William Chin

Prime ideals in differential operator rings and crossed products of infinite groups

Abstract We use a version of the Martindale ring of quotients to study prime ideals in extensions of a ringR corresponding to two cases. IfG is a group of automorphisms ofR we form the crossed product R ∗ G . If g is a Lie algebra of derivations of R we have the twisted differential operator ring, denoted byR ∗ g (sometimes known as the “twisted” smash product R#t u ( g )). We obtain analogues of Incomparability for crossed products of nilpotent groups, and differential operator rings of solvable Lie algebras. In the case of crossed products, the incomparability result of D. S. Passman and M. Lorenz for the infinite cyclic group (also proved by G. Bergman) is generalized.

The Coradical Filtration for Quantized Enveloping Algebras

We determine the coradical filtration on a quantized enveloping algebra t/9(g) = U defined over Q(q). As applications, we determine the biideals of U and describe the group of Hopf algebra automorphisms of U.

Multiparameter quantum enveloping algebras

Let A = A(p, λ) be the multiparameter deformation of the coordinate algebra of n × n matrices as described by Artin, et al. (1991). Let U be the quantum enveloping algebra which is associated to A, in the sense of Faddeev, Reshetikhin and Takhtadzhyan. We prove a PBW theorem for U and establish a presentation by generators and relations, when λ is not a root of unity. Our approach depends on a cocycle twisting method which reduces many arguments to the standard one-parameter deformation.

Rings graded by polycyclic-by-finite groups

We use the duality between group gradings and group actions to study polycyclic-by-finite group-graded rings. We show that, for such rings, graded Noetherian implies Noetherian and relate the graded Krull dimension to the Krull dimension. In addition we find a bound on the length of chains of prime ideals not containing homogeneous elements when the grading group is nilpotent-by-finite. These results have suitable corollaries for strongly groupgraded rings. Our work extends several results on skew group rings, crossed products and group-graded rings. Introduction. An associative ring with identity is said to be graded by the group G if R = E R(x) xEG is a direct sum of additive subgroups R(x) with R(x)R(y) C R(xy). It follows that 1R E R(1). R is said to be strongly G-graded if R(x)R(y) = R(xy) for all x, y E G. The group G is said to be polycyclic-by-finite if G has a subnormal series 1 = Go a G, a * * a Gn = G, where Gi/Gi-, is either infinite cyclic or finite. If G has a normal series of this type, then G is said to be strongly polycyclic-by-finite. The number of infinite cyclic factors which occur in this series is called the Hirsch number of G and is denoted by h(G). Since any two series have isomorphic refinements, h(G) is a nonzero integer invariant of G. In their paper [6] M. Cohen and S. Montgomery proved a duality theorem for finite group gradings and group actions on rings. Their methods provided a way to translate results on finite crossed products to more general group-graded rings. Their construction was developed concretely by the second author in [11], where further applications were given. In addition the approach there works for infinite groups. Here we use this method to study Krull dimension and chains of prime ideals in rings graded by polycyclic-by-finite groups. The construction of [11] for infinite groups is sketched in ?1. We comment that the duality theorem of Cohen and Montgomery has been extended by R. Blattner and S. Montgomery [4] to handle certain infinite groups as a corollary to a more general theorem on Hopf algebra actions. Other studies of the duality have been made by M. van den Bergh [2] and J. Osterburg [10]. We now describe the main results of this paper. Let R be graded by a polycyclicby-finite group G. In ?2 we show that a graded R-module M, which is graded Noetherian, is in fact Noetherian. Also in this situation, we show that the Krull Received by the editors October 7, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A03, 16A55, 16A33; Secondary 16A24. ?1988 American Mathematical Society 0002-9939/88

HEREDITARY AND PATH COALGEBRAS

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REPRESENTATION THEORY OF LIFTINGS OF QUANTUM PLANES

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Linear cellular automata with boundary conditions

Introduction It is a well-known that, over an algebraically closed field and up to Morita equivalence, finite dimensional hereditary algebras are exactly the finite dimensional path algebras (see e.g. [Be]). In this note, we give short proof of the dual of this fundamental fact, which holds for arbitrary coalgebras and path coalgebras of quivers, without finiteness assumptions. The results of this article were recently obtained differently in [JMLS] where hereditary coalgebras are studied via a notion of formal smoothness. Here we present a short and direct approach. Background material may be found in [C,Mo,Sw]. We let Q be a quiver and let kQ denote the path coalgebra (see [CMo] or [C]) over the field k, with comultiplication ∆. For a path p, s(p) denotes the starting vertex. Let {Cn} denote the coradical filtration of the coalgebra C. By the quiver of C, we mean the Ext-quiver as in [C,CMo]. A coalgebra is said to be hereditary [NTZ] if homomorphic images of injective comodules are injective. Our main result is

Spectra of smash products

We systematically determine the regular representations, quivers and representation type of all liftings of two-dimensional quantum linear spaces.

Prime ideals in restricted differential operator rings

The main results of the paper concern graphs of linear cellular automata with boundary conditions. We show that the connected components of such graphs are direct sums of trees and cycles, and we provide a complete characterization of the trees, as well as enumerate the cycles of various lengths. Our work generalizes and clarifies results obtained previously in special cases.

Injective comodules for 2 × 2 quantum matrices

LetT=R #H be a smash product whereH is a finite dimensional Hopf algebra. We show that ideals ofT invariant under the dualH* ofH are extended fromH-invariant ideals ofR. This allows us to transport the study of ideals inT to invariant ideals. When the Hopf algebra is pointed the relationship between an ideal and its invariant ideal is shown to be manageable. Restricting to prime ideals, this yields results on the prime spectra ofR andT. We obtain Krull relations forR ⊆T for someH, including Incomparability wheneverH is commutative (or more generally whenH* is pointed after base extension). The results generalize and unify a number of results known in the context of group and restricted Lie actions.