X-Inner Automorphisms of Semi-Commutative Quantum Algebras

Abstract

Many important quantum algebras such as quantum symplectic space, quantum Euclidean space, quantum matrices, q-analogs of the Heisenberg algebra, and the quantum Weyl algebra are semi-commutative. In addition, enveloping algebras U(L+) of even Lie color algebras are also semi-commutative. In this paper, we generalize work of Montgomery and examine the X-inner automorphisms of such algebras. The theorems and examples in our paper show that for algebras R of this type, the non-identity X-inner automorphisms of R tend to have infinite order. Thus if G is a finite group of automorphisms of R, then the action of G will be X-outer and this immediately gives useful information about crossed products R∗tG.