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Interactions of quantum affine algebras with
cluster algebras, current algebras and categorification

 Last modified on June 7, 2018.

Schedule



All talks will take place in Aquinas Hall; room numbers are given below.

Saturday, June 2 Location: 
8:30-9:30 Registration and refreshments Lobby
9:30 Dr. Kiran Bhutani 102
9:35-11:05 Alistair Savage 102
String diagrams and categorification I
Tea break
11:30-1:00 Bernard Leclerc 102
Quantum affine algebras and cluster algebras I
Sunday, June 3 Location: 
9:00-9:30 Refreshments Lobby
9:30-11:00 Alistair Savage 102
String diagrams and categorification II
Tea break
11:30-1:00 Bernard Leclerc 102
Quantum affine algebras and cluster algebras II
Monday, June 4 Location: 
8:30-9:00 Refreshments Lobby
9:00-10:30 Alistair Savage 102
String diagrams and categorification III
Tea break
10:30-12:30 Bernard Leclerc 102
Quantum affine algebras and cluster algebras III
Lunch break
Contributed talks
Session I: Aquinas 102 Session II: Aquinas 201
2:30-2:45 Molly Lynch Margaret Rahmoeller
2:50-3:05 Max Gurevich Jonathan Brown
3:10-3:25 Bach Nguyen Xiao He
3:30-3:45 Man-Wai Cheung Rebecca Jayne
Tea break
4:10-4:25 Karl Schmidt Kayla Murray
4:30-4:45 JiaRui Fei Lauren Grimley
4:50-5:05 Henry Kvinge Krishanu Roy
5:10-5:25 Michael Reeks Rajendran Venkatesh
Tuesday, June 5 Location:
8:30-9:00 Registration and Refreshments Lobby
9:00-9:50 Shrawan Kumar 102
10:00-10:50 Anthony Joseph 102
Tea break
11:20-12:10 Aaron Lauda 102
Lunch break
2-2:50 Eric Vasserot 102
Contributed talks
Session I: Aquinas 102 Session II: Aquinas 201
3:10-3:25 Mrigendra Singh Kushwaha Mee Seong Im
3:30-3:45 Punita Batra Xinli Xiao
3:50-4:05 Matthew Lee Suzanne Crifo
Tea break
4:40-5:30 Cristian Lenart 102
Wednesday, June 6 - all events in Aquinas 102
9:00-9:30 Refreshments
9:30-10:20 Masaki Kashiwara
10:30-11:20 David Hernandez
Tea break
11:50-12:30 Yiqiang Li
Lunch break
2:30-3:20 Bogdan Ion
Tea break
3:40-4:30 Erhard Neher
6 - 7 Reception - Caldwell Auditorium
7 - 9  Banquet Dinner - Caldwell Auditorium
Thursday, June 7  Location: 
8:30-9:00 Refreshments Lobby
9:00-9:50 Georgia Benkart 102
10:00-10:50 Monica Vazirani 102
Tea break
11:20-12:10 Adriano Moura 102
Lunch break
2-2:50 Michael Lau 102
Contributed talks
Session I - Aquinas 102 Session II - Aquinas 201
3:10-3:25 Aleksei Ilin Arik Wilbert
3:30-3:45 Matheus Brito Joanna Meinel
3:50-4:05 Bing Duan Sachin S. Sharma
4:10-4:25 Léa Bittmann Tanusree Khandai
Tea break
4:50-5:40 Weiqiang Wang
Friday, June 8 - all events in Aquinas 102
8:30-9:00 Refreshments
9:00-9:50 Alistair Savage
10:00-10:50 Evgeny Mukhin
Tea break
11:20-12:10 Venkatramani Lakshmibai



Summer School mini-courses


 

Bernard Leclerc (Université de Caen, France)

Quantum affine algebras and cluster algebras

Ten years ago, it was observed that cluster algebras play a role in the representation theory of quantum affine algebras. Certain monoidal categories of finite-dimensional representations were shown to have a Grothendieck ring with a natural structure of a cluster algebra of finite type $A$, $D$, $E$. Since then, cluster algebras have developed into a new combinatorial tool for studying finite-dimensional simple modules over quantum affine algebras. For instance the special class of real simple modules, which conjecturally correspond to cluster monomials, have been shown to enjoy interesting properties. This minicourse will provide an introduction to these ideas, following the works of many people: Hernandez-Leclerc, Nakajima, Kimura-Qin, Kang-Kashiwara-Kim-Oh, Qin, Brito-Chari-Moura, ...


Alistair Savage (University of Ottawa, Canada)

String diagrams and categorification

Diagrammatic methods have come to play a vital role in the modern theory of categorification. The goal of this mini-course is to introduce students to these methods. We will discuss the string diagram formalism for monoidal categories, where morphisms are represented by planar diagrams, and will see how the concept of adjunction is extremely natural from this point of view. We will then discuss applications to categorification, emphasizing the importance of pivotal categories, where one has a natural topological notion of duals.




