The dynamical <mml:math id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>U</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> quantum group

Koelink, Erik; Van Norden, Yvette

doi:https://doi.org/10.1155/IJMMS/2006/65279

International Journal of Mathematics and Mathematical Sciences

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Volume 2006 | Article ID 065279 | https://doi.org/10.1155/IJMMS/2006/65279

The dynamical U(n) quantum group

Erik Koelink¹and Yvette Van Norden¹

Received04 Aug 2005

Revised01 May 2006

Accepted10 May 2006

Published17 Aug 2006

Abstract

We study the dynamical analogue of the matrix algebra M(n), constructed from a dynamical R-matrix given by Etingof and Varchenko. 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 GL(n) quantum group associated to the dynamical R-matrix. We study a ∗-structure leading to the dynamical U(n) quantum group, and we obtain results for the canonical pairing arising from the R-matrix.

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Copyright © 2006 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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