Primitive Ideals and Automorphisms of Quantum Matrices

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,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 automorphism 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 automorphism group of O q (M m,n ) is isomorphic to the torus (K * ) m+n−1 when m = n and (m, n) = (1, 3), (3, 1).

Automorphisms and derivations of Uq(sl4+)Launois1, Lopes2 2007Journal of Pure and Applied Algebra12020View full textAdd to dashboardCite

“…By (17), u 2 = 0 ⇐⇒ u 4 = 0 and if this occurs thenq X 1 X 4 u 6 = q X 2 u 6 , on account of (19). If u 6 = 0 the latter impliesq X 1 X 4 = q X 2 , which is false as the X i form a PBW basis.…”

“…The hypotheses of [19,Prop. 3.2] can be slightly weakened to yield, with essentially the same proof, the following proposition.…”

“…To obtain the final contradiction, we just have to look at the degrees occurring in (19). Indeed, deg(X 1 X 4 u 6 ) = 2 + d 6 < 1 + d 4 + d 6 = deg(X 1 u 4 u 6 ); similarly, deg(X 1 u 4 X 6 ) < deg(X 1 u 4 u 6 ), deg(X 2 u 6 ) < deg(u 2 u 6 ) and deg(u 2 X 6 ) < deg(u 2 u 6 ).…”

The Prime Spectrum and Simple Modules Over the Quantum Spatial Ageing Algebra

“…The prime or/and primitive ideals of various quantum algebras (and their classification) are considered in [16,17,22,25,26,27,28,35,38,40,43]. The automorphism group of some (quantum) algebras are considered in [1,2,14,23,33,36,37,44].…”

The first Hochschild cohomology group of quantum matrices and the quantum special linear group

“…In an earlier paper [10] we have calculated the automorphism group of O q .M m;n / in the case that m ¤ n. Partial results were obtained for the square case O q .M n /, but technicalities prevented a resolution of the problem in this case. In a subsequent paper, we intend to use the results obtained in this paper to finish the calculation of the automorphism group of O q .M n /.…”