Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 026, 10 pages      arXiv:1709.08394

Contravariant Form on Tensor Product of Highest Weight Modules

Andrey I. Mudrov
Department of Mathematics, University of Leicester, University Road, LE1 7RH Leicester, UK

Received August 23, 2018, in final form March 25, 2019; Published online April 07, 2019

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.

Key words: highest weight modules; contravariant form; tensor product; complete reducibility.

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