ENG 
ESP Session S16 - Quantum symmetries
Talks
Wednesday, July 14, 11:-1 ~ 11:24 UTC-3
The Poisson spectrum of the symmetric algebra of the Virasoro algebra
Susan Sierra
University of Edinburgh, UK - This email address is being protected from spambots. You need JavaScript enabled to view it.
Let $W$ be the Witt algebra of polynomial vector fields on the punctured complex plane, and let $Vir$ be the Virasoro algebra, the unique nontrivial central extension of $W$. We discuss joint work with Alexey Petukhov to analyse Poisson ideals of the symmetric algebra of $Vir$.
We focus on understanding Poisson primitive ideals, which can be given as the Poisson cores of maximal ideals of $\operatorname{Sym}(Vir)$ and of $\operatorname{Sym}(W)$. We give a complete classification of maximal ideals of $\operatorname{Sym}(W)$ which have nontrivial Poisson cores. We then lift this classification to $\operatorname{Sym}(Vir)$, and use it to show that if $\lambda \neq 0$, then $(z-\lambda)$ is a Poisson primitive ideal of $\operatorname{Sym}(Vir)$. We give applications to the Poisson geometry of the space of functions on $Vir$.
Joint work with Alexey Petukhov (Institute for Information Transmission Problems, Russia).
Wednesday, July 14, 11:35 ~ 12:00 UTC-3
Finiteness of the first chiral homology group
Reimundo Heluani
IMPA , Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
For a vertex algebra V and an elliptic curve X, it follows from Zhu's work that the degree zero chiral homology of X with coefficients in V is finite dimensional if V is $C_2$ cofinite. In this talk I will give a generalization of this condition to the degree 1 case.
Joint work with Jethro Van Ekeren (UFF, Niteroi).
Wednesday, July 14, 12:10 ~ 12:35 UTC-3
Markov chains from Weyl modules for quantum $sl_2$
Georgia Benkart
University of Wisconsin-Madison, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
The McKay matrix $M_V$ records the result of tensoring the simple modules with a fixed finite-dimensional module $V$. It is the adjacency matrix of the quiver determined by tensoring with $V$. For a finite group, the eigenvectors for $M_V$ are the columns of the character table, and the eigenvalues come from evaluating the character of $V$ on conjugacy class representatives.
Tensoring determines a Markov chain, and the McKay matrix $M_V$ is closely related to the transition matrix, which tells us the probability of going from one site to another on the chain. We describe two different Markov chains obtained from taking tensor products of the Weyl modules with the two-dimensional Weyl module $V$ for the quantum group $U_q(sl_2)$, when $q^2$ is a primitive $\ell$th root of unity for $\ell$ an odd integer $\ge 3$.
Joint work with Samuel A. Lopes (Universidade do Porto, Portugal).
Wednesday, July 14, 12:45 ~ 13:10 UTC-3
Weyl modules and fusion products for the current superalgebra sl(1|2)[t]
Lucas Calixto
Federal University of Minas Gerais, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
We study a class modules, called Chari-Venkatesh modules, for the current superalgebra sl(1|2)[t]. This class contains other important modules, such as graded local Weyl, truncated local Weyl and Demazure-type modules. We prove that Chari-Venkatesh modules can be realized as fusion products of generalized Kac modules. In particular, this proves Feigin and Loktev's conjecture, that fusion products are independent of their fusion parameters, in the case where the fusion factors are generalized Kac modules. As an application of our results, we obtain bases, dimension and character formulas for Chari-Venkatesh modules
Joint work with Tiago Macedo (Federal University of São Paulo, Brazil) and Matheus Brito (Federal University of Paraná, Brazil).
Wednesday, July 14, 13:45 ~ 14:10 UTC-3
Some non-semisimple modular categories constructed from super quantum groups
Guillermo Sanmarco
Universidad Nacional de Córdoba, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.
We will report on work in progress. We construct a class of super quantum groups associated to Nichols algebras of super type A and prove that their representation categories provide examples of non-semisimple modular tensor categories. The main tools come from recent work by Laugwitz-Walton, where braided Drinfeld doubles and relative monoidal centers are employed to construct such modular categories.
Joint work with Robert Laugwitz (The University of Nottingham).
Wednesday, July 14, 14:20 ~ 14:45 UTC-3
Twisted deformations vs. cocycle deformations for quantum groups
Gastón Andrés García
Universidad Nacional de La Plata, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.