 

Invited talks

Georgia Benkart (University of Wisconsin-Madison, USA)

McKay quivers and connections with quantum groups

A finite group $\mathsf{G}$ and a finite-dimensional $\mathsf{G}$-module $\mathsf{V}$ can be used to construct a McKay quiver. When $\mathsf{G}$ is a finite subgroup of $\mathsf{SU}_2$, and $\mathsf{V}$ is the 2-dimensional natural module of $\mathsf{G}$, the resulting quiver is a simply laced affine Dynkin diagram. This talk will focus on properties exhibited by such quivers that are constructed from finite groups and from finite-dimensional Hopf algebras such as quantum groups at roots of unity. The quivers lead to Markov chains that have interesting properties.


David Hernandez (Université Paris-Diderot Paris 7, IMJ-PRG, France)

Quantum Grothendieck ring isomorphims, cluster algebras and Kazhdan-Lusztig algorithm

Quantum Grothendieck rings are natural $t$-deformations of representations rings of quantum affine algebras. Important families of such quantum Grothendieck are known to have a structure of a quantum cluster algebra. Using these structures, we establish ring isomorphisms between certain quantum Grothendieck rings in types $A$ and $B$. Combining we recent results of Kashiwara-Kim-Oh, we prove for the corresponding categories in type $B$ a conjecture formulated by the speaker in 2002 : the multiplicities of simple modules in standard modules are given by the evaluation of certain analogues of Kazhdan-Lusztig polynomials and the coefficients of these polynomials are positive (joint work with Hironori Oya; supported by the ERC Grant Agreement number 647353 Qaffine).


Bogdan Ion (University of Pittsburg, USA)

Artin groups of extended affine Lie algebras

I will introduce (topologically) the concept of Artin group associated to an $n$-EALA and show that for $n=2$ these coincide with the double affine Artin groups. Thus, the Artin groups associated to $n$-EALAs can be regarded as $n$-affine Artin groups. I will describe some of their properties and, in particular, explain the emergence of outer actions of congruence groups on $n$-affine Artin groups. This is joint work with S. Sahi.


Anthony Joseph (The Weizmann Institute of Science, Israel)

Trails and $S$-graphs

Kashiwara defined a crystal $B(\infty)$ corresponding to the Verma module of a Kac-Moody Lie algebra $\mathfrak g$. When the corresponding Weyl group is finite, then relative to a reduced decomposition of the longest element, Berenstein and Zelevinski showed that this crystal could be described as a polyhedral set given by linear functions defined by «trails» following the reduced decomposition.

In general trails are not combinatorially defined. We review progress in attempting to describe trails in terms of $S$-graphs and to describe $B(\infty)$ for all $\mathfrak g$.

In joint work with S. Zelikson we show that there is a particular choice of reduced decomposition such that for $\mathfrak g$ simple and classical, the set of all trails is itself a crystal.


Masaki Kashiwara (Research Institute for Mathematical Sciences, Kyoto University, Japan)

Categorification of the cluster algebra structure via quiver Hecke algebras

The quantum unipotent coordinate ring has a cluster algebra structure. On the other hand, this ring is isomorphic to the Grothendieck ring of the module category of quiver Hecke algebras. We can prove that the module corresponding to the cluster monomials is a real simple module. This is a joint work with Seok-Jin Kang, Myungho Kim and Se-jin Oh.


Shrawan Kumar (University of North Carolina, Chapel Hill, USA)

A complete set of intertwiners for arbitrary tensor product representations via current algebras

Let $\mathfrak g$ be a reductive Lie algebra and let $\vec V(\vec\lambda)$ be a tensor product of $k$ copies of finite dimensional irreducible $\mathfrak g$-modules. Choosing $k$ points in $\mathbb C$, $\vec V(\vec \lambda)$ acquires a natural structure of the current algebra $\mathfrak g\otimes\mathbb C[t]$-module. Following a work of Rao, we produce an explicit and complete set of $\mathfrak g$-module intertwiners of $\vec V(\vec\lambda)$ in terms of the action of the current algebra.


Venkatramani Lakshmibai (Northeastern University, USA)

Cotangent bundle to the flag variety

I shall first recall some results concerning the Schubert varieties inside the Flag Grassmannian variety. Then I shall show some important varieties sitting as open subsets of suitable Schubert varieties. I shall then present the main result on the cotangent bundle, realizing the cotangent bundle sitting canonically inside an affine Schubert variety.


Michael Lau (Université Laval, Canada)

Weight modules for current algebras

Generalized current algebras replace the loop of affine Kac-Moody theory with more general algebraic varieties. In this talk, we will describe the simple weight modules (with finite-dimensional weight spaces) for these Lie algebras. We close with some remarks on simple integrable weight modules with infinite-dimensional weight spaces.


Aaron Lauda (University of Southern California, USA)

Current Algebras and Categorification

Many important algebraic objects such as quantum groups, Hecke algebras, and Heisenberg algebras admit diagrammatically defined categorifications. These categorifications are certain monoidal categories defined by planar diagrams modulo local relations. Typically, decategorification is the operation of reducing a monoidal category to an algebra by taking the Grothendieck ring of the category. However, the diagrammatic nature of these categorifications suggests an alternative decategorification, the 'trace', which can be thought of as the result of restricting to diagrams that live on an annulus. In this talk we will explain how the theory of categorified quantum groups recovers the current algebra and some of its representations via its trace decategorification. A basic consequence of this theory is that any 2-representation of the categorified quantum group automatically gives a representation of the current algebra on a part of the 2-representation.


