
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:
In general, xy is the product of x and y, and the operation is called vector multiplication. However, several special kinds of algebras go by different names.
Algebras can also more generally be defined over any commutative ring K: we need a module A over K and a multiplication operation which satisfies the same identities as above; then A is a Kalgebra, and K is the base ring of A.
Vector multiplication is a bilinear operator from A × A to A, and is therefore completely determined by the multiplication of basis elements of A. Conversely, once a basis for A has been choses, 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 n^{3} structure coefficients c_{i,j,k}, which are scalars. These structure coefficients determine the multiplication in A via the following rule:
In mathematical physics, the structure coefficients are often written c_{i,j}^{k}, and their defining rule is written using the Einstein summation convention as
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.
Kinds of algebras and examples
The most important types of algebras are:
In geometric quantisation[?], one considers Poisson algebras[?], which carry two multiplications, turning them into commutative algebras and Lie algebras in different ways.
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