Summary: Version 1.21, February, 1998. To appear in Trans. Amer. Math. Soc.
THE DIXMIER­MOEGLIN EQUIVALENCE IN QUANTUM
COORDINATE RINGS AND QUANTIZED WEYL ALGEBRAS
K. R. Goodearl and E. S. Letzter
Abstract. We study prime and primitive ideals in a unified setting applicable to quanti­
zations (at nonroots of unity) of n \Theta n matrices, of Weyl algebras, and of Euclidean and
symplectic space. The framework for this analysis is based upon certain iterated skew poly­
nomial algebras A over infinite fields k of arbitrary characteristic. Our main result is the
verification, for A, of a characterization of primitivity established by Dixmier and Moeglin
for complex enveloping algebras. Namely, we show that a prime ideal P of A is primitive if
and only if the center of the Goldie quotient ring of A=P is algebraic over k, if and only if P
is a locally closed point -- with respect to the Jacobson topology -- in the prime spectrum of
A.
These equivalences are established with the aid of a suitable group H acting as automor­
phisms of A. The prime spectrum of A is then partitioned into finitely many ``H­strata'' (two
prime ideals lie in the same H­stratum if the intersections of their H­orbits coincide), and we
show that a prime ideal P of A is primitive exactly when P is maximal within its H­stratum.
This approach relies on a theorem of Moeglin­Rentschler (recently extended to positive char­
acteristic by Vonessen), which provides conditions under which H acts transitively on the set
of rational ideals within each H­stratum. In addition, we give detailed descriptions of the

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

Collections: Mathematics