Cristian Lenart (SUNY at Albany, USA)

Lusztig's $t$-analogue of weight multiplicity via crystals

Lusztig defined the Kostka-Foulkes polynomial $K_{\lambda\mu}(t)$ as a $t$-analogue of the multiplicity of a dominant weight $\mu$ in the irreducible representation of highest weight $\lambda$ of a semisimple Lie algebra. This polynomial has remarkable properties, such as being an affine Kazhdan-Lusztig polynomial. Finding combinatorial formulas for $K_{\lambda\mu}(t)$ beyond type $A_n$ has been a long-standing problem. In joint work with Cédric Lecouvey, we give the first such formula, for $K_{\lambda,0}(t)$ in type $C_n$, using combinatorics of Kashiwara's crystal graphs; the special case $\mu=0$ is, in fact, the most complex one. Related aspects and applications will be discussed. I will also mention the so-called atomic decomposition of Kostka-Foulkes polynomials, as well as its relevance to the geometric construction of representations given by the Satake correspondence


Yiqiang Li (University of Buffalo, USA)

Quiver varieties and symmetric pairs

To an $ADE$ Dynkin diagram, one can attach a simply-laced complex simple Lie algebra, say $\mathfrak g$, and a class of Nakajima's quiver varieties. The latter provides a natural home for a geometric representation theory of the former. If the algebra $\mathfrak g$ is further equipped with an involution, it leads to a so-called symmetric pair $(\mathfrak g,\mathfrak k)$ where $\mathfrak k$ is the fixed-point subalgebra under the involution. In this talk, I’ll present bridges at several levels between symmetric pairs and Nakajima varieties.


Adriano Moura (UNICAMP, Brazil)

Irregular minimal affinizations

Minimal affinizations of representations of quantum groups, introduced by Chari in 1995, are the central objects in the project of understanding the smallest representations of quantum affine algebras. After the initial work of Chari and Pressley, it remained to classify the Drinfeld polynomials of the so called irregular minimal affinizations for types $D$ and $E$. More precisely, the minimal affinizations whose highest weights are supported in all three sides of trivalent node, but not on the trivalent node itself.

The restriction of the Drinfeld polynomial of any minimal affinization to a connected subdiagram that remains connected afer removing the trivalent node must correspond to a minimal affinization of type $A$. We refer to a Drinfeld polynomial with this property as being preminimal and define its minimality order to be the number of maximal connected subdiagrams of type $A$ for which its restriction corresponds to a minimal affinization. If the highest weight is not supported on all three sides of the trivalent node, such Drinfeld polynomial corresponds to a minimal affinization if and only if its minimality order is $3$. Otherwise, the minimality order can be at most $2$ and, in the regular case, it corresponds to a minimal affinization if and only if its minimality order is $2$.

In type $D_4$, Chari and Pressley conjectured that all irregular preminimal affinizations of minimality order $2$ are indeed minimal affinizations. Moreover, they showed that all other minimal affinizations have minimality order $1$ and classified them. We subdivide the preminimal affinizations of minimality order $2$ in two kinds: coherent and incoherent ones. The latter exist only in the irregular case. We conjecture that, whenever incoherent Drinfeld polynomials exist, they must correspond to minimal affinizations which are strictly smaller than the ones associated to the coherent counterparts. We will discuss the results that lead to a proof of this for type $D$ as well as for type $E$ under certain technical restrictions on the highest weight. In particular, for type $D_4$, only half of the $6$ equivalence classes of irregular preminimal affinizations of minimality order $2$ are in fact minimal affinizations. We finish the talk presenting the families of irregular minimal affinizations for type $D$ with minimality order $1$ and $0$.


Evgeny Mukhin (Indiana University-Purdue University, USA)

The affine $\mathfrak{gl}(n)-\mathfrak{gl}(m)$ duality

We discuss $\mathfrak{gl}(n)-\mathfrak{gl}(m)$ dualities in representation theory and in integrable systems. The new example is the duality of representations of quantum affine algebras and of integrable systems constructed from transfer matrices of quantum toroidal algebras. The $\mathfrak{gl}(1)-\mathfrak{gl}(2)$ case gives several conjectures about local and non-local integrals of motion of the celebrated quantum KdV model.


Erhard Neher (University of Ottawa, Canada)

Integrable representations of weight-graded Lie algebras

A weight-graded Lie algebra is a complex Lie algebra $L$, which is a weight module with respect to the adjoint action of a finite-dimensional semisimple subalgebra $\mathfrak g$. Some conditions on the occurring weight system will be assumed, e.g. that the weights are the same as the root system of $\mathfrak g$. Several types of Lie algebras recently studied can be viewed as weight graded Lie algebras (current algebras, parabolic subalgebras of affine Lie algebras, some equivariant map algebras). We will present results on the structure of integrable representations of $L$ whose weights are bounded by a dominant weight of $\mathfrak g$. In particular, we will link the category of such representations to the module category of an associative, not necessarily commutative algebra. The talk is a based on my joint paper with Manning and Salmasian.


