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Primitive Ideals and Automorphisms of Quantum Matrices

Authors:	S Launois T H Lenagan

Institution:	(1) Maxwell Institute for Mathematical Sciences, School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, Scotland;(2) Present address: Institute of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, CT2 7NF, UK

Abstract:	Let $\mathbb{K}$ be a field and q be a nonzero element of $\mathbb{K}$ that is not a root of unity. We give a criterion for 〈0〉 to be a primitive ideal of the algebra ${\mathcal{O}}_{q} {\left( {M_{{m,n}} } \right)}$ of quantum matrices. Next, we describe all height one primes of ${\mathcal{O}}_{q} {\left( {M_{{m,n}} } \right)}$ ; these two problems are actually interlinked since it turns out that 〈0〉 is a primitive ideal of ${\mathcal{O}}_{q} {\left( {M_{{m,n}} } \right)}$ whenever ${\mathcal{O}}_{q} {\left( {M_{{m,n}} } \right)}$ has only finitely many height one primes. Finally, we compute the automorphism group of ${\mathcal{O}}_{q} {\left( {M_{{m,n}} } \right)}$ in the case where m ≠ n. In order to do this, we first study the action of this group on the prime spectrum of ${\mathcal{O}}_{q} {\left( {M_{{m,n}} } \right)}$ . Then, by using the preferred basis of ${\mathcal{O}}_{q} {\left( {M_{{m,n}} } \right)}$ and PBW bases, we prove that the automorphism group of ${\mathcal{O}}_{q} {\left( {M_{{m,n}} } \right)}$ is isomorphic to the torus ${\left( {\mathbb{K}*} \right)}^{{m + n - 1}}$ when m ≠ n and (m,n) ≠ (1, 3),(3, 1). This research was supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme and by Leverhulme Research Interchange Grant F/00158/X.

Keywords:	Quantum matrices Quantum minors Prime ideals Primitive ideals Automorphisms
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