On quantum methods in the representation theory of reductive Lie algebras (Q1896956)

The author applies quantum methods to the reduction problem, i.e., for a reductive Lie algebra \(\mathfrak g\) over \(\mathbb{C}\) and a subalgebra \({\mathfrak g}_0\) which is reductively embedded in \(\mathfrak g\) and whose system of simple roots is embedded in that of \(\mathfrak g\), the study of the decomposition of a simple \(\mathfrak g\)-module \(E(\lambda)\) with highest weight \(\lambda\) as a sum of simple \({\mathfrak g}_0\)-modules. In the main theorem a basis is constructed for a certain quotient of a crystal lattice of \(E(\lambda)\) which is orthonormal with respect to a bilinear form determined by \({\mathfrak g}_0\). No detailed proofs are given. In the English translation the author makes some corrections to the original Russian version.