Abstracts

Honglian Zhang:

Title: Quantum N-toroidal algebras and extended quantized GIM algebras of N-fold affinization.

 Abstract：We introduce the notion of quantum N-toroidal algebras as natural generalization of the quantum toroidal algebras as well as extended quantized GIM algebras of Nfold affinization. We show that the quantum N-toroidal algebras are quotients of the extended quantized GIM algebras of N-fold affinization, which generalizes a well known result of Berman and Moody for Lie algebras. This talk is based on the joint work with Prof. Y. Gao, N. Jing and L. Xia

Nina Yu:

Title: On Fusion Products of Twisted Modules in Permutation Orbifolds 

Abstract: Orbifold theory examines a vertex operator algebra under the action of a finite group. The primary focus lies in understanding the representation theory for the fixed-point vertex operator subalgebra. Permutation orbifolds investigate the action of the symmetric group of degree n on the n-tensor product of a vertex operator algebra. In this talk, I will talk about our recent work on fusion products of twisted modules in permutation orbifolds. This is a joint work with C. Dong and F. Xu.

Elena Aladova Chestakov

Title: Automorphisms of category of free finitely generated algebras with 

Abstract: Let $\Theta$ be an arbitrary variety of algebras and $\Theta^{0}$ the category of all free finitely generated algebras in $\Theta$. The group $\operatorname{Aut}\left(\Theta^{0}\right)$ of automorphisms of the category $\Theta^{0}$ plays an important role in universal algebraic geometry. We refer to $[2,3]$ for relevant notions in universal algebraic geometry.

It turns out that for a wide class of varieties, the group $\operatorname{Aut}\left(\Theta^{0}\right)$ can be decomposed into a product of the normal subgroup $\operatorname{Inn}\left(\Theta^{0}\right)$ of inner automorphisms and the subgroup $\operatorname{St}\left(\Theta^{0}\right)$ of strongly stable automorphisms. The method of verbal operations provides a machinery to calculate the group $\operatorname{St}\left(\Theta^{0}\right)$ of strongly stable automorphism (see $[4,5,6]$ ). This allows us to find out the structure of the group of outer automorphisms $\operatorname{Out}\left(\Theta^{0}\right)=\operatorname{Aut}\left(\Theta^{0}\right) / \operatorname{Inn}\left(\Theta^{0}\right)$, which, in some sense, measures the difference between two types of geometrical equivalences of algebras (see [3]).

In this talk we give some clarifying remarks describing the place of $\Theta^{0}$ and $\operatorname{Aut}\left(\Theta^{0}\right)$ in the general set up of the universal algebraic geometry and discuss new results concerning the group of strongly stable automorphisms for the variety of non-associative algebras with unit (see [1]).

References

[1] E. Aladova, J.S. Sousa de Lima Oliveira, Automorphisms of the Category of Free Non-Associative Algebras with Unit. J. Algebra Appl., (2024) accepted.

[2] B. Plotkin, Seven lectures on the universal algebraic geometry. In: Groups, Algebras and Identities. Contemporary Mathematics AMS, Vol. 726 (2019) $143-215$.

[3] B. Plotkin, Some results and problems related to universal algebraic geometry, Int. J. Algebra Comput., Vol. 17(56) (2007) 1133-1164.

[4] B. Plotkin, G. Zhitomirski, Automorphisms of categories of free algebras of some varieties, J. Algebra, Vol. 306(2) (2006) 344-367.

[5] A. Tsurkov, Automorphic equivalence of algebras, Int. J. Algebra Comput., Vol. 17(5-6) (2007) 1263-1271.

[6] G. Zhitomirski, On automorphisms of categories with applications to universal algebraic geometry, Algebra Univers., Vol. 84(31) (2023) 24 pp.

