Structure constant

In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two basis vectors in the algebra as a linear combination. Given the structure constants, the resulting product is bilinear and can be uniquely extended to all vectors in the vector space, thus uniquely determining the product for the algebra.

Structure constants are used whenever an explicit form for the algebra must be given. Thus, they are frequently used when discussing Lie algebras in physics, as the basis vectors indicate specific directions in physical space, or correspond to specific particles. Recall that Lie algebras are algebras over a field, with the bilinear product being given by the Lie bracket or commutator.

Given a set of basis vectors ${\displaystyle \{\mathbf {e} _{i}\}}$ for the underlying vector space of the algebra, the structure constants or structure coefficients ${\displaystyle c_{ij}^{\;k}}$ express the multiplication ${\displaystyle \cdot }$ of pairs of vectors as a linear combination:

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