International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 65279, 30 pages
doi:10.1155/IJMMS/2006/65279
The dynamical U(n) quantum group
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.
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.