Summary: quo82.tex August 26, 1998
QUANTUM n­SPACE AS A QUOTIENT OF CLASSICAL n­SPACE
K. R. Goodearl and E. S. Letzter
Abstract. The prime and primitive spectra of Oq (k n ), the multiparameter quantized coor­
dinate ring of affine n­space over an algebraically closed field k, are shown to be topological
quotients of the corresponding classical spectra, spec O(k n ) and maxO(k n ) ß k n , provided
the multiplicative group generated by the entries of q avoids \Gamma1.
Introduction
In the representation theory of a noncommutative ring A, the natural analog of the
maximal spectrum of a commutative ring is prim A, the set of primitive ideals, equipped
with the Zariski (Jacobson) topology. Thus, if A is a quantization of a classical coordinate
ring O(V ) over an algebraically closed field, one can view primA as a quantization of the
variety V . The question then naturally arises, how are primA and V related? Some ``piece­
wise'' relations are known in many cases. For instance, if G is a connected, semisimple,
complex algebraic group and H is a maximal torus of G, then various (generic) quantiza­
tions A of O(G) exhibit the following properties: H acts on A via automorphisms; there
are only finitely many H­orbits in prim A, and they are locally closed; each H­orbit in
prim A is homeomorphic to a torus; and each H­orbit in prim A is a set­theoretic quotient
of a locally closed subset of G that is stable under translation by H (see [8, 9, 11, 12,
10]). Similar pictures have been observed to hold for quantized coordinate rings of affine

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

Collections: Mathematics