5 O ct 2 00 1 PRIME IDEALS INVARIANT UNDER WINDING AUTOMORPHISMS IN QUANTUM MATRICES

The main goal of the paper is to establish the existence of tensor product decompositions for those prime ideals P of the algebra A = Oq(Mn(k)) of quantum n× n matrices which are invariant under winding automorphisms of A, in the generic case (q not a root of unity). More specifically, every such P is the kernel of a map of the form A −→ A ⊗ A −→ A ⊗ A −→ (A/P)⊗ (A/P) where A → A ⊗ A is the comultiplication, A and A are suitable localized factor algebras of A, and P is a prime ideal of A invariant under winding automorphisms. Further, the algebras A, which vary with P , can be chosen so that the correspondence (P, P) 7→ P is a bijection. The main theorem is applied, in a sequel to this paper, to completely determine the winding-invariant prime ideals in the generic quantum 3 × 3 matrix algebra. Introduction This paper represents part of an ongoing project to determine the prime and primitive spectra of the generic quantized coordinate ring of n× n matrices, Oq(Mn(k)). Here k is an arbitrary field and q ∈ k is a non-root of unity. The current intermediate goal is to determine the prime ideals of Oq(Mn(k)) invariant under all winding automorphisms. (See below for a discussion of the relations between these winding-invariant primes and the full prime spectrum of Oq(Mn(k)).) Our main result exhibits a bijection between these primes and pairs of winding-invariant primes from certain ‘localized step-triangular factors’ of Oq(Mn(k)), namely the algebras R r = ( Oq(Mn(k))/〈Xij | j > t or i < rj〉 ) [X −1 r11, . . . , X −1 rtt ] R c = ( Oq(Mn(k))/〈Xij | i > t or j < ci〉 ) [X −1 1c1 , . . . , X −1 tct ] This research was partially supported by NSF research grants DMS-9622876 and DMS-9970159 and by NATO Collaborative Research Grant CRG.960250. Some of the research was done while both authors were visiting the Mathematical Sciences Research Institute in Berkeley during the winter of 2000, and they thank MSRI for its support. Typeset by AMS-TEX 1 2 K. R. GOODEARL AND T. H. LENAGAN where r = (r1, . . . , rt) and c = (c1, . . . , ct) are strictly increasing sequences of integers in the range 1, 2, . . . , n. In particular, since each R r and R − c can be presented as a skewLaurent extension of a localized factor algebra of Oq(Mn−1(k)), the above bijection can be used to obtain descriptions (as pullbacks of primes in the algebras R r ⊗ R − c ) of the winding-invariant primes of Oq(Mn(k)) from those of Oq(Mn−1(k)). In a sequel [7] to this paper, we follow the route just sketched to develop a complete list, with sets of generators, of the winding-invariant primes in Oq(M3(k)). The theorem indicated above depends on some detailed structural results concerning Oq(Mn(k)) and on some general work with primes in tensor product algebras invariant under group actions. First, we construct a partition of specOq(Mn(k)) indexed by pairs (r, c) as above, together with localized factor algebras Ar,c of Oq(Mn(k)), such that the portion of specOq(Mn(k)) indexed by (r, c) is Zariski-homeomorphic to specAr,c. We next prove that Ar,c is isomorphic to a subalgebra Br,c of R + r ⊗R − c , identify the structure of Br,c, and show that R + r ⊗R − c is a skew-Laurent extension of Br,c. Finally, with the help of some general work on tensor products, we prove that each winding-invariant prime of Br,c extends uniquely to a winding-invariant prime of R + r ⊗R − c , and that the latter primes can be uniquely expressed in the form (P ⊗ R c ) + (R + r ⊗ P ) where P (respectively, P) is a winding-invariant prime in R r (respectively, R − c ). We thus conclude that every winding-invariant prime of Oq(Mn(k)) can be uniquely expressed as the kernel of a map Oq(Mn(k)) −→ Oq(Mn(k))⊗Oq(Mn(k)) −→ R + r ⊗R − c −→ (R + r /P )⊗ (R c /P ), where the first arrow is comultiplication and the others are tensor products of localization or quotient maps. Algebraic background. The algebra Oq(Mn(k)) has standard generators Xij for i, j = 1, . . . , n and relations which we recall in (5.1)(a), along with the bialgebra structure of this algebra. The latter structure allows us to define left and right winding automorphisms corresponding to those characters (k-algebra homomorphisms Oq(Mn(k)) → k) which are invertible in Oq(Mn(k)) ∗ with respect to the convolution product (cf. [1, (I.9.25)] or [12, (1.3.4)] for the Hopf algebra case). It is well known that the collection of left (respectively, right) winding automorphisms of Oq(Mn(k)) forms a group isomorphic to the diagonal subgroup of GLn(k), whose action on the matrix of generators (Xij) is given by left (respectively, right) multiplication. We combine these actions to obtain an action of the group H = (k) × (k) on Oq(Mn(k)) by k-algebra automorphisms satisfying the rule (∗) (α1, . . . , αn, β1, . . . , βn).Xij = αiβjXij . One indication of the extent of the symmetry given by this action is the fact that there are only finitely many (actually, at most 2 2 ) primes of Oq(Mn(k)) invariant under H [9, (5.7)(i)]. The quoted result also shows that all H-primes of this algebra are prime, and so the H-primes coincide with the winding-invariant primes in Oq(Mn(k)). (Recall that the definition of an H-prime ideal is obtained from the standard ideal-theoretic definition of a prime ideal by restricting to H-invariant ideals.) WINDING-INVARIANT PRIME IDEALS IN QUANTUM MATRICES 3 In [9, Theorem 6.6], Letzter and the first author showed that the overall picture of the prime spectrum of an algebra with certain basic features like those of Oq(Mn(k)) is determined to a great extent by the primes invariant under a suitable group action. We quote the improved version of this picture presented in [1, Theorem II.2.13]. Let A be a noetherian algebra over an infinite field k, and let H = (k) be (the group of k-points of) an algebraic torus acting rationally on A by k-algebra automorphisms. Each H-prime of A is a prime ideal, and specA is the disjoint union of the sets specJ A := {P ∈ specA | ⋂