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A Schur double centralizer theorem for cotriangular Hopf algebras and generalized Lie algebras. (English) Zbl 0818.16031
In the paper of M. Cohen, D. Fischman and S. Westreich [Proc. Am. Math. Soc. 122, No. 1, 19-29 (1994; see the preceding review Zbl 0818.16030)] Schur’s double centralizer theorem was generalized to the case of the symmetric monoidal category $$_H{\mathcal M}$$ over a triangular Hopf algebra $$H$$. In this paper similar results are obtained for the category of comodules $$^H{\mathcal M}$$ over a cotriangular Hopf algebra $$H$$. A certain version of Radford’s biproduct construction for Hopf algebras is used.
Many examples of cotriangular structures on the group ring $$kG$$ are given by bicharacters $$\alpha$$. As a corollary, a generalization of Schur’s double centralizer theorem for Lie color algebras $$\text{gl}_ \alpha (V)$$ is given. Classically, Schur’s theorem is used to study the representation theory of $$\text{GL}_ n (k)$$ by using the representation theory of $$S_ n$$. The explicit description of the centralizer of $$U( \text{gl}_ \alpha (V))$$ may give a more direct way of studying representations of $$\text{gl}_ \alpha (V)$$.

##### MSC:
 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 17B70 Graded Lie (super)algebras 16S40 Smash products of general Hopf actions 17B37 Quantum groups (quantized enveloping algebras) and related deformations 20G05 Representation theory for linear algebraic groups 19D23 Symmetric monoidal categories
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