Alistair Savage (University of Ottawa, Canada)

Advances in Heisenberg categorification

The Heisenberg algebra plays a vital role in many areas of mathematics and physics. In this talk, we will discuss recent advances related to its categorification. In particular, we will explain how one can unify and generalize many existing modifications of Khovanov’s original Heisenberg category. This unification involves defining a Heisenberg category depending on a choice of graded Frobenius superalgebra and “central charge”. We will then discuss work in progress with Jon Brundan on a quantum analogue of these general Heisenberg categories.


Éric Vasserot (Université Paris-Diderot Paris 7, IMJ-PRG, France)

Cohomological Hall algebras of quivers, curves and surfaces

Cohomological Hall algebras are a new class of algebras which are closely related to quantum affine algebras and are attached to several abelian categories such that representations of preprojective algebras or coherent sheaves on curves and surfaces. We will review a few basic facts concerning them and explain some open problems, in particular in the case of curves and surfaces .


Monica Vazirani (University of California Davis, USA)

The «Springer» representation of the DAHA

Building on the work of Calaque–Enriquez–Etingof, Lyubashenko– Majid, and Arakawa–Suzuki, Jordan constructed a functor from quantum D-modules on the special linear group to representations of the double affine Hecke algebra (DAHA) in type $A$. Our preliminary findings show when we input the so-called quantum Springer sheaf at parameter $kN$ the output is roughly a $k$-thickened version of the regular representation of $SN$. Part of this work is defining what the quantum Springer sheaf is, and through Jordan’s functor, understanding its structure. Further, we gain a greater understanding of the category of strongly equivariant quantum $D$-modules. This is joint work with David Jordan.


Weiqiang Wang (University of Virginia, USA)

Canonical bases arising from quantum symmetric pairs

A quantum symmetric pair (QSP) consists of $(U,U^i)$, where $U$ is a quantum group and $U^i$ is a coideal subalgebra corresponding to a Lie subalgebra fixed by an involution. (The classification of QSP’s of finite type corresponds to the classification of real simple Lie algebras.) We shall present a theory of $i$-canonical bases on the modified coideal subalgebras of finite type and the tensor product $U$-modules. In a special case of QSP of type AIII, the $i$-canonical bases admit a geometric interpretation, positivity properties, and applications to super Kazhdan-Lusztig theory. This is joint work with Huanchen Bao (Maryland), and it is also partly based on work with Yiqiang Li (Buffalo).




 

Contributed talks

Punita Batra (Harish-Chandra Research Institute Allahabad, India)

Classification Of Integrable Modules Of Twisted Full Toroidal Lie Algebras

Twisted full toroidal Lie algebras are extensions of multiloop algebras twisted by several finite order automorphisms. I will try to classify irreducible integrable representations for these Lie algebras.


Léa Bittmann (Université Paris-Diderot, France)

$t$-DEFORMATIONS OF GROTHENDIECK RINGS AS QUANTUM CLUSTER ALGEBRAS.

It is known that some Grothendieck rings of categories of representations of quantum affine algebras can be endowed with cluster algebras structures. This is true for example for certain categories $\mathscr O$ containing the category of finite-dimensional representations. On the other hand, certain Grothendieck rings of categories of finite dimensional representations admit remarkable $t$-deformations, which are linked to quiver varieties and are useful to compute characters. The aim of this work is to obtain such $t$-deformations in the context of categories $\mathscr O$. Our approach is based on quantum cluster algebras. (Supported by the ERC Grant Agreement no. 647353 «Qaffine»).


Matheus Brito (Universidade Federal do Paraná, Brazil)

PIERI RULES AND $q$-CHARACTERS OF HERNANDEZ-LECLERC MODULES FOR QUANTUM AFFINE $sl_{n+1} $

In 2009 Hernandez and Leclerc defined a family of prime representations of quantum affine $sl_{n+1}$ by using an $A_n$- quiver. In the case of the sink-source quiver and the monotonic quiver they proved that the associated subcategory of finite–dimensional representations of the quantum affine algebra was a monoidal categorification of a cluster algebra with the prime representations corresponding to cluster variables. In this talk we shall work with an arbitrary quiver and prove a Pieri rule for these prime representations. As a consequence of this rule we generalize the monoidal categorification result of HL to any $A_n$-quiver and moreover we give a $q$-character formula for these prime representations.


Jonathan Brown (SUNY Oneonta, USA)

PRIMITIVE IDEALS AND FINITE W-ALGEBRAS OF LOW RANK.

Finite W-algebras are certain deformations of the centralizers of nilpotent elements in semisimple Lie algebras. Their representation theory is intimately related with the infinite-dimensional representation theory of Lie algebras. In this talk I will go over how the primitive ideals of finite W-algebras is related to the primitive ideals of universal enveloping algebras. Then I will explain recent work in which primitive ideals of finite W-algebras are calculated which in turn classifies many of the multiplicity-free primitive ideals in universal enveloping algebras of Lie algebras. This has led to two examples of multiplicity-free primitive ideals in universal enveloping algebras which cannot be induced from completely prime primitive ideals of a Levi-subalgebras in the Lie algebras.