This talk is based on a joint project with F. Gavarini where we study different versions of multiparameter quantum groups. Here we focus on the study of quantum universal enveloping algebras, in short QUEA's, for which we consider those arising from twisted deformations (in short TwQUEA's) and those arising from $ 2 $--cocycle deformations, usually called multiparameter QUEA's (in short MpQUEA's). Up to technicalities, we show that the two deformation methods are equivalent, in that they eventually provide isomorphic outputs, which are deformations (of either kinds) of the ``canonical'', well-known one-parameter QUEA by Jimbo and Lusztig. Thus, the two notions of TwQUEA's and of MpQUEA's --- which, in Hopf algebra theoretical terms are naturally dual to each other --- actually coincide.
Joint work with Fabio Gavarini (Univesitá degli studi di Roma "Tor Vergata").
Wednesday, July 14, 14:55 ~ 15:20 UTC-3
Computing the truncated Gerstenhaber-Schack bialgebra cohomology
Mitja Mastnak
Saint Mary's University, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.
The classification of various certain kinds of pointed Hopf algebras (and more generally Hopf Algebras with the Chevalley property) involves first describing a graded (over non-negative integers) Hopf algebra and then describing all its liftings, i.e., filtered Hopf algebras whose associated graded Hopf algebra is one of the fixed graded Hopf algebra we found in step one. In my talk I will present some results involved in a cohomological approach to computing liftings of a fixed graded Hopf algebra. In particular I will describe some recently developed tools for computing the second and third truncated Gerstenhaber-Schack cohomologies of graded bialgebras.
Wednesday, July 14, 15:30 ~ 15:55 UTC-3
Dendriform operads and brace algebras
Maria Ronco
Universidad de Talca, Chile - This email address is being protected from spambots. You need JavaScript enabled to view it.
Dendriform algebras were introduced by J.-L. Loday in [8], even if the first example of them was yet described by S. Eilenberg and S. MacLane in [1]. On the other hand, a brace algebra $A$ is equivalent to a formal deformation $*$ of the shuffle product on $T(A)$, which makes $(A, *, \Delta^c)$ a Hopf algebra, for the deconcatenation coproduct, and is given by a family of $n+1$-ary products $M_{1,n}$, for $n\geq 1$.
The operad of dendriform algebras is regular and Hopf, which means that the tensor product of two dendriform algebras is dendriform and the notion of dendriform bialgebra is well-defined. In [10], we defined a functor from dendriform algebras to brace algebras, and proved that the subspace of primitive elements of a dendriform bialgebra are closed under the brace structure, more precisely any conilpotent dendriform bialgebra is isomorphic to the {\it dendriform} enveloping algebra of the brace algebra of its primitive elements.
The construction of the functor from dendriform algebras to brace algebras has been applied to several combinatorial Hopf algebras recently, see for instance [2], [4], [5], [6], [7], [11], [3] for the last two years. There exist new families of examples, concerning constructions on the faces of associahedra and $m$-versions of planar binary trees, partitions and permutations (see [11]and [9]) which involve more operations than the dendriform structure. In these cases we still have a notion of bialgebra, and therefore we are able to compute the operad of primitive elements associated to these algebras, but most results published on triples of operads or rigidity structure theorems failed to provide Cartier-Milnor-Moore structure theorems in these cases.
In a joint work with M. Livernet, we give a description of dendriform in terms of generators and relations, envolving braces. This result allows us to give an adjoint {\it free} functor to ${\mbox{Dend-alg}}\longrightarrow {\mbox{Brace-alg}}$, and to get structure theorems for dendriform algebras equipped with some additional structure, like tridendriform and $m$-Dyck algebras.
{\bf References}
[1] S. Eilenberg, S. Maclane, On the groups $H(\Pi, n)$, I}, Annals of Maths., Vol. 58}, (1) (1953), 55--106
[2] K Ebrahimi-Fard, F Patras, Shuffle group laws: applications in free probability, Proceedings of the London Math. Soc. 119 (3) (2019)
[3] K. Ebrahimi-Fard, L. Foissy, J. Kock, F. Patras, Operads of (noncrossing) partitions, interacting bialgebras, and moment-cumulant relations, Adv. in Math. 369 (2020).
[4] L Foissy, Algebraic structures associated to operads, preprint arXiv:1702.05344 (2017).
[5] M. N. Hounkonnou, G. D. Houndedji, Solutions of associative Yang-Baxter equation and equation in low dimensions and associated Frobenius algebras and Connes cocycles, J. of Algebra and Its Applications, Vol. 17, No. 1 (2018).
[6] Y. Li, Q. Mo, X. Zhao, The Freiheitssatz and automorphisms for free brace algebras, Comm. in Algebra 47(10) (2019) 4125--4136
[7] Y. Li, Q. Mo, L.A. Bokut, Gr\"obner-Shirshov bases for symmetric brace algebras, Comm. in Algebra 49 (3) (2021) 1368--1369.
