Primitive Ideals and Automorphisms of Quantum Matrices

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).