N ov 2 00 5 Primitive ideals and automorphisms of quantum matrices

Let q be a nonzero complex number 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,n) of quantum matrices. Next, we describe all height one primes of O q (M m,n); these two problems are actually interlinked since it turns out that 0 is a primitive ideal of O q (M m,n) whenever O q (M m,n) has only finitely many height one primes. Finally, we compute the au-tomorphism group of O q (M m,n) in the case where m = n. In order to do this, we first study the action of this group on the prime spectrum of O q (M m,n). Then, by using the preferred basis of O q (M m,n) and PBW bases, we prove that the automor-phism group of O q (M m,n) is isomorphic to the torus (C *) m+n−1 when m = n and