[8] J.-L. Loday, Dialgebras in Dialgebras and related operads, Lecture Notes in Math., 1763, Springer, Berlin, 2001.7--66
[9] D. L\'opez, L.-F. Pr\'eville-Ratelle, M. Ronco, A simplicial complex splitting associativity, J. of Pure and Applied Algebra, Vol 224, (5) (2020).
[10] M. Ronco, Eulerian idempotents and Milnor-Moore theorem for certain non-cocommutative Hopf algebras, J. of Algebra. 254(1) (2002)152-172.
[11] Y. Zhang, X. Gao, D. Manchon, {\it Free (tri) dendriform family algebras}, J. of Algebra 547 (2020) 456--493
Joint work with Muriel Livernet, Univ. Paris-Diderot..
Friday, July 16, 11:00 ~ 11:25 UTC-3
Frobenius exact symmetric tensor categories
Pavel Etingof
MIT, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
I will report on a joint work with K. Coulembier and V. Ostrik. We show that a symmetric tensor category in characteristic p>0 admits a fiber functor to the Verlinde category (semisimplification of Rep$(Z/p)$) if and only if it has moderate growth and its Frobenius functor (an analog of the classical Frobenius in the representation theory of algebraic group) is exact. For example, for p=2 and 3 this implies that any such category is (super)-Tannakian. We also give a characterization of super-Tannakian categories for p>3. This generalizes Deligne's theorem that any symmetric tensor category over C of moderate growth is super-Tannakian to characteristic p. At the end I'll discuss applications of this result to modular representation theory.
Joint work with Kevin Coulembier and Victor Ostrik.
Friday, July 16, 11:35 ~ 12:00 UTC-3
The adjoint algebra for 2-categories
NOELIA BORTOLUSSI
Universidad Nacional de San Luis, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.
In 2017, Kenichi Shimizu introduced the notion of the adjoint algebra $\mathcal{A}_{\mathcal{C}}$ and the space of class functions CF($\mathcal{C}$) for a finite tensor category. The adjoint algebra is defined as the end $\int_{X \in \mathcal{C}} X \otimes X^*$. In fact, $\mathcal{A}_\mathcal{C}$ is an algebra in the Drinfeld center $\mathcal{Z}(\mathcal{C})$. Both, the adjoint algebra and the space of class functions generalize the well know adjoint representation and character algebra for a finite group. Later, in 2019, the author generalized these concepts for any left $\mathcal{C}$-module category $\mathcal{M}$.
In this talk, I will present a generalization of the adjoint algebra to the realm of 2-categories. For any 0-cell B in a 2-category $\mathcal{B}$, I will define the adjoint algebra$\mathcal{Ad}_B$ wich is an algebra in the center of $\mathcal{B}$. For $\mathcal{C}$ a finite tensor category, we will see that when we apply this notion to the 2-category ${}_\mathcal{C} \mathcal{Mod}$ of $\mathcal{C}$-module categories, it coincides with the one introduced by Shimizu, and how it behaves under 2-equivalences.
References:
K. Shimizu. "The monoidal center and the character algebra". J. Pure Appl. Algebra 221, No. 9, 2338–2371 (2017).
K. Shimizu, "Further results on the structure of (Co)ends in fintite tensor categories", Appl. Categor. Struct. (2019). https://doi.org/10.1007/s10485-019-09577-7
N. Bortolussi, M. Mombelli. "The adjoint algebra for 2-categories". Kyoto Journal of Mathematics, to appear. arXiv:2005.05271.
Joint work with Martín Mombelli (Universidad Nacional de Córdoba, Argentina).
Friday, July 16, 12:10 ~ 12:35 UTC-3
Filtered Frobenius algebras in monoidal categories
Chelsea Walton
Rice University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
We develop filtered-graded techniques for algebras in monoidal categories with the goal of establishing a categorical version of Bongale's 1967 result: A filtered deformation of a Frobenius algebra over a field is Frobenius as well. Towards the goal, we construct a monoidal associated graded functor, building on prior works of Ardizzoni-Menini, of Galatius et al., and of Gwillian-Pavlov. We then produce equivalent conditions for an algebra in a rigid monoidal category to be Frobenius in terms of the existence of categorical Frobenius form. These two results of independent interest are used to achieve our goal. We illustrate these results by defining braided Clifford algebras, which are filtered deformations of Bespalov et al.'s braided exterior algebras, and show that these are Frobenius algebras in braided, rigid monoidal categories.
Joint work with Harshit Yadav (Rice University).
Friday, July 16, 12:45 ~ 13:10 UTC-3
Doubly-rational finite groups
Andrew Schopieray
Pacific Institute for the Mathematical Sciences/University of Alberta, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.
