Department of Mathematics

Algebra Seminar

UCCS University Photo

"Rings and Wings"

The Colorado Springs Algebra Seminar typically meets every other Wednesday, from 3:45 until 5:00. Often the meeting place is the campus of the University of Colorado at Colorado Springs, but other venues have been used as well.

We encourage talks from all areas of algebra.

Talks are typically attended by math faculty from throughout the Pikes Peak region, including the University of Colorado at Colorado Springs, The Colorado College, and Colorado State University - Pueblo. Talks are also often attended by graduate students and advanced undergraduate students.

Talks are typically given by those who typically attend, as well as any out-of-town visiting algebraists who may happen along ...

It is now traditional that, on completion of the presentation, those who are interested head to a local eatery / watering hole (often Clyde's on the UCCS campus) for dinner or liquid refreshment or snacks (e.g., Wings?).

Contact the seminar organizer, Gene Abrams if you are interested in participating.

On this page we will also typically include other talks which will happen in the Pikes Peak region which may be of interest to algebraists.

Spring 2016 Schedule

All talks in ENGR 201 (Engineering Dean's Conference Room) on the UCCS campus, unless otherwise indicated. (note: this is a different room than the one we've used in the past, but the same room as we used in Spring 2015...) All "Rings and Wings" talks begin at 3:45pm, unless otherwise indicated. Contact Gene Abrams (preferably a few days in advance of the talk) if you need a UCCS parking permit.

Cristobal Gil Universidad de Malaga - Spain

The commutative core of a Leavitt path algebra

Leavitt path algebras are the algebraic version of Cuntz-Krieger graph $C^*$-algebras. The relation between these two classes of graph algebras has been mutually beneficial. The algebraic and analytic theories share important similarities, but also present some remarkable differences. This is the case for the topic discussed in this talk: the analytic result was given by Nagy and Reznikoff for the $C^*$-algebras $C^*(E)$, and we give here the algebraic analogue for the Leavitt path algebras $L_R(E)$. We will introduce the commutative core $M_R(E)$: a maximal commutative subalgebra inside $L_R(E)$, which in particular contains the diagonal subalgebra. Furthermore, we will characterize injectivity of representations which gives a generalization of the Cuntz-Krieger uniqueness theorem: the result says that a representation of $L_R(E)$ is injective if and only if it is injective on $M_R(E)$. On the other hand, we will generalize and simplify the result about commutative Leavitt path algebras over fields. This work was done jointly with Alireza Nasr-Isfahani.

Sergio Lopez-Permouth Ohio University

Modules over Infinite Dimensional Algebras

Let A be an infinite dimensional K- algebra, where K is a field and let B be a basis for A. In this talk we explore a property of the basis B that guarantees that K^B (the direct product of copies indexed by B of the field K) can be made into an A-module in a natural way. We call bases satisfying that property "amenable" and we show that not all amenable bases yield isomorphic A-modules. Then we consider a relation (which we name congeniality) that guarantees that two different bases yield isomorphic A-module structures on K^B. We will look at several examples in the familiar setting of the algebra K[x] of polynomials with coefficients in K. Finally, if time allows we will mention some results regarding these notions in the context of Leavitt Path Algebras (joint work with Lulwah A-Essa and Najat Muthana).

Be'eri Greenfeld, Bar Ilan Univ., Israel

Growth, Geometry and Representations of some Non-commutative Graded Algebras

The investigation of growth of finitely generated groups has been a fruitful, well studied area for mathematicians, since Gromov's remarkable theorem (that a finitely generated group with polynomially bounded growth is virtually nilpotent), through Grigorchuk's inspiring example of a group with intermediate (i.e. super-polynomial but subexponential) growth, and until recent results of Tao and others.

The parallel study for finitely generated (associative, not necessarily commutative) algebras turns out to be very much different. For example, we show how many super-polynomial functions are realized as the growth types of prime algebras, improving a recent result of Bartholdi and Smoktunowicz. For algebras of polynomially bounded growth, the degree of the polynomial (the GK-dimension) turns out to be an important invariant of the algebra. It naturally appears in the study of holonomicity in the theory of D-modules, and also enabled Artin and Van den Bergh to investigate NC (non-commutative) projective schemes -- which are essentially graded domains with polynomial growth. While NC projective curves were classified, the conjectured classification of NC projective surfaces is still open. We study the behavior of the classical (namely, Krull) dimension for surfaces, and show it is always bounded (even for non-noetherian surfaces).

From a representation-theoretic point of view, we finally show that prime, graded algebras with restricted growth either satisfy a polynomial identity or act faithfully on a simple module; this provides an affirmative answer to a graded version of a long-standing question raised by Braun and Small.

This is partially based on joint work with Leroy, Smoktunowicz and Ziembowski and on separate sole research by the speaker.

Speaker: Greg Oman, UCCS

Title / Abstract: Polynomial and power series rings with finite quotients

An old textbook problem posed by Kaplansky is to show that the group Z of integers is the unique infinite abelian group G with the property that G/H is finite for every nonzero subgroup H of G. In the literature, rings R with the property that R/I is finite for every nonzero two-sided ideal I are called rings with finite quotients (alternatively, residually finite rings). In this talk, we classify the rings R (assumed only to be associative, not necessarily with 1) for which the polynomial ring R[X] (respectively, power series ring R[[X]]) has finite quotients. An analogous problem for one-sided ideals will also be discussed. This is joint work with Adam Salminen of The University of Evansville.

Daniel Goncalves, UFSC - SC - Brasil

Title / Abstract: Simplicity of partial skew group rings with applications to Leavitt path algebras

In this talk I will make an introduction to algebraic partial actions and their associated partial skew group rings, describing a simplicity criteria for certain partial skew group rings. As a motivating example I will show how to realize Leavitt path algebras as partial skew group rings and derive the simplicity criteria for Leavitt path algebras from partial skew ring theory.

Michael Penn, Colorado College

Tutt Science Building, Room 229

Title / Abstract:Vertex Operator Algebras: Motivation, Definition, and Examples Part II

Speaker: Michael Penn, Colorado College

Tutt Science Building, Room 229

Title / Abstract: Vertex Operator Algebras: Motivation, Definition, and Examples Part I

Vertex operator algebras(VOA) are a relatively new class of algebraic objects which have found uses across many branches of mathematics and physics: representation theory, modular forms and q-series, the study of finite simple groups, string theory, and topological quantum field theory. In this talk we will explore three equivalent formulations of a vertex algebra and provide a few examples — the Heisenberg(VOA) and lattice VOAs. The Heisenberg VOA can be considered a “quantum” version of the classical polynomial algebra, while lattice VOAs will form the basis of a second talk where we will cover several recent results.

Speaker: Kulumani Rangaswamy, UCCS

On prime ideals in Leavitt path algebras

In this talk I present some information about prime factorization of ideals in a Leavitt path algebra L. This may include some (preliminary) results showing how the behavior of graded ideals influence that of the non-graded ideals in L.