Some Applications of Semitensor Product of Matrices in Algebra

A review on semitensor product (STP) of matrices is given. It is a generalization of the conventional matrix product for the case when the dimensions of the factor matrices do not satisfy the requirement. Using it, we investigate some structure-related properties of algebras. First, we consider when an algebra is a Lie algebra. The result reveals the topological structure of all finite-dimensional Lie algebras as the variety of a set of polynomial equations. Then we investigate the invertibility of algebras. Invertibility condition is expressed via STP. Finally, the tensor product of algebras is investigated.