arXiv subjects

E. S. Letzter

Publications and source records attributed to E. S. Letzter.

6 recordsLinked to original sources

arXiv2026

Skew fields not finitely generated as algebras

An associative division algebra D is said to be _affine_ over a central subfield k if D is finitely generated as a k-algebra. In 1956 Amitsur famously proved that, when k is uncountable, D cannot be k-affine unless D is algebraic over k. In this paper we consider affineness -- and nonaffineness -- for certain naturally occurring classes of division algebras over arbitrary fields. The primary applications are to division algebras of fractions of suitably conditioned iterated skew polynomial rings over k, including many examples naturally arising in Lie theoretic and quantum group settings. Many transcendental division algebras are thus verified to be nonaffine over k. Division algebras of fractions of Weyl algebras and quantum affine spaces are determined to be affine over their centers exactly when they are finite dimensional over their centers.

math.RA↗

arXiv2008

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.

math.QA↗

arXiv2005

The Closed-Point Zariski Topology for Irreducible Representations

In previous work, the second author introduced a topology, for spaces of irreducible representations, that reduces to the classical Zariski topology over commutative rings but provides a proper refinement in various noncommutative settings. In this paper, a concise and elementary description of this refined Zariski topology is presented, under certain hypotheses, for the space of simple left modules over a ring R. Namely, if R is left noetherian (or satisfies the ascending chain condition for semiprimitive ideals), and if R is either a countable dimensional algebra (over a field) or a ring whose (Gabriel-Rentschler) Krull dimension is a countable ordinal, then each closed set of the refined Zariski topology is the union of a finite set with a Zariski closed set. The approach requires certain auxiliary results guaranteeing embeddings of factor rings into direct products of simple modules. Analysis of these embeddings mimics earlier work of the first author and Zimmermann-Huisgen on products of torsion modules.

math.RA↗

arXiv1999

Quantum n-space as a quotient of classical n-space

Let $A$ denote the commutative polynomial ring in $n$ variables, over an algebraically closed field $k$, and let $R$ denote the standard multiparameter quantization of $A$ determined by a multiplicatively antisymmetric $n\times n$ matrix $(q_{ij})$. In this paper we prove, when -1 cannot be multiplicatively generated by the $q_{ij}$, that the primitive spectrum of $R$ is a topological quotient of $k^n$. Under the same hypothesis, we further prove that the prime spectrum of $R$ is a topological quotient of the prime spectrum of $A$.

math.RA↗

arXiv1999

Noetherian Centralizing Hopf Algebra Extensions and Finite Morphisms of Quantum Groups

Let $A \subset H$ be a finite centralizing extension of noetherian Hopf algebras, and let $X$ denote the group of characters of $H$ that restrict to the augmentation map on $A$. Our main results provide necessary and sufficient conditions for the fibers of the canonical surjection from $spec H$ onto $spec A$ to coincide with the $X$-orbits in $spec H$. In particular, all of the fibers are $X$-orbits if and only if the fiber 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.

math.RA↗

arXiv1999

Module Extensions Over Classical Lie Superalgebras

We study certain filtrations of indecomposable injective modules over classical Lie superalgebras, applying a general approach for noetherian rings developed by Brown, Jategaonkar, Lenagan, and Warfield. To indicate the consequences of our analysis, suppose that $g$ is a complex classical simple Lie superalgebra and that $E$ is an indecomposable injective $g$-module with nonzero (and so necessarily simple) socle $L$. (Recall that every essential extension of $L$, and in particular every nonsplit extension of $L$ by a simple module, can be formed from $g$-subfactors of $E$.) A direct transposition of the Lie algebra theory to this setting is impossible. However, we are able to present a finite upper bound, easily calculated and dependent only on $g$, for the number of isomorphism classes of simple highest weight $g$-modules appearing as $g$-subfactors of $E$.

math.RA↗