The Defining Relations of Quantum n × n Matrices

Let q ( M n ) denote the coordinate ring of quantum n × n matrices. We show there exists a subvariety n of P( M n ) and an automorphism σ n of n such that q ( M n ) determines, and is determined by, the geometric data { n , σ n }; the linear span of the defining relations of q ( M n ) is the set of all those elements of M n ∗ ⊗ M n ∗ that vanish on the graph of σ n . Moreover, if q 2 ≠ 1, the variety n is independent of q . Our main result is that there are two natural descriptions of n . Firstly, if q ∈ k × , there is a natural bijection between n and the point modules over q ( M n ), and the automorphism σ n is the shift functor on point modules. Secondly, since q ( M n ) is a graded flat deformation of 1 ( M n ) the polynomial ring ( M n ), there is a homogeneous Poisson bracket on ( M n ) and an associated Poisson structure on P( M n ). In this context, if q 2 ≠ 1, the variety n consists of those points of P( M n ) which are the zero‐dimensional symplectic leaves with respect to this Poisson structure.