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Braided matrix structure of the Sklyanin algebra and of the quantum Lorentz group

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Abstract

Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided group versions of the standard quantum groupsU q (g). They have the same FRT generatorsl ± but a matrix braided-coproductΔL=L⊗L, whereL=l + Sl , and are self-dual. As an application, the degenerate Sklyanin algebra is shown to be isomorphic to the braided matricesBM q(2); it is a braided-commutative bialgebra in a braided category. As a second application, we show that the quantum doubleD(U q (sl 2)) (also known as the “quantum Lorentz group”) is the semidirect product as an algebra of two copies ofU q (sl 2), and also a semidirect product as a coalgebra if we use braid statistics. We find various results of this type for the doubles of general quantum groups and their semi-classical limits as doubles of the Lie algebras of Poisson Lie groups.

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Communicated by N. Yu. Reshetikhin

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Majid, S. Braided matrix structure of the Sklyanin algebra and of the quantum Lorentz group. Commun.Math. Phys. 156, 607–638 (1993). https://doi.org/10.1007/BF02096865

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