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HOMOLOGICAL PROPERTIES OF QUANTIZED COORDINATE RINGS OF SEMISIMPLE GROUPS
 

Summary: HOMOLOGICAL PROPERTIES OF QUANTIZED COORDINATE
RINGS OF SEMISIMPLE GROUPS
K.R. GOODEARL AND J.J. ZHANG
Abstract. We prove that the generic quantized coordinate ring Oq(G) is Aus-
lander-regular, Cohen-Macaulay, and catenary for every connected semisimple
Lie group G. This answers questions raised by Brown, Lenagan, and the first
author. We also prove that under certain hypotheses concerning the existence
of normal elements, a noetherian Hopf algebra is Auslander-Gorenstein and
Cohen-Macaulay. This provides a new set of positive cases for a question of
Brown and the first author.
0. Introduction
A guiding principle in the study of quantized coordinate rings has been that
these algebras should enjoy noncommutative versions of the algebraic properties
of their classical analogs. Moreover, based on the types of properties that have
been established, one also conjectures that quantized coordinate rings should enjoy
properties similar to the enveloping algebras of solvable Lie algebras. A property
of the latter type is the catenary condition (namely, that all saturated chains of
prime ideals between any two fixed primes should have the same length), which was
established for a number of quantized coordinate rings by Lenagan and the first
author [11]. However, among the quantized coordinate rings Oq(G) for semisimple

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics