LD-stability for Goldie rings

Abstract

The lower transcendence degree , introduced by J. J Zhang, is an important non-commutative invariant in ring theory and non-commutative geometry strongly connected to the classical Gelfand-Kirillov transcendence degree. For LD-stable algebras , the lower transcendence degree coincides with the Gelfand-Kirillov dimension. We show that the following algebras are LD -stable and compute their lower transcendence degrees: rings of differential operators of affine domains, universal enveloping algebras of finite dimensional Lie superalgebras, symplectic reflection algebras and their spherical subalgebras , finite W -algebras of type A , generalized Weyl algebras over Noetherian domain (under a mild condition), some quantum groups . We show that the lower transcendence degree behaves well with respect to the invariants by finite groups, and with respect to the Morita equivalence. Applications of these results are given.