Man-Wai Cheung (Harvard, USA)

QUIVER REPRESENTATIONS AND THETA FUNCTIONS

Scattering diagrams theta functions and broken lines were developed in order to describe toric degenerations of Calabi-Yau varieties and construct mirror pairs. Later, Gross-Hacking-Keel-Kontsevich unravel the relation of those objects with cluster algebras. In the talk, we will discuss how we can combine the representation theory with these objects. We will also see how the broken lines on scattering diagram give a stratification of quiver Grassmannians using this setting.


Suzanne Crifo (North Carolina State University, USA)

Maximal Dominant Weights for Affine Lie Algebra Representations

Affine Lie algebras are infinite dimensional analogs of finite dimensional simple Lie algebras. It is known there are finitely many maximal dominant weights for any integrable highest weight representation of an affine Lie algebra. However, determining these maximal dominant weights is a nontrival task. So far only the descriptions of these weights are known for affine Lie algebra $A_n^{(1)}$. In this talk we will discuss the maximal dominant weights of the integrable highest weight representation of any affine Lie algebra with highest weight $k\Lambda_0$.


Bing Duan (Lanzhou university, China, and University of Connecticut, USA)

CLUSTER ALGEBRA AND SNAKE MODULES

We recall the notation of snake module and prime snake modules of type $A_n$ and $B_n$. We prove that prime snake modules are real and HL conjecture is true for prime snake modules. More precisely, we give a mutation sequence of prime snake modules and equation systems satisfied by $q$-characters of prime snake modules.


JiaRui Fei (Shanghai Jiao Tong University, China)

TENSOR PRODUCT MULTIPLICITY VIA UPPER CLUSTER ALGEBRAS

By tensor product multiplicity we mean the multiplicities in the tensor product of any two finite-dimensional irreducible representations of a simply connected Lie group. Finding their polyhedral models is a long-standing problem. The problem asks to express the multiplicity as the number of lattice points in some convex polytope. Accumulating from the works of Gelfand, Berenstein and Zelevinsky since 1970’s, around 1999 Knutson and Tao invented their hive model for the type A cases, which led to the solution of the saturation conjecture. Outside type $A$, Berenstein and Zelevinsky’s models are still the only known polyhedral models up to now. Those models lose a few nice features of the hive model. In this talk, I will explain how to use upper cluster algebras, an interesting class of commutative algebras introduced by Berenstein-Fomin-Zelevinsky, to discover new polyhedral models for all Dynkin types. Those new models improve the ones of Berenstein-Zelevinsky’s, or in some sense generalize the hive model. It turns out that the quivers of relevant upper cluster algebras are related to the Auslander-Reiten theory of presentations, which can be viewed as a categorification of these quivers. The upper cluster algebras are graded by triple dominant weights, and the dimension of each graded component counts the corresponding tensor multiplicity. The proof also invokes another categorification – Derksen-Weyman-Zelevinsky’s quiver-with-potential model for the cluster algebra. The bases of these upper cluster algebras are parametrized by $\mu$-supported $g$-vectors. The polytopes will be described via stability conditions. The talk is based on the preprint arXiv:1603.02521.


Lauren Grimley (Spring Hill College, USA)

QUANTUM EXTERIOR ALGEBRAS EXTENDED BY GROUPS

Quantum Drinfeld Hecke algebras arise as deformations of the quantum polynomial algebra extended by finite groups. Previous work has established a dictionary between the Poincare-Birkhoff-Witt conditions and conditions on Hochschild cohomology in the case of group extensions of (quantum) polynomial algebras. In this talk, we will characterize the Hochschild cohomology of quantum exterior algebras extended by groups in an effort to expand the dictionary for algebras with additional relations.


Max Gurevich (National University of Singapore, Singapore)

BRANCHING LAWS FOR REPRESENTATIONS OF P-ADIC GROUPS

The quantum affine Schur-Weyl duality, as formulated by Chari-Pressley, allows for a transfer of problems and tools between the domains of quantum affine algebras and affine Hecke algebras in type $A$. The latter are intimately related to representations of $p$-adic groups. I will describe how a recent result of Hernandez on tensor products of representations of quantum affine algebras can be applied to prove a newly extended version of the celebrated Gan-Gross-Prasad conjectures. Namely, we explore which irreducible quotients appear when restricting an irreducible representation of $p$-adic $GL_n$ to the subgroup $GL_{n-1}$. The GGP conjectures produce an aesthetic rule with which such restrictions must comply in the class of unitarizable representations.


Xiao He (Université Laval, Québec, Canada)

$W$-ALGEBRAS ASSOCIATED TO TRUNCATED CURRENT LIE ALGEBRAS

Given a finite-dimensional semi-simple Lie algebra $\mathfrak g$ and a non-zero nilpotent element $e\in \mathfrak g$, one can construct various (classical and quantum, finite and affine) $W$-algebras. In these constructions, a non-degenerate invariant bilinear form and a good $\mathbb Z$-grading of the Lie algebra play essential roles.

