Representations and Lie Theory Seminar

Schedule of talks:

For our international speakers, we will schedule the seminar on Thursdays at 10AM, to compensate for the time difference. These exceptions are coloured in red.

Abstracts

January 20. In this talk we'll discuss relations between two families of algebras coming from mathematical physics: shifted Yangians on the hand, and Coulomb branches of quiver gauge theories on the other. Coulomb branches are a newcomer mathematically speaking, only recently given a mathematical definition by Braverman, Finkelberg and Nakajima. Meanwhile Yangians have been studied since the 1980's beginning in the study of integrable systems, and shifted Yangians are a variant inspired by the work of Brundan and Kleshchev on finite W-algebras. We'll discuss recent results relating these two families of algebras, and in particular how Coulomb branches of finite ADE type arise as quotients of shifted Yangians, by ideals with representation-theoretic significance.

January 27. Spherical subalgebra of Cherednik's double affine Hecke algebra of type A admits a polynomial representation in which its generators act via elementary symmetric functions and Macdonald operators. Recognizing the elementary symmetric functions as eigenfunctions of quantum Toda Hamiltonians, and applying (the inverse of) the Toda spectral transform, one obtains a new representation of spherical DAHA. In this talk, I will discuss how this new representation gives rise to an injective homomorphism from the spherical DAHA into a quantum cluster algebra in such a way that the action of the modular group on the former is realized via cluster transformations. The talk is based on a joint work in progress with Philippe Di Francesco, Rinat Kedem, and Gus Schrader.

February 4. I will report on a joint work with Bart Vlaar aimed at producing solutions of the reflection equation on category O integrable representations of quantum Kac-Moody algebras. In the affine case, our constructions apply to the category of finite-dimensional representations of quantum loop algebras, producing a parameter-dependent (and conjecturally meromorphic) operator satisfying the spectral reflection equation.

February 11. In this talk we will discuss the conjecture that for any representation of Y(gl(n)) which is a tensor product of evaluation modules a Bethe subalgebra corresponding to a regular semisimple element (or to some extended parameter space) has a cyclic vector. One can think about this property of Bethe subalgebras as a necessary condition for completeness of the Bethe ansatz.

February 17. Skew representations (corresponding to skew Young diagrams) of Yangian and quantum affine algebra of type A were introduced by Cherednik and extensively studied by Nazarov and Tarasov. In this talk, we will discuss some known results about skew representations of super Yangian of type A such as Jacobi-Trudi identities, Drinfeld functor, irreducibility conditions of tensor products, and extended T-systems. We also discuss some open problems related to tame modules of super Yangian. Some essential differences comparing to the even case will be discussed as well.

March 4. Associated to a finite-dimensional complex simple Lie algebra is a family of algebras, the shifted Yangians. In this talk we are interested in a category O of modules over these shifted Yangians. We establish cyclicity and cocyclicity properties for the tensor product of a "signed" module (which we will explain in the talk) with an arbitrary irreducible module, for all choices of spectral parameters. These properties enable us to construct the R-matrices for these tensor products and imply furthermore that these R-matrices are polynomial functions of the spectral parameter. As applications, we prove that in category O: any irreducible module factorizes through a truncated shifted Yangian; the class of modules of finite representation length is stable under tensor product. Based on a joint work with David Hernandez.

March 17. iQuantum groups arising from quantum symmetric pairs are a natural generalization of Drinfeld-Jimbo quantum groups. Several fundamental constructions for quantum groups (such as R-matrix, canonical basis, Hall algebra realization) have been generalized to quantum groups recently. In this talk, I will review the basics on i-quantum groups. I will explain the Serre presentation and higher order Serre (= Serre-Lusztig) relations for i-quantum groups, which are expressed neatly via the i-divided powers. Joint work with Xinhong Chen and Ming Lu.

March 24. The Drinfeld (loop) presentation has played a fundamental role in the representation theory of affine quantum groups . The i-quantum groups are coideal subalgebras of quantum groups arising from quantum symmetric pairs, and they are natural generalizations of quantum groups. Recently, Lu and Wang formulated a Drinfeld type presentation for affine i-quantum groups of split ADE types, and I generalized their work to split BCFG types. In this talk, I will present these constructions.

April 14. Classical q-shuffle algebras provide combinatorial models for the positive half U_q(n) of a finite quantum group. We define a loop version of this construction, yielding a combinatorial model for the positive half U_q(Ln) of a quantum loop group. In particular, we construct a PBW basis of U_q(Ln) indexed by standard Lyndon words, generalizing the work of Lalonde-Ram, Leclerc and Rosso in the U_q(n) case. We also connect this to Enriquez' degeneration A of the elliptic algebras of Feigin-Odesskii, proving a conjecture that describes the image of the embedding of U_q(Ln) into A, in terms of pole and wheel conditions. Joint work with Andrei Negut.

April 28. Deformed double current algebras form a family of double affine quantum algebras related to quantum toroidal algebras and affine Yangians. I will give a survey of the few results, old and new, that are known about these.

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