As a young student of algebra, one might be happy to discover that the character tables of some finite groups consist only of integers. These are sometimes known as rational groups. There are many sporadic results in the classification of rational groups but it very much remains an open area of inquiry. In this talk we will discuss the search for quantum doubles of finite groups whose (generalized) character tables consist only of integers. It would be sensible to call these doubly-rational groups. Our task is a strict subset of the classical problem of characterizing rational groups, but also a strict subset of the modern problem of characterizing certain modular tensor categories. As such, this talk should be of interest to a broad audience.
Joint work with Terry Gannon (University of Alberta).
Friday, July 16, 13:45 ~ 14:10 UTC-3
On odd-dimensional modular tensor categories
Agustina Czenky
University of Oregon, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk we will discuss odd-dimensional modular tensor categories and maximally non-self dual (MNSD) modular tensor categories of low rank. We will give lower bounds for the ranks of modular tensor categories in terms of the rank of the adjoint subcategory and the order of the group of invertible objects. As an application of these results, we will see that MNSD modular tensor categories of ranks 13 and 15 are pointed. In addition, we will show that MNSD tensor categories of ranks 17, 19, 21 and 23 are either pointed or perfect. This talk is based on arXiv:2007.01477.
Joint work with Julia Plavnik (Indiana University, United States).
Friday, July 16, 14:20 ~ 14:45 UTC-3
Universal (co)acting Hopf algebras
Ana Agore
Simion Stoilow Institute of Mathematics of the Romanian Academy, Romania - This email address is being protected from spambots. You need JavaScript enabled to view it.
We introduce the notion of support equivalence for (co)module algebras (over Hopf algebras), which generalizes in a natural way (weak) equivalence of gradings. We show that for each equivalence class of (co)module algebra structures on a given algebra $A$, there exists a unique universal Hopf algebra $H$ together with an $H$-(co)module structure on $A$ such that any other equivalent (co)module algebra structure on $A$ factors through the action of $H$. We study support equivalence and the universal Hopf algebras mentioned above for group gradings, Hopf--Galois extensions, actions of algebraic groups and cocommutative Hopf algebras. We show how the notion of support equivalence can be used to reduce the classification problem of Hopf algebra (co)actions.
Joint work with Alexey Gordienko (M. V. Lomonosov Moscow State University, Russia) and Joost Vercruysse (Universite Libre de Bruxelles, Belgium).
Friday, July 16, 14:55 ~ 15:20 UTC-3
Partial Smash Coproduct of Multiplier Hopf Algebras
Eneilson Fontes
Universidade Federal do Rio Grande (FURG), Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this work we define partial (co)actions on multiplier Hopf algebras, we also present some examples and properties. From a partial comodule coalgebra we construct a partial smash coproduct generalizing the constructions made by the L. Delvaux, E. Batista and J. Vercruysse.
Joint work with Grasiela Martini (Universidade Federal do Rio Grande, Brazil), and Graziela Fonseca (Instituto Federal Sul-Rio-Riograndense, Brazil).
Friday, July 16, 15:30 ~ 15:55 UTC-3
New Examples of Semisimple Hopf Algebras
Yorck Sommerhäuser
Memorial University of Newfoundland, Canada - This email address is being protected from spambots. You need JavaScript enabled to view it.
In joint work with Yevgenia Kashina, the speaker recently constructed two series of Yetter-Drinfel'd Hopf algebras over the Klein four-group in order to refute certain conjectures. All algebras in these series have dimension 8 and depend on a not necessarily primitive fourth root of unity. The algebras in the first series are commutative, while the algebras in the second series are noncommutative.
Via the Radford biproduct construction, these Yetter-Drinfel'd Hopf algebras give rise to semisimple Hopf algebras of dimension 32 that appear not to have been considered in the literature before. We discuss the properties of these Hopf algebras and discuss in particular how many non-isomorphic Hopf algebras arise in this way. For example, it turns out that the biproducts coming from the second series are isomorphic to biproducts coming from the first series.
Joint work with Yevgenia Kashina (DePaul University, USA).
Monday, July 19, 16:00 ~ 16:25 UTC-3
Braided Zesting and its applications
Cesar Galindo
Universidad de Los Andes, Colombia - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk, I will introduce a construction of new braided fusion categories from a given category known as zesting. This method has been used to provide categorifications of new fusion rule algebras, modular data, and minimal modular extensions of super-modular categories. I will present a complete obstruction theory and parameterization approach to the construction and illustrate its utility with several examples.
Joint work with COLLEEN DELANEY (Indiana University), JULIA PLAVNIK (Indiana University), ERIC C. ROWELL (Texas A&M, USA), and AND QING ZHANG (Purdue University, USA).
