Contravariant forms and extremal projectors

TL;DR

This paper presents a criterion for the complete reducibility of tensor products of irreducible modules over semi-simple quantum groups, using contravariant forms and extremal projectors for computational verification.

Contribution

It introduces a new criterion based on extremal projectors and contravariant forms to determine when tensor products are semi-simple, enhancing computational methods in quantum group representation theory.

Findings

Tensor product is semi-simple iff the restricted contravariant form is non-degenerate.

Expresses the restriction via extremal projector for computational feasibility.

Provides a practical criterion for reducibility in quantum group modules.

Abstract

Tensor product of irreducible modules of highest weight over a semi-simple quantum group is semi-simple if and only if a natural contravariant form is non-degenerate when restricted to the span of singular vectors. We express this restriction through the extremal projector of the quantum group providing a computationally feasible criterion for complete reducibility of tensor products.