is an Associative algebra together with a Lie bracket that also satisfies Leibniz' law; that is the bracket In Mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given Classical theory. They carry two multiplications, turning them into commutative algebras and Lie algebras in different ways.

Algebras and rings

The definition of an associative K-algebra with unit is also frequently given in an alternative way. In this case, an algebra over a field K is a ring A together with a ring homomorphism

$\eta\colon K\to Z(A),$

where Z(A) is the center of A. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication The term center or centre is used in various contexts in Abstract algebra to denote the set of all those elements that commute with all other elements Since η is a ring morphism, then one must have either that A is the trivial ring, or that η is injective. This definition is equivalent to that above, with scalar multiplication

$K\times A \to A$

given by

$(k,a) \mapsto \eta(k) a.$

Given two such associative unital K-algebras A and B, a unital K-algebra morphism f: A → B is a ring morphism that commutes with the scalar multiplication defined by η, which one may write as

f (k a) = k f (a)

for all $k\in K$ and $a \in A$ . In Mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative Identity element (or unit) i In other words, the following diagram commutes:

$\begin{matrix} && K && \\ & \eta_A \swarrow & \, & \eta_B \searrow & \\ A && \begin{matrix} f \\ \longrightarrow \end{matrix} && B \end{matrix}$

Structure coefficients

For algebras over a field, the bilinear multiplication from A × A to A is completely determined by the multiplication of basis elements of A. Basis vector redirects here For basis vector in the context of crystals see Crystal structure. 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.

Thus, given the field K, any algebra can be specified up to isomorphism by giving its dimension (say n), and specifying n³ structure coefficients c_i,j,k, which are scalars. In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, the dimension of a Vector space V is the cardinality (i In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication These structure coefficients determine the multiplication in A via the following rule:

$\mathbf{e}_{i} \mathbf{e}_{j} = \sum_{k=1}^n c_{i,j,k} \mathbf{e}_{k}$

where e₁,. . . ,e_n form a basis of A. The only requirement on the structure coefficients is that, if the dimension n is infinite, then this sum must always converge (in whatever sense is appropriate for the situation). Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with

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. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined Specifically, lower indices are covariant indices, and transform via pullbacks, while upper indices are contravariant, transforming under pushforwards. Suppose that φ: M → N is a Smooth map between smooth manifolds M and N; then there is an associated Linear map from The notion of pushforward in Mathematics is "dual" to the notion of Pullback, and can mean a number of different but closely related things Thus, in mathematical physics, the structure coefficients are often written c_i,j^k, and their defining rule is written using the Einstein notation as

e_ie_j = c_i,j^ke_k. Mathematical physics is the scientific discipline concerned with the interface of Mathematics and Physics. In Mathematics, especially in applications of Linear algebra to Physics, the Einstein notation or Einstein summation convention is a notational

If you apply this to vectors written in index notation, then this becomes

(xy)^k = c_i,j^kxⁱy^j. Index notation is used in Mathematics to refer to the elements of matrices or the components of a vector.

If K is only a commutative ring and not a field, then the same process works if A is a free module over K. In Mathematics, a free module is a Free object in the category of modules Given a set S, a free module on S is a (particular construction 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, the expressions generator, generate, generated by and generating set can have several closely related technical meanings

Contents

Definitions

Kinds of algebras and examples

Algebras and rings

Structure coefficients

See also