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The dynamical \(U(n)\) quantum group. (English) Zbl 1163.17016

Summary: We study the dynamical analogue of the matrix algebra \(M(n)\), constructed from a dynamical \(R\)-matrix given by P. Etingof and A. Varchenko [Commun. Math. Phys. 205, No. 1, 19–52 (1999; Zbl 0943.17010)]. A left and a right corepresentation of this algebra, which can be seen as analogues of the exterior algebra representation, are defined and this defines dynamical quantum minor determinants as the matrix elements of these corepresentations. These elements are studied in more detail, especially the action of the comultiplication and Laplace expansions. Using the Laplace expansions we can prove that the dynamical quantum determinant is almost central, and adjoining an inverse the antipode can be defined. This results in the dynamical \(\text{GL}(n)\) quantum group associated to the dynamical \(R\)-matrix. We study a \(\ast \)-structure leading to the dynamical \(U(n)\) quantum group, and we obtain results for the canonical pairing arising from the \(R\)-matrix.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics

Citations:

Zbl 0943.17010
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References:

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