Monday, July 19, 16:35 ~ 17:00 UTC-3
Frobenius-Schur indicators for some families of quadratic fusion categories
Henry Tucker
University of California, Riverside, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
The family of quadratic fusion categories provides most examples of ``exotic'' fusion categories, i.e. not coming from finite, Lie, or quantum groups. Recently, Izumi and Grossman families of modular data that are conjectured to give the modular data of Drinfel'd centers of the quadratic fusion categories in general. (In fact, it is true for all known examples.) Using this new modular data, we compute the categorical Frobenius-Schur indicators for these families, an important categorical invariant for fusion categories. Moreover, we look more closely at the relationship between indicators in the fusion category and indicators in its center. This is a preliminary report.
Monday, July 19, 17:10 ~ 17:35 UTC-3
Modular categories with transitive Galois actions
Qing Zhang
Purdue University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
The absolute Galois group acts on the set of isomorphism classes of simple objects of any modular category via the characters of the fusion ring. In this talk, we consider transitive modular categories, which are modular categories with transitive Galois actions. We prove every transitive modular category admits a unique prime factorization up to the permutation of factors. The prime transitive modular categories are completely classified, which are given by certain quantum group modular categories with prime Frobenius-Schur exponents. This talk is based on joint work with S.-H. Ng and Y. Wang.
Joint work with Siu-Hung Ng (Louisiana State University) and Yilong Wang (Louisiana State University).
Monday, July 19, 17:45 ~ 18:10 UTC-3
Cocommutative q-cycle coalgebra structures on the dual of the truncated polynomial algebra
Christian Valqui
Pontificia Universidad Católica del Perú, Perú - This email address is being protected from spambots. You need JavaScript enabled to view it.
In order to construct solutions of the braid equation we consider biijective left non-degenerate set-theoretic type solutions, which correspond to regular q-cycle coalgebras. We obtain a partial classification of the different q-cycle coalgebra structures on the dual coalgebra of $K[y]/\langle y^n\rangle$, the truncated polynomial algebra. We obtain an interesting family of involutive q-cycle coalgebras which we call Standard Cycle Coalgebras. They are parameterized by free parameters $\{p_1,...,p_{n-1}\}$ and in order to verify that they are compatible with the braid equation, we have to verify that certain differential operators $\partial^j$ on formal power series in two variables $K[ [x, y] ]$ satisfy the condition $(\partial^j G)_i = (\partial^i G)_j$ for all $i, j$, where $G$ is a formal power series associated to the given q-cycle coalgebra. It would be interesting to find out the relation of these operators with the operators given by Yang in the context with 2-dimensional quantum field theories, which was one of the origins of the Yang-Baxter equation.
Joint work with Juan José Guccione (Universidad de Buenos Aires) and Jorge Guccione (Universidad de Buenos Aires).
Monday, July 19, 18:45 ~ 19:10 UTC-3
Diagonalizing the full twist braid in finite characteristic
Ben Elias
University of Oregon, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
We give a brief introduction to the categorical diagonalization of the full twist braid in the Hecke category. We then indicate some of the interesting changes which occur in finite characteristic.
Joint work with Matt Hogancamp (Northeastern University) and Lars Thorge Jensen (unaffiliated).
Monday, July 19, 19:20 ~ 19:45 UTC-3
Multipermutation solutions to the Yang-Baxter equation
Emiliano Acri
IMAS-UBA, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this work, we study the structure group of a certain important class of solutions to the Yang-Baxter equation, the so-called multipermutation solutions. The structure group of one of these solutions admits a left order. We explore the relation between this class of solutions and groups with the unique product property. We also provide an algorithm that allows us to check if a group has not this property, at least for low order solutions. We develop a notion of nilpotency for skew braces in order to generalize the study of multipermutation solutions.
Joint work with R. Lutowski (Gdańsk, Poland) and L. Vendramin (IMAS-UBA, Argentina).
Monday, July 19, 19:55 ~ 20:20 UTC-3
Decomposition theorems for involutive solutions to the Yang-Baxter equation
Santiago Ramírez
Universidad de Buenos Aires, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.
There is a classic paper where Rump proved a conjecture of Gateva-Ivanova stating that all square-free solutions of the set theoretical Yang-Baxter equation are decomposable. These solutions can be characterized as those where a certain combinatorial invariant is the identity. Inspired by the result of Rump and using a recent theorem of Cedo, Jespers and Okninski we present similar results on (in)decomposable solutions in terms of the combinatorial invariant.
Joint work with Leandro Vendramin (Universidad de Buenos Aires, Argentina).
Monday, July 19, 20:30 ~ 20:55 UTC-3
Charge Conserving Yang-Baxter Operators
Eric Rowell
Texas A&M University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
An operator $R$ on $V\otimes V$ is charge conserving if the span of $e_i\otimes e_j,e_j\otimes e_i$ is $R$-invariant. For example, the universal $R$-matrix associated with $U_q\mathfrak{sl}_2$ is charge conserving. We study the problem of classifying charge conserving Yang-Baxter operators in all dimensions. Using special symmetries we can give a characterization of all such $n^2\times n^2$ solutions, and discover a remarkable combinatorial relationship with hierarchical orderings on $n$ individuals.
