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Contravariant Form on Tensor Product of Highest Weight Modules
Symmetry, integrability and geometry: methods and applications, Tome 15 (2019) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: highest weight modules; contravariant form; tensor product; complete reducibility. 
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     author = {Andrey I. Mudrov},
     title = {Contravariant {Form} on {Tensor} {Product} of {Highest} {Weight} {Modules}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a25/}
}
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Andrey I. Mudrov. Contravariant Form on Tensor Product of Highest Weight Modules. Symmetry, integrability and geometry: methods and applications, Tome 15 (2019). http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a25/

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