Vol. 8, No. 8, 2014

Download this article
Download this article For screen
For printing
Recent Issues

Volume 14
Issue 8, 2001–2294
Issue 7, 1669–1999
Issue 6, 1331–1667
Issue 5, 1055–1329
Issue 4, 815–1054
Issue 3, 545–813
Issue 2, 275–544
Issue 1, 1–274

Volume 13, 10 issues

Volume 12, 10 issues

Volume 11, 10 issues

Volume 10, 10 issues

Volume 9, 10 issues

Volume 8, 10 issues

Volume 7, 10 issues

Volume 6, 8 issues

Volume 5, 8 issues

Volume 4, 8 issues

Volume 3, 8 issues

Volume 2, 8 issues

Volume 1, 4 issues

The Journal
About the Journal
Editorial Board
Editors' Interests
Submission Guidelines
Submission Form
Editorial Login
Ethics Statement
ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Author Index
To Appear
Other MSP Journals
Quantum matrices by paths

Karel Casteels

Vol. 8 (2014), No. 8, 1857–1912

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.

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