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Quantum supergroups of $GL(n|m)$ type: differential forms, Koszul complexes and Berezinians

Volodymyr Lyubashenko·A. Sudbery
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Abstract

We introduce and study the Koszul complex for a Hecke $R$-matrix. Its cohomologies, called the Berezinian, are used to define quantum superdeterminant for a Hecke $R$-matrix. Their behaviour with respect to Hecke sum of $R$-matrices is studied. Given a Hecke $R$-matrix in $n$-dimensional vector space, we construct a Hecke $R$-matrix in $2n$-dimensional vector space commuting with a differential. The notion of a quantum differential supergroup is derived. Its algebra of functions is a differential coquasitriangular Hopf algebra, having the usual algebra of differential forms as a quotient. Examples of superdeterminants related to these algebras are calculated. Several remarks about Woronowicz's theory are made.

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Volodymyr Lyubashenko, A. Sudbery. 2008-02-20. Quantum supergroups of $GL(n|m)$ type: differential forms, Koszul complexes and Berezinians. https://arxiv.org/abs/hep-th/9311095

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