# Primitive ideals and automorphisms of quantum matrices.

Launois, Stephane, Lenagan, T.H. (2007) Primitive ideals and automorphisms of quantum matrices. Algebras and Representation Theory, 10 (4). pp. 339-365. ISSN 1386-923X. (doi:10.1007/s10468-007-9059-0) (KAR id:2167)

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http://dx.doi.org/10.1007/s10468-007-9059-0

## Abstract

Let K be a field and q be a nonzero element of K that is not a root of unity. We give a criterion for (0) to be a primitive ideal of the algebra O-q(M-m,M-n) of quantum matrices. Next, we describe all height one primes of these two problems are actually interlinked since it turns out that (0) is a primitive ideal of O-q(M-m,M-n) whenever O-q(M-m,M-n) has only finitely many height one primes. Finally, we compute the automorphism group of O-q(M-m,M-n) in the case where m not equal n. In order to do this, we first study the action of this group on the prime spectrum of O-q(M-m,M-n). Then, by using the preferred basis of O-q(M-m,M-n) and PBW bases, we prove that the automorphism group of O-q(M-m,M-n) is isomorphic to the torus (K*)(m+n=1) when m not equal n and (m, n) not equal (1, 3) (3, 1).

Item Type: Article 10.1007/s10468-007-9059-0 quantum matrices; quantum minors; prime ideals; primitive ideals; automorphisms Q Science > QA Mathematics (inc Computing science) Faculties > Sciences > School of Mathematics Statistics and Actuarial Science Anna Thomas-4 19 Dec 2007 19:31 UTC 06 May 2020 03:00 UTC https://kar.kent.ac.uk/id/eprint/2167 (The current URI for this page, for reference purposes) https://orcid.org/0000-0001-7252-8515