We study $W$-algebras associated to truncated current Lie algebras, where the level $p$ truncated current Lie algebra $\mathfrak g_p$ is the quotient $\mathfrak g \otimes \mathbb C[t]/\mathfrak g \otimes t^{p+1}\mathbb C[t]$ of the current Lie algebra $\mathfrak g \otimes \mathbb C[t$]. We show that non-degenerate invariant bilinear forms exist on truncated current Lie algebras and a good $\mathbb Z$-grading on $\mathfrak g$ induces a good $\mathbb Z$-grading on $\mathfrak g_p$. Finite $W$-algebras associated to truncated current Lie algebras were then defined similarly to the semi-simple case. We prove that these finite $W$-algebras are quantizations of the jet schemes of ordinary Slodowy slices and the celebrated Skryabin equivalence and Kostant’s theorem hold for them. We also introduce affine $W$-algebras associated to truncated current Lie algebras and we might explain some relation between finite $W$-algebras and higher level Zhu algebras of affine $W$-algebras.


Aleksei Ilin (Higher school of economics, Moscow, Russia)

DEGENERATION OF BETHE SUBALGEBRAS IN THE YANGIANS

We study degenerations of Bethe subalgebras $B(C)$ in the Yangian $Y(\mathfrak{gl}_n)$, where $C$ is a regular diagonal matrix. We show that closure of the parameter space of the family of Bethe subalgebras, which parametrizes all possible degenerations, is the Deligne-Mumford moduli space of stable rational curves $M_{0,n+2}$. All subalgebras corresponding to the points of $M_{0,n+2}$ are free and maximal commutative. We describe explicitly the “simplest” degenerations and show that every degeneration is the composition of the simplest ones. We also discuss Bethe subalgebras for Yangians of arbitrary simple Lie algebra.


Mee Seong Im (United States Military Academy, USA)

ON THE AFFINE VW SUPERCATEGORY

I will first give an introduction to the construction of what is known as periplectic Lie superalgebras $\mathfrak p(n)$. A construction of the affine VW supercategory arose from our study of the representation theory of $\mathfrak p(n)$. Letting $V$ to be a superspace with $\mathbb Z/2\mathbb Z$-grading and $M$ to be a $\mathfrak p(n)$-module, we construct a super version of the degenerate BMW algebra in the process of examining higher Schur-Weyl duality for the tensor product of $M$ with finitely-many copies of $V$. I will discuss affine VW superalgebras and their center, and the affine VW supercategory and its connection to Brauer supercategory. This is joint with M. Balagovic, Z. Daugherty, I. Entova-Aizenbud, I. Halacheva, J. Hennig, G. Letzter, E. Norton, V. Serganova, and C. Stroppel.


Rebecca Jayne (Hampden-Sydney College, USA)

MULTIPLICITIES OF MAXIMAL DOMINANT WEIGHTS OF INTEGRABLE $\widehat{sl}(n)$-MODULES

For $n, k \geq 2$, we study the multiplicities of certain maximal dominant weights of the irreducible highest weight $\widehat{\mathfrak{sl}}(n)$-module $V(k\Lambda_0)$. We give the multiplicity of the weight $k\Lambda_0 -\sum_{i=0}^{\ell}(\ell -i)(\alpha_i +\alpha_{n-i})$ by the number of certain admissible sequences of $k-1$ lattice paths in a colored $\ell \times \ell$ square. In turn, we find that the number of such admissible sequences of lattice paths is given by the sum of squares of the number of standard Young tableaux of shape $\lambda\vdash \ell$ with $l(\lambda) \leq k$, a value that can be calculated using the well-known Frame-Robinson-Thrall hook length formula. (joint work with Kailash C. Misra)


Tanusree Khandai (Indian Institute of Science Education and Research, Mohali, India)

A DECOMPOSITION OF A CATEGORY OF A CLASS OF INTEGRABLE REPRESENTATIONS OF TOROIDAL LIE ALGEBRAS

The block decomposition of the graded level zero integrable representations of the affine Kac-Moody Lie algebras has been given by V Chari and J Greenstein. In this talk, starting from the description of the irreducible positive level integrable representations of toroidal Lie algebras, we obtain a decomposition of the subcategory of finite-length objects of the category of positive level integrable representations of toroidal Lie algebras. In the case when the finite type Lie algebra associated with the toroidal Lie algebra is of type $A_n$, $D_n$, $E_7$, $E_8$, $F_4$, we parametrize the blocks by a set of functions.