Joint work with Paul Martin (University of Leeds).
Posters
Algebraic properties of Universal Quantum Semigroupoids
Fabio Calderón
Universidad Nacional de Colombia, Colombia - This email address is being protected from spambots. You need JavaScript enabled to view it.
The study of quantum symmetries over a non-connected graded algebra $A$ leads to the concept of (co)actions of weak bialgebras and, in particular, the Universal Quantum Semigroupoids (UQSGs). A remarkable set-up in which these UQSGs arise is when the algebra $A=\Bbbk Q$ is a path algebra over a finite quiver $Q$.
Last year, H. Huang, C. Walton, E. Wicks and R. Won proved two relevant results: 1. the associated UQSG of $A=\Bbbk Q$ is isomorphic to the Hayashi's face algebra $\mathfrak{H}(Q)$ attached to $Q$, and 2. when $Q$ is an extended Dynkin quiver the associated UQSG of the preprojective algebra $A=\Pi_Q$ attached to $Q$ is isomorphic to a certain quotient of $\mathfrak{H}(Q)$.
In joint work with Chelsea Walton, we study some ring-theoretic and homological properties of $\mathfrak{H}(Q)$ using the previous results. We will provide the motivation behind this research, the results already obtained and the following steps of our study.
Joint work with Chelsea Walton (Rice University, TX, USA).
Pre-Nichols algebras of Cartan, super and standard type with finite Gelfand-Kirillov dimension.
Emiliano Campagnolo
FAMAF Universidad Nacional de Córdoba, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.
Abstract: The lift method proposes a recipe to classify dotted Hopf algebras by fixing the coradical and the infinitesimal braid. In the non-finite-dimensional context, the pre-Nichols algebras use this recipe. Previously Andruskiewitsch-Sanmarco reduced the problem of classifying pre-Nichols algebras of finite dimension Gelfand-Kirillov when the braid was of Cartan type by introducing the eminent pre-Nichols algebra and showing its relation to the introduced distinguished pre-Nichols algebra. by Angiono. Following the same direction, we will present how we continue the work when the braid is of the super or standard type.
Joint work with Iván Angiono (CONICET-Universidad Nacional de Córdoba). and Guillermo Sanmarco (CONICET-Universidad Nacional de Córdoba)..
Weak Smash Coproduct
Graziela Fonseca
Instituto Federal Sul Rio-grandense, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
Motivated by the notion of coaction of weak bialgebras on coalgebras, in this work the definition of coaction of weak Hopf algebras on coalgebras is introduced and several examples are presented. Furhtermore, the conditions on the weak smash coproduct are established in order to construct a weak Hopf algebra.
Joint work with Eneilson Fontes (Universidade Federal do Rio Grande, Brazil) and Grasiela Martini (Universidade Federal do Rio Grande, Brazil).
A method to compute partial actions
Martini Grasiela
Universidade Federal do Rio Grande, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this work a method to compute partial actions of a given Hopf algebra on its base field is developed and, as an application, we exhibit all partial actions of such type for some families of Hopf algebras.
Joint work with Antonio Paques (Universidade Federal do Rio Grande do Sul, Brazil) and Leonardo Silva (Universidade Federal de Santa Maria, Brazil).
Knots and Links in Modular Tensor Categories
Sung Kim
University of Southern California, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
Modular Tensor Categories (MTCs) are algebraic structures that are equivalent to $(2+1)$-topological quantum field theories. Knot theory has become a powerful practical tool to help us understand and distinguish MTCs. Since the advent of MTCs, it has been conjectured that these categories are classified just by their modular data $(S, T)$ where $S$ and $T$ are invariants derived from the Hopf link and once-twisted unknot respectively. However, Mignard and Schauenburg recently disproved this conjecture by studying a specific class of MTCs known as the twisted quantum double of finite groups. As a result, the study of other knot and link invariants beyond the modular data is important to advance in the classification of MTCs. In this poster, I will provide a basic introduction to MTCs and a new construction technique known as zesting. I will also discuss a particular link invariant, the $W$-matrix, that is derived from the Whitehead link and how zesting affects knot and link invariants.
Joint work with Julia Plavnik (Indiana University Bloomington) and Colleen Delaney (Indiana University Bloomington).
Zesting of Fusion Categories from quantum groups of Type $A$
Hector Giovanny Mora Diaz
Universidad de los Andes, Colombia - This email address is being protected from spambots. You need JavaScript enabled to view it.
Zesting is a construction of a new braided fusion category from a given braided fusion category by defining a new tensor product.
