International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 65279, 30 pages

The dynamical U(n) quantum group

Erik Koelink and Yvette Van Norden

DIAM, Faculteit Elektrotechniek, Wiskunde en Informatica (EWI), Technische Universiteit Delft, Postbus 5031, Delft 2600, GA, The Netherlands

Received 4 August 2005; Revised 1 May 2006; Accepted 10 May 2006

Copyright © 2006 Erik Koelink and Yvette Van Norden. 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.


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.