Algebra Over A Field - Structure Coefficients

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. 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 the resulting multiplication satisfies 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. These structure coefficients determine the multiplication in A via the following rule:

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).

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 c_i,jk, and their defining rule is written using the Einstein notation as

e_ie_j = c_i,jke_k.

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

(xy)k = c_i,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 set that spans 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.