Vol. 8, No. 8, 2014

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Quantum matrices by paths

Karel Casteels

Vol. 8 (2014), No. 8, 1857–1912
Abstract

We study, from a combinatorial viewpoint, the quantized coordinate ring of m × n matrices Oq(m,n(K)) over an infinite field K (often simply called quantum matrices).The first part of this paper shows that Oq(m,n(K)), which is traditionally defined by generators and relations, can be seen as a subalgebra of a quantum torus by using paths in a certain directed graph. Roughly speaking, we view each generator of Oq(m,n(K)) as a sum over paths in the graph, each path being assigned an element of the quantum torus. The Oq(m,n(K)) relations then arise naturally by considering intersecting paths. This viewpoint is closely related to Cauchon’s deleting derivations algorithm.

The second part of this paper applies the above to the theory of torus-invariant prime ideals of Oq(m,n(K)). We prove a conjecture of Goodearl and Lenagan that all such prime ideals, when the quantum parameter q is a non-root of unity, have generating sets consisting of quantum minors. Previously, this result was known to hold only when char(K) = 0 and with q transcendental over . Our strategy is to prove the stronger result that the quantum minors in a given torus-invariant ideal form a Gröbner basis.

Keywords
quantum algebra, quantum groups, quantum matrices, combinatorial noncommutative algebra, noncommutative Gröbner bases
Mathematical Subject Classification 2010
Primary: 16T20
Secondary: 16T30
Milestones
Received: 12 February 2014
Revised: 6 August 2014
Accepted: 12 September 2014
Published: 28 November 2014
Authors
Karel Casteels
School of Mathematics, Statistics and Actuarial Science
University of Kent
Canterbury
Kent, CT2 7NF
United Kingdom