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.
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×A→A, 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.:
- (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 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.
Two algebras A and B over K are isomorphic if there exists a bijective K-linear map f : A → B 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.
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.
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:
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).
Note however that several different sets of structure coefficients can give rise to isomorphic algebras.
- eiej = ci,jkek.
If you apply this to vectors written in index notation, then this becomes
- (xy)k = ci,jkxiyj.
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.
Kinds of algebras and examples
- 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[x] of all polynomials over K
- 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. 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-star-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:
Lie algebras, for which we require the Jacobi identity (xy)z + (yz)x + (zx)y = 0 and anticommutativity: xy + yx = 0. For these algebras the product is called the Lie bracket and is 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 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 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:
- 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 algebra s, 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:
- The Poisson algebras are considered in geometric quantization . They carry two multiplications, turning them into commutative algebras and Lie algebras in different ways.