The Cartan product (Q1774316)

The Cartan product generalizes the symmetric product of two highest weight vectors in the following sense. Let \(L\) be a semisimple Lie algebra over the complex numbers, and suppose \(V\) and \(W\) are finite dimensional irreducible representations of \(L\) of highest weight \(\lambda\) and \(\mu\), respectively, and corresponding vectors \(v\) and \(w\). Then the Cartan product of \(V\) and \(W\) has highest weight \(\lambda+\mu\), and is generated by \(v\otimes w\). The author of the paper under review discusses the fundamental properties of the Cartan product. Furthermore he gives a lot of examples, and shows how one uses the Cartan product in the construction of important algebras.