Mrigendra Singh Kushwaha (The Institute of Mathematical Sciences, Chennai, India)

A PATH APPROACH TO KOSTANT MODULES

Let $V(\lambda)$ and $V(\mu)$ be two (finite dimensional) irreducible representations of a complex semi-simple lie algebra with respective highest weights $\lambda$ and $\mu$. For an element $w$ of the Weyl group $W$, the corresponding Kostant module is the cyclic submodule of $V(\lambda) \otimes V(\mu)$, generated by $v_\lambda \otimes v_{w \mu}$, where $v_\lambda$ is highest weight vector of $V(\lambda)$, and $v_{w \mu}$ is a non-zero vector of weight space of weight $w \mu$ of $V(\mu)$. The Kostant modules form a filtration of $V_\lambda\otimes V_\mu$ parametrized by the double coset space $W_\lambda\backslash W/W_\mu$. Let $B_\lambda$ and $B_\mu$ respectively be the sets of Lakshmibai-Seshadri (LS) paths of shape $\lambda$ and $\mu$. We give, in spirit of Littelmann's Littlewood-Richardson rule, a decomposition rule for Kostant modules in terms of of LS paths. We give a filtration on the set $B_\lambda\star B_\mu$ of concatenated LS paths for which the parametrizing set is again $W_\lambda\backslash W/W_\mu$. Recall that $B_\lambda\star B_\mu$ is a path model for $V(\lambda)\otimes V(\mu)$. We show that the filtration on $B_\lambda\star B_\mu$ mirrors the filtration by Kostant modules of $V(\lambda)\otimes V(\mu)$, so that we get path models for Kostant modules.And lastly, we prove analogues for Kostant modules of Kumar's refined PRV theorem (joint work with K.~N.~Raghavan and Sankaran Viswanath).


Henry Kvinge (Colorado State University, USA)

HEISENBERG CATEGORIES, TOWERS OF ALGEBRAS, AND SYMMETRIC FUNCTIONS.

Heisenberg categories are monoidal categories built from planar diagrams which (conjecturally) categorify the Heisenberg algebra. Each of these categories is attached to a tower of algebras and in many cases the center of this category (the closed diagrams) controls the centers of all algebras in the tower simultaneously. In this talk we will discuss examples of this phenomenon, focusing on the case where the tower of algebras is either group algebras of symmetric groups or the Sergeev superalgebras. We will show that there is a natural way to identify the center of a Heisenberg category with symmetric functions (or their shifted analogues), and we will explore how this construction can give useful information about the centers of the associated tower of algebras. Finally, we will discuss a surprising connection between centers of Heisenberg categories and Markov processes on branching graphs for the associated towers of algebras.


Matthew Lee (University of California Riverside, USA)

GLOBAL WEYL MODULES AND MAXIMAL PARABOLICS OF TWISTED AFFINE LIE ALGEBRAS

In this talk I will discuss the structure of non-standard maximal parabolics of twisted affine Lie algebras, global Weyl modules and the associated commutative associative algebra $A_\lambda$. These modules are one analog of Verma modules in the affine setting and were defined in 2001 by Chari and Pressley. Since the global Weyl modules associated with the standard maximal parabolics have found many applications the hope is that these non-standard maximal parabolics will lead to different, but equally interesting applications.


Molly Lynch (North Carolina State University, USA)

RELATIONS AMONG CRYSTAL OPERATORS AND THE MÖBIUS FUNCTION

Many crystals have a natural poset structure. In this talk, we explain how to define a partial order corresponding to a crystal of type $A_{n}$ arising from a representation. We then use a tool from topological combinatorics called lexicographic discrete Morse theory to better understand possible relations among crystal operators in a given interval of our crystal poset. We explore how the Möbius function of the poset interval gives an indication as to what types of relations among crystal operators will occur, obtaining an assortment of results in this way.


Joanna Meinel (Universität Bonn, Germany)

BOSONIC PARTICLE CONFIGURATIONS

We give a short introduction to bosonic particle configurations consisting of a finite number of particles arranged on a line segment or a circle, and we explain the connection with crystals of affine type A. On these configurations, the classical resp. affine local plactic algebra acts. This action is not faithful, and in the classical case we describe the additional relations and define the partic algebra to be the corresponding quotient of the classical local plactic algebra. We obtain faithfulness, a very simple normal form for monomials in the algebra, and we describe the center of the partic algebra. The affine case is much harder to describe, but we give a conjecturally full list of relations in order to obtain faithfulness of the action on bosonic particle configurations on a circle.


Kayla Murray (UC Riverside, USA)

GRADED REPRESENTATIONS OF CURRENT ALGEBRAS

One motivation for studying graded representations of current algebras is the desire to understand irreducible representations for quantum affine Lie algebras. I will focus on the indecomposable representations for current algebras associated to a partition, $\xi$, called $V(\xi),$ which were first defined by Chari and Venkatesh. I will explain what we know about the structure of these representations


Bach Nguyen (Louisiana State University, USA)

QUANTUM CLUSTER ALGEBRA AT ROOTS OF UNITY AND DISCRIMINANT FORMULA

As a noncommutative analog of cluster algebras, quantum cluster algebras were defined by Berenstein and Zelevinsky in 2005. Since then, such algebras have been an active research area with important applications in the study of canonical bases, combinatorics and representation theory. In this talk, we will define quantum cluster algebra at roots of unity. Then we will provide an explicit formula for discriminant of this algebra. As application, we will demonstrate this on various class of quantum nilpotent algebras. This is a joint work with Trampel and Yakimov.


