Contravariant Form on Tensor Product of Highest Weight Modules

Abstract

We give a criterion for complete reducibility of tensor product $V\otimes Z$ of two irreducible highest weight modules $V$ and $Z$ over a classical or quantum semi-simple group in terms of a contravariant symmetric bilinear form on $V\otimes Z$. This form is the product of the canonical contravariant forms on $V$ and $Z$. Then $V\otimes Z$ is completely reducible if and only if the form is non-degenerate when restricted to the sum of all highest weight submodules in $V\otimes Z$ or equivalently to the span of singular vectors.

The previous version is split into two parts. The part concerning quantum vector bundles on projective spaces has been moved to a separate paper. The remining part on complete reducibility of tensor product is deeply revised and reformulated in terms of contravariant form. An error in former Lemma 3.10 corrected

References in corpus (1)

• Equivariant vector bundles over quantum projective spaces

Cited by in corpus (1)

• Equivariant vector bundles over quantum projective spaces