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PRIMITIVE IDEALS IN QUANTUM SL3 AND GL3 K.R. GOODEARL AND T.H. LENAGAN
 

Summary: PRIMITIVE IDEALS IN QUANTUM SL3 AND GL3
K.R. GOODEARL AND T.H. LENAGAN
Abstract. Explicit generating sets are found for all primitive ideals in the
generic quantized coordinate rings of SL3 and GL3 over an arbitrary alge-
braically closed field k. (Previously, generators were only known up to cer-
tain localizations.) These generating sets form polynormal regular sequences,
from which it follows that all primitive factor algebras of Oq(SL3(k)) and
Oq(GL3(k)) are Auslander-Gorenstein and Cohen-Macaulay.
0. Introduction
The primitive ideals of quantum SL3 were first classified by Hodges and Lev-
asseur in the case of Oq(SL3(C)) [6, Theorems 4.2.2, 4.3.1, 4.4.1 and 4.5]. (Here
and throughout, we consider only generic quantized coordinate rings, meaning that
quantizing parameters such as q are not roots of unity.) This classification was
extended to Oq(SLn(C)) in [7], to Oq(G) for semisimple groups G and q tran-
scendental in [9], and to multiparameter quantizations Oq,p(G) over C in [8]. In
these classifications, the primitive ideals appear as pullbacks of maximal ideals from
certain localizations, and it is only in the localizations that generating sets are cal-
culated (assuming the base field is algebraically closed). The only case in which
generating sets for primitive ideals have been explicitly determined is the easy case
of Oq(SL2(C)) [6, Theorem B.1.1]. Of course, once the primitive (or prime) ideals

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara
Lenagan, Tom - Department of Mathematics & Statistics, University of Edinburgh

 

Collections: Mathematics