Margaret Rahmoeller (Roanoke College, USA)

A NOTE ON $U_q(\widehat{\mathfrak{sl}(n)})$-DEMAZURE CRYSTALS

In 1991, Kang, Kashiwara, Misra, Miwa, Nakashima, and Nakayashiki gave the path realizations of affine crystals as a semi-infinite tensor product of some finite crystals called perfect crystals. The crystal for a Demazure module of the quantum affine algebra $U_q(\widehat{\mathfrak{sl}(n)})$ is called a Demazure crystal. A Demazure crystal is a suitable subset of the crystal for the associated integrable module of $U_q(\widehat{\mathfrak{sl}(n)})$. These Demazure crystals have a path realization. For a fixed sequence of Weyl group elements we show that certain $U_q(\widehat{\mathfrak{sl}(n)})$- Demazure crystals have tensor product-like structure.


Rajendran Venkatesh (Indian Institute of Science, Bangalore, India)

Borel-de Siebenthal theory for real affine root systems

We completely classify and give explicit descriptions of the maximal closed subroot systems of real affine root systems. As an application we describe a procedure to get the classification of all regular subalgebras of affine Kac Moody algebras in terms of their root systems. It is a joint work with Krishanu Roy.


Michael Reeks (University of Virginia, USA)

TRACE AND CENTER OF THE TWISTED HEISENBERG CATEGORY

The twisted Heisenberg category introduced by Cautis and Sussan encodes much interesting representation-theoretic and combinatorial information about Sergeev algebras, which are closely related to the spin representation theory of symmetric groups. In this talk, we study decategorifications of the twisted Heisenberg category. We show that the trace of this category is isomorphic to a classical type subalgebra of $W_{1+\infty}$ originally studied by Kac, Wang and Yan. Then we establish an isomorphism between the center of the twisted Heisenberg category and the subalgebra $\Gamma$ of the symmetric functions generated by odd power sums. This isomorphism sends bubbles in the category to certain symmetric functions introduced by Petrov which encode data related to the asymptotic representation theory of Sergeev algebras.


Krishanu Roy (The Institute of Mathematical Sciences, Chennai, India)

Weyl orbits of $\pi$-systems in Kac-Moody algebras

Given a symmetrizable Kac-Moody algebra $\mathfrak{g}$, a $\pi$-system of $\mathfrak{g}$ is a subset of its {\em real roots} such that pairwise differences are not roots. Dynkin showed that linearly independent $\pi$--systems arise precisely as simple systems of regular semisimple subalgebras of finite dimensional semisimple Lie algebra $\mathfrak{\mathring{g}}$ over $\mathbb{C}$. He also computed the number of Weyl group orbits for each $\pi$-system in $\mathfrak{\mathring{g}}$. We prove if any symmetrizable Kac-Moody algebra $\mathfrak{g}$ admits a linearly independent $\pi$-system of affine type, then the number of Weyl orbits of $\pi$-systems of this type is necessarily infinite. We also prove if $\mathfrak{g}$ is simply-laced and the $\pi$-system is (simply-laced) of overextended type, then the number of Weyl group orbits is finite, and can in fact be obtained as a sum of the number of orbits of certain finite type $\pi$-systems inside finite root systems.


Karl Schmidt (University of Oregon, Eugene, USA)

BASED QUANTUM CLUSTER ALGEBRAS

In this talk, we attempt to meaningfully marry the notion of a $U_q(\mathfrak{g})$-module with a specified basis (e.g. dual canonical) and the notion of a quantum cluster algebra. We investigate braided tensor products of such based quantum cluster algebras. In particular, we produce a compatible anti-linear anti-involution and basis in the tensor product, then initiate the search for compatible quantum cluster algebra structures. Explicit quantum cluster structures are exhibited in some cases.


Sachin S. Sharma (IIT Kanpur, India)

INTEGRABLE MODULES FOR LIE TORI.

Centerless Lie tori play an important role in explicitly constructing the extended affine Lie algebras; they play similar role as derived algebras modulo center play in the realization of affine Kac-Moody algebras. In this talk we consider the universal central extension of a centerless Lie torus and classify its irreducible integrable modules when the center acts non-trivially. They turn out to be highest weight modules for the direct sum of finitely many affine Lie algebras upto an automorphism.


Arik Wilbert (Melbourne, Australia)

EXOTIC SPRINGER FIBERS AND TWO-BOUNDARY TEMPERLEY-LIEB ALGEBRAS

In this talk we will study the geometry and topology of a certain family of exotic Springer fibers. These algebraic varieties appear as the fibers under a resolution of singularities of the exotic nilpotent cone which plays a prominent role in Kato’s Deligne-Langlands type classification of simple modules for multiparameter Hecke algebras of type $C$. We describe our results in terms of the combinatorics of the two-boundary Temperley-Lieb algebra. This relates the exotic Springer fibers to interesting categorification problems arising in low-dimensional topology.


Xinli Xiao (UC Riverside, USA)

COHA REALIZATION OF SYMMETRIC POLYNOMIALS

In the quiver case the cohomological Hall algebra (COHA for short) is built on the algebra of symmetric polynomials. We study the $\lambda$-ring structure of the COHA and construct the relations between different bases of symmetric polynomial algebra.



Funded by the National Science Foundation

Partially supported by the European Research Council with the ERC Grant "Qaffine"