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 $\mathbb {R}$-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.