Qing Wang (Xiamen)

Title : Affine vertex operator superalgebra $L_{\widehat{osp(1|2)}}(l,0)$ at admissible level

Abstract： We present our recent results on affine vertex operator superalgebra $L_{\widehat{osp(1|2)}}(l,0)$ at admissible level $l$.  We prove that the category of weak $L_{\widehat{osp(1|2)}}(l,0)$-modules on which the positive part of $\widehat{osp(1|2)}$ acts locally nilpotent is semisimple. Then we prove that Q-graded vertex operator superalgebras $(L_{\widehat{osp(1|2)}}(l,0),\omega_\xi)$ with a new Virasoro element $\omega_\xi$ are rational and the irreducible modules are exactly the admissible modules for $\widehat{osp(1|2)}$, where $0<\xi<1$ is a rational number. Furthermore, we determine the Zhu's algebras $A(L_{\widehat{osp(1|2)}}(l,0))$ and their bimodules $A(L(l,j))$ for $(L_{\widehat{osp(1|2)}}(l,0),\omega_\xi)$, where $j$ is the admissible weight. As an application, we calculate the fusion rules among the irreducible ordinary modules of $(L_{\widehat{osp(1|2)}}(l,0),\omega_\xi)$. This is a joint work with Huaimin Li.

 Zhaobing Fan

Title: Geometric approach to quantum algebras

Abstract: In the past thirty years, geometric representation theory has made rapid development, and the geometric approach to quantum algebras is one of directions, which can  be regarded as a categorization of quantum algebras. In this talk, I will briefly review some geometric approaches to quantum algebras and  recent progress.

Yuly Billig (Carleton University)

Title: Gelfand-Fuks cohomology of vector fields on algebraic varieties

Abstract: Gelfand-Fuks cohomology of vector fields was developed in the 1970s in the setting of C^\infty manifolds. In this talk, we explain how to adapt it to the algebraic setting of affine varieties, and more generally, to Lie-Rinehart algebras. As an illustration of our methods, we compute algebraic Gelfand-Fuks cohomology with values in tensor modules for the affine space and for the torus.

David Ridout (Melbourne, Australia)

Title: Weight modules for affine vertex operator algebras with finite

and infinite multiplicities.

Abstract: We investigate the representation theory of the simple affine vertex operator algebra $L_k(\mathfrak{g})$ at admissible levels $k$. For $\mathfrak{g}=\mathfrak{sl}_2$, the irreducible weight modules all have finite multiplicities, while for higher ranks, infinite multiplicities are encountered for most $k$. This leads to infinite-multiplicity weight $\mathfrak{g}$-modules that appear to be different to those constructed by other means (eg, as Gelfand--Tsetlin modules).

Jiansai Sun:

Title：General twisting of of nonlocal vertex algebras

Abstract：In this talk we introduce and study the concept of twistor for a nonlocal vertex algebra. This concept provides a unifying for various constructions of nonlocal vertex algebras, such as twisted tensor products of nonlocal vertex algebras, iterated twisted tensor products of nonlocal vertex algebras and L-R- twisted tensor products of nonlocal vertex algebras.

V.Dotsenko  (University of Strasbourg, France)

Title: Finite basis property and distributivity questions for varieties of Novikov algebras

Abstract: Novikov algebras are important nonassociative algebras that originally appeared in the study of Hamiltonian operators in the formal calculus of variations by Gelfand and Dorfman, and then rediscovered by Balinskii and Novikov in the context of classification of linear Poisson brackets of hydrodynamical type. In this talk, I shall present two recent results about varieties of Novikov algebras over a field of zero characteristic: first, I shall explain that every variety of Novikov algebras is defined by finitely many identities (that is, has the Specht property), and second, I shall present a classification of all varieties of Novikov algebras whose lattice of subvarieties is distributive. These results are obtained in collaboration with N. Ismailov and U. Umirbaev (the first one) and with B. Zhakhayev (the second one).

José Victor Gomes Teixeira (UFRN / IME - USP)

Title: The Group of Outer Automorphisms of the Category of Finitely Generated Nilpotent Free Algebras of Degree n