Structure constants

Remarks

By bilinearity, for $x = \sum_i x_i e_i$ and $y = \sum_j y_j e_j$ the product is $xy = \sum_{i,j,k} x_i y_j c_{ij}^k e_k$. Conversely, any family of $n^3$ scalars $c_{ij}^k$ defines an algebra structure on $K^n$ in this way. The algebra is commutative if and only if $c_{ij}^k = c_{ji}^k$ for all $i, j, k$. It is associative if and only if $\sum_l c_{ij}^l c_{lk}^m = \sum_l c_{jk}^l c_{il}^m$ for all $i, j, k, m$. Example: for $\R^3$ with the cross product and the standard basis, $c_{ij}^k = \varepsilon_{ijk}$, the Levi-Civita symbol; for instance $e_1 \times e_2 = e_3$ gives $c_{12}^3 = 1$. The structure constants depend on the basis: in another basis of the same algebra they are in general different numbers.