If $\mathcal{C}=\bigoplus_{a \in A} \mathcal{C}_a$ is an $A$-graded fusion category where $A$ is an abelian group, we define a new tensor product $X_a\stackrel{\lambda}{\otimes} Y_b:=(X_a\,\otimes\, Y_b)\otimes\,\lambda(a,b)$, where $X_a\in\mathcal{C}_a$ and $Y_b\in\mathcal{C}_b$ are simple objects in their corresponding graded components and $ \lambda(a,b)\in\mathcal{C}_e$ is an invertible object in the trivial component.
In this work, I present a total description of the Braided Zestings from the modular category $SU(N)_k$ obtained from the quantum group $U_q (\mathfrak{sl}_n)$ extending the work on [2] by giving explicit calculations for finding of all the braided zestings obtained from $SU(N)_k$. \textbf{ References:} [1] C. {Delaney} and C. {Galindo} and J. {Plavnik} and E. {Rowell} and Q. {Zhang}, \textit{Braided Zestings and Aplications}. Communication in Mathematical Physics. [2] H. Mora, \textit{Zestings de categorías modulares construidas a partir de grupos cu\'anticos de tipo A}. Masther Thesis. Universidad de los Andes, Colombia.
On the combinatorial rank of quantum groups
Vanusa Moreira Dylewksi
Universidade Federal do Rio Grande do Sul, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this work we present the definition of combinatorial rank and the first examples obtained by Kharchenko and collaborators, who calculated the combinatorial rank of the positive part $u_q^+(\mathfrak{g})$ of the multi-parameter version of the Lusztig quantum group, where $q$ is a root of the unity and $\mathfrak{g}$ is a simple Lie algebra of type $A_n$, $B_n$, $C_n$ and $D_n$. As a continuation of this study, we provide the combinatorial rank for quantum groups of type $G_2$ and $F_4$.
Joint work with Bárbara Pogorelsky(Universidade Federal do Rio Grande do Sul, Brazil) and Carolina Renz(Universidade Federal de Ciências da Saúde de Porto Alegre, Brazil).
Hopf actions of some quantum groups on path algebras
Amrei Oswald
University of Iowa, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
We investigate examples of quantum symmetry by studying Hopf actions of $U_q(\mathfrak{b})$, $U_q(\mathfrak{sl}_2)$, generalized Taft algebras, and the small quantum group on path algebras. We begin by parametrizing these actions using linear algebraic data. Then, we attempt to describe the "building blocks" of these actions by viewing path algebras as tensor algebras in the tensor category $\mathsf{rep}(H)$ for the appropriate quantum group $H$ and attempting to classify the minimal, faithful tensor algebras. We construct an equivalence between categories of bimodules in $\mathsf{rep}(H)$ and a subcategory of certain finite-dimensional representations of associative algebras, explicitly given in terms of quivers with relations. This allows us to determine whether classification of indecomposable bimodules in these categories is feasible based on the representation types of the quivers.
Joint work with Ryan Kinser (University of Iowa).
The Drinfeld double of the Jordan plane, the super Jordan plane and their restricted versions
Héctor Martín Peña Pollastri
Universidad Nacional de Córdoba, Argentina - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this work, we consider Hopf algebras naturally associated with the Jordan plane, the super Jordan plane, and their restricted versions in characteristic odd. Specifically, we consider the Drinfeld double of their bosonizations. We describe these algebras in various ways and we compute their simple modules.
The Jordan plane is a well-known quadratic algebra in non-commutative geometry and the theory of quantum groups. Cibils, Lauda, and Witherspoon observed this is a Nichols algebra in characteristic 0 and they compute the corresponding Nichols algebra in positive characteristic. In this case, the resulting algebra is finite-dimensional and we call it the restricted Jordan plane. The super Jordan plane introduced by Andruskiewitsch, Angiono, and Heckenberger in 2016, is an algebra with two generators, one quadratic and one cubic relation. In characteristic 0 it is a Nichols algebra. In 2019 the same authors computed the corresponding Nichols algebra in positive characteristic. This finite-dimensional algebra is called the restricted super Jordan plane. These two algebras are important in the classification of the Nichols algebras with finite Gelfand-Kirillov dimension.
We first consider the restricted Jordan plane, its bosonization $H$ by a cyclic group and its corresponding Drinfeld double $D(H)$. We present the latter with generators and relations. It follows from this presentation that $D(H)$ is an abelian extension (as Hopf algebra) of the restricted enveloping algebra of $\mathfrak{u}(\mathfrak{sl}_2)$ by a local Hopf algebra. This allows us to deduce the simple modules of $D(H)$ are the same as the ones of $\mathfrak{u}(\mathfrak{sl}_2)$. Afterward, we construct an infinite-dimensional covering $\widetilde{D}$ de $D(H)$ and we prove that this is an abelian extension of the enveloping algebra $U(\mathfrak{sl}_2)$ by the algebra of regular functions of a soluble algebraic group. Hence, we obtain a commutative diagram of nine Hopf algebras with three exact rows and three exact columns. The vertical arrows are a (kind of) quantum Frobenius morphisms.
