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Interrelations between quantum groups and reflection equation (braided) algebras. (English) Zbl 0878.17012

It is shown that a differential complex \(\Omega_B\) over the braided matrix algebra \(\text{BM}_q(N)\) is a comodule algebra over the Hopf algebra \(\Omega_A\) which is a differential extension of \(\text{GL}_q(N)\). On other hand \(\Omega_A\) is a bicomodule over two versions \(\Omega_B\) and \(\overline\Omega_B\) of a braided differential complex. The results are true for an \(R\)-matrix of Hecke type. This leads the author to the conjecture that the differential algebras \(\Omega_A\) and \(\Omega_B\) are related by Majid’s transmutation procedure (like their zero-components \(\text{GL}_q(N)\) and \(\text{BM}_q(N)\)). As a geometrical application the author considers an \(\Omega_B\)-comodule algebra structure on the algebra generated by noncommutative one-form connections \(A\) and the curvature two-form \(F=\text{ d}A-A^2\).

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
46L85 Noncommutative topology
46L87 Noncommutative differential geometry
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References:

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