Edward S. Letzter

Complete sets of representations of classical Lie superalgebras

Descriptions of the complete sets of irreducible highest-weight modules over complex classical simple Lie superalgebras are recorded. It is further shown that the finite-dimensional irreducible modules over a (not necessarily classical simple) finite-dimensional complex Lie superalgebra form a complete set if and only if the even part of the Lie superalgebra is reductive and the universal enveloping superalgebra is semiprime.

Finite correspondence of spectra in Noetherian ring extensions

Let R S be an embedding of associative noetherian rings such that S is finitely generated as a right R-module. There is a correspondence from the prime spectrum of S to the prime spectrum of R obtained by associating to a given prime ideal P of S the prime ideals of R minimal over P n R. The prime and primitive ideal theories for several specific noncommutative noetherian rings, including group algebras, PI algebras, and enveloping algebras, depend on understanding instances of this correspondence. We prove that the correspondence has finite fibers for a class of noetherian ring extensions that unites these examples.

Semiclassical limits of quantum affine spaces

Semiclassical limits of generic multiparameter quantized coordinate rings A = O_q(k^n) of affine spaces are constructed and related to A, for k an algebraically closed field of characteristic zero and q a multiplicatively antisymmetric matrix whose entries generate a torsionfree subgroup of k*. A semiclassical limit of A is a Poisson algebra structure on the corresponding classical coordinate ring R = O(k^n), and results of Oh, Park, Shin and the authors are used to construct homeomorphisms from the Poisson prime and Poisson primitive spectra of R onto the prime and primitive spectra of A. The Poisson primitive spectrum of R is then identified with the space of symplectic cores in k^n in the sense of Brown and Gordon, and an example is presented (over the complex numbers) for which the Poisson primitive spectrum of R is not homeomorphic to the space of symplectic leaves in k^n. Finally, these results are extended from quantum affine spaces to quantum affine toric varieties.

The fundamental prime ideals of a noetherian prime PI ring

Let R be a noetherian prime PI ring and let P be a prime ideal of R. There is a set of prime ideals, the fundamental prime ideals, associated with the injective hull of R/P and denoted by Fund(P). The set Fund(P) is finite, by a result of Miiller. In this paper we give a natural description of Fund(P) in terms of the trace ring of R which strengthens Miillers result by establishing a uniform bound for the size of Fund(P) for all primes P in the ring,

Goldie Ranks of Skew Power Series Rings of Automorphic Type

Let A be a semprime, right noetherian ring equipped with an automorphism α, and let B: = A[[y; α]] denote the corresponding skew power series ring (which is also semiprime and right noetherian). We prove that the Goldie ranks of A and B are equal. We also record applications to induced ideals.

Skew polynomial extensions of commutative Noetherian Jacobson rings

The Jacobson condition (i.e., that all prime ideals are semiprimitive) is proved to pass from a commutative noetherian ring R to a skew polynomial ring R[y; , 3], assuming only that T is an automorphism.

ON CONTINUOUS AND ADJOINT MORPHISMS BETWEEN NON-COMMUTATIVE PRIME SPECTRA

We study topological properties of the correspondence of prime spectra associated to a noncommutative ring homomorphism R→ S. Our main result provides criteria for the adjointness of certain functors between the categories of Zariski closed subsets of SpecR and SpecS; these functors arise naturally from restriction and extension of scalars. When R and S are left noetherian, adjointness occurs only for centralizing and “nearly centralizing” homomorphisms.

An affine PI Hopf algebra not finite over a normal commutative Hopf subalgebra

In formulating a generalized framework to study certain noncommutative algebras naturally arising in representation theory, K. A. Brown asked if every finitely generated Hopf algebra satisfying a polynomial identity was finite over a normal commutative Hopf subalgebra. In this note we show that Radfords biproduct, applied to the enveloping algebra of the Lie superalgebra pl(1, 1), provides a noetherian prime counterexample.

Noetherianity of the space of irreducible representations

LetR be an associative ring with identity. We study an elementary generalization of the classical Zariski topology, applied to the set of isomorphism classes of simple leftR-modules (or, more generally, simple objects in a complete abelian category). Under this topology the points are closed, and whenR is left noetherian the corresponding topological space is noetherian. IfR is commutative (or PI, or FBN) the corresponding topological space is naturally homeomorphic to the maximal spectrum, equipped with the Zariski topology. WhenR is the first Weyl algebra (in characteristic zero) we obtain a one-dimensional irreducible noetherian topological space. Comparisons with topologies induced from those on A. L. Rosenberg’s spectra are briefly noted.

Noetherian Centralizing Hopf Algebra Extensions and Finite Morphisms of Quantum Groups

We study finite centralizing extensions A ⊂ H of noetherian Hopf algebras. Our main results provide necessary and sufficient conditions for the fibres of the surjection spec H [Rarr ]spec A to coincide with the X -orbits in spec H , where X denotes the finite group of characters of H that restrict to the counit of A . In particular, all of the fibres are X -orbits if and only if the fibre over the augmentation ideal of A is an X -orbit. An application to the representation theory of quantum function algebras, at roots of unity, is presented.