We also obtain similar results with the restricted super Jordan plane but related with the category of super vector spaces. The Drinfeld double constructed $E$ is a bosonization of a Hopf superalgebra $\mathcal E$; that is $E \simeq \mathcal E \# \Bbbk C_2$. Furthermore, $\mathcal E$ is a super abelian extension of the restricted enveloping algebra of the Lie superalgebra $\mathfrak{osp}(1|2)$ by a local Hopf algebra. Hence the simple modules are the same as the ones from $\mathfrak{u}(\mathfrak{osp}(1|2))$. Afterward, we construct an infinite-dimensional covering $\widetilde{\mathcal E}$ of $\mathcal E$. The latter is a super abelian extension of the enveloping algebra $U(\mathfrak{osp}(1|2))$ by the algebra of functions of an algebraic supergroup. We then obtain a similar commutative diagram of nine Hopf superalgebras to the one obtained for the Jordan plane.
Joint work with Nicolás Andruskiewitsch (Universidad Nacional de Córdoba).
Real quantum error correction
Diego Arturo Romero Fonseca
Universidad de los Andes, Colombia - This email address is being protected from spambots. You need JavaScript enabled to view it.
The poster aims to show part of the theory developed for the correction of errors in quantum codes, showing first the necessary and sufficient conditions that make a code error-correcting. Later we move to some of the construction of subspaces of n-cubit that can be protected against certain sets of errors; to arrive at these results we will provide the set of error operators with a symplectic space structure. We will describe its isotropic subspaces, group of symplectic transformations, among other characteristics.
Finally, since errors on a quantum codes are the action of complex unitary matrices, we will show how the results change when we consider only real errors. In this case, the symplectic structure will be replaced by a quadratic space and Its orthogonal group will take the role of the symplectic group. These changes result in a theory of great harmony where the real errors do not need to pass into the complex world to be corrected.
Algebraic structures in group-theoretical fusion categories
Ana Ros Camacho
Cardiff University, Wales - This email address is being protected from spambots. You need JavaScript enabled to view it.
Algebra objects (or simply, algebras) in categories with a tensor product are interesting mathematical objects that appear in a range of research topics in mathematical physics, like e.g. boundary conditions of rational conformal field theories, VOA extensions or spin topological field theories. In this talk, we generalize results from Ostrik and Natale that describe algebras in pointed fusion categories to the case of group-theoretical fusion categories. These algebras also have very nice properties that we will describe in detail.
Joint work with Yiby Morales (Universidad de Los Andes, Colombia), Monique Mueller (Universidade Federal de Sao Joao del-Rei, Brazil), Julia Plavnik (Indiana University, Bloomington; USA), Angela Tabiri (AIMS, Ghana) and Chelsea Walton (Rice University, USA).
Tambara-Yamagami Categories over the Reals: The Non-Split Case
Sean Sanford
Indiana University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
In 1998, Daisuke Tambara and Shigeru Yamagami investigated a simple set of fusion rules, and proved under which circumstances those rules could be given a coherent associator. This elementary yet difficult computation remains an outlier in the subject, as such associators are rarely computed by hand.
In this project, we are investigating a generalization of such fusion rules to the setting where the simple objects are no longer required to be absolutely simple. Over the real numbers, this means that objects are either real, complex or quaternionic. This flexibility has interesting implications that we will explore through a series of examples.
Joint work with Julia Plavnik (Indiana University) and Dalton Sconce (Indiana University).
Partial (co)actions of Taft and Nichols Hopf algebras on their base fields
Leonardo Silva
Universidade Federal de Santa Maria, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
Using the method developed in [1] to compute partial actions of a given Hopf algebra on its base field, in this paper we determine all partial actions and partial coactions of Taft and Nichols Hopf algebras on their base fields. Furthermore, we prove that all such partial (co)actions are symmetric. These results are developed in [2].
[1] G. Martini, A. Paques and L. Silva. Deformations of Hopf algebras by partial actions, arXiv e-prints (2020), arXiv:2009.08540
[2] G. Fonseca, G. Martini and L. Silva. Classifying partial (co)actions of Taft and Nichols Hopf algebras on their base fields, arXiv e-prints (2020), arXiv:2012.04106
Joint work with Grasiela Martini (Universidade Federal do Rio Grande, Brazil) and Graziela Fonseca (Instituto Federal Sul Rio-Grandense, Brazil).