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The Defining Relations of Quantum n × n Matrices
Author(s) -
Vancliff M.
Publication year - 1995
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/52.2.255
Subject(s) - automorphism , bijection , mathematics , combinatorics , symplectic geometry , subvariety , functor , affine variety , poisson manifold , context (archaeology) , variety (cybernetics) , discrete mathematics , pure mathematics , affine transformation , paleontology , biology , statistics
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.