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Journal of the London Mathematical Society 1995 52(2):255-262; doi:10.1112/jlms/52.2.255
© 1995 by London Mathematical Society
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© The London Mathematical Society

The Defining Relations of Quantum n x n Matrices

M. Vancliff

Department of Mathematics, University of Washington Seattle, Washington 98195, USA

Received 20 October 1993. Revision received 13 January 1994.

Let Oq(Mn) denote the coordinate ring of quantum n x n matrices. We show there exists a subvariety Pn of P(Mn) and an automorphism {sigma}n of Pn such that Oq(Mn) determines, and is determined by, the geometric data {Pn, {sigma}n}; the linear span of the defining relations of Oq(Mn) is the set of all those elements of Formula {otimes} Formula that vanish on the graph of {sigma}n. Moreover, if q2 != 1, the variety Pn is independent of q. Our main result is that there are two natural descriptions of Pn. Firstly, if q isin kx, there is a natural bijection between Pn and the point modules over Oq(Mn), and the automorphism {sigma}n is the shift functor on point modules. Secondly, since Oq(Mn) is a graded flat deformation of O1(Mn) the polynomial ring O(Mn), there is a homogeneous Poisson bracket on O(Mn) and an associated Poisson structure on P(Mn). In this context, if q2 != 1, the variety Pn consists of those points of P(Mn) which are the zero-dimensional symplectic leaves with respect to this Poisson structure.

Current Address: Department of Mathematics, UIA, 2610 Wilrijk, Antwerp, Belgium E-mail: vancliff{at}wins.uia.ac.be


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