Structure constants for a bilinear map

Definition
Suppose $$R$$ is a ring and $$A,B,C$$ are $$R$$-modules. A $$R$$-bilinear map $$f:A \times B \to C$$ can be described in terms of certain elements of $$R$$, termed its structure constants. These are defined in terms of choices of generating sets for $$R$$. Note that the structure constants are uniquely determined by the $$R$$-bilinear map only in the case that $$C$$ is a free $$R$$-module and the chosen generating set for $$C$$ is a freely generating set; however, even if it is not, the structure constants still uniquely determine the $$R$$-bilinear map.

The structure constant $$\lambda_{ij}^k$$ is defined as the coefficient of $$c_k$$ in the image $$f(a_i,b_j)$$ where $$a_i,b_j,c_k$$ are the elements of the generating sets of $$A,B,C$$ respectively.

The structure constants for a bilinear map play essentially the same role as the coefficients of a matrix do for a linear map. In fact, they can be viewed s matrix coefficients if we think of the bilinear map as a linear map $$A \otimes B \to C$$.

Structure constants are particularly useful when describing the multiplication in a $$k$$-algebra, or of a Lie algebra over $$k$$.

Some examples of structure constants are:


 * The Christoffel symbols give, at each point, the structure constants of the connection, which is a $$\R$$-bilinear map
 * The structure constants of a Lie algebra

In group theory, structure constants arise when we are looking at the representation ring or the character ring for instance.