Lie Theory Workshop, U. C. Berkeley, May 13-14, 2017

Infinite Dimensional Lie Theory

Conference in Honor of Ivan Penkov

Titles and Abstracts

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Alex Chirvasitu Universal representation categories

Abstract: We discuss tensor categories of representations for infinite-dimensional Lie algebras arising as unions of simple Lie subalgebras. These come in several flavors based on the types of algebras involved, but they share the property of being definable, as tensor categories, by a handful of ``generators and relations'' in the sense that they are universal objects in the bicategory of abelian tensor categories with left exact tensor functors as morphisms. Universality allows us to better understand the internal structure of the categories in question. For example, they contain subcategories whose inclusion behaves like the restriction functor from representations of a reductive algebraic group to representations of a parabolic subgroup. (joint with I. Penkov)

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Ivan Dimitrov Integrable weight modules of finitary Lie algebras Abstract: In this talk I will discuss integrable weight modules with finite dimensional weight spaces over the finitary Lie algebras $A_\infty$, $B_\infty$, $C_\infty$, and $D_\infty$. While these modules are the most natural analogues of finite dimensional modules over finite dimensional simple Lie algebras, it is known that they are not necessarily highest weight modules. It turns out, however, that the irreducible integrable weight modules with finite dimensional weight spaces are parameterized by pairs $(\lambda, [\mu])$, where $\lambda$ is a dominant integral weight and $[\mu]$ is an equivalence class of weights dominated by $\lambda$. I will provide a description of the resulting modules and the condition under which modules corresponding to different pairs $(\lambda', [\mu'])$ and $(\lambda’’, [\mu''])$ are isomorphic.
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Edward Frenkel Manifestations of the Quantum Langlands Duality
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Dimitar Grantcharov Singular Gelfand-Tsetlin Modules Abstract: Every irreducible finite-dimensional module of $gl(n)$ can be described with the aid of the classical Gelfand-Tsetlin formulas. The same formulas can be used to define a $gl(n)$-module structure on some infinite-dimensional modules - the so-called generic (nonsingular) Gelfand-Tsetlin modules. In this talk we will discuss results on a special type of singular Gelfand-Tsetlin $gl(n)$-modules - those of index 2. The talk is based on a joint work with V. Futorny and L. E. Ramirez.
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Crystal Hoyt A new category of $\mathfrak{sl}(\infty)$-modules related to Lie superalgebras

Abstract: The (reduced) Grothendieck group of the category of finite-dimensional representations of the Lie superalgebra $\mathfrak{gl}(m|n)$ is an $\mathfrak{sl}(\infty)$-module with the action defined via translation functors, as shown by Brundan and Stroppel. This module is indecomposable and integrable, but does not lie in the tensor category, in other words, it is not a subquotient of the tensor algebra generated by finitely many copies of the natural and conatural $\mathfrak{sl}(\infty)$-modules. In this talk, we will introduce a new category of $\mathfrak{sl}(\infty)$-modules in which this module is injective, and describe the socle filtration of this module. Joint with: I. Penkov, V. Serganova

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Alexej Petukhov Primitive ideals and infinite-dimensional Lie algebras

Abstract: In my talk I wish to discuss several joint works with Ivan Penkov on two-sided ideals of the universal enveloping algebra of a locally simple Lie algebra. The final output is that all such Lie algebras can be divided into 3 classes. For the Lie algebras of the first class, the universal enveloping algebras have only one proper two-sided ideal (the augmentation ideal). For the Lie algebras of the second class, the universal enveloping algebras have countably many two-sided ideals, and we can provide a combinatorial description of these ideals. The third class consists of Lie algebras $sl(\infty), o(\infty), sp(\infty)$, and we can show that the corresponding universal enveloping algebras have only countably many radical two-sided ideals.

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Vera Serganova Representations of direct limit Lie algebras Abstract: The goal of this talk is to review different results about structure and representation theory of direct limits of finite-dimensional Lie algebras. I hope it will serve as an introduction to other talks on the topic.
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Gregg Zuckerman Algebraic constructions of $\mathfrak k$-admissible $\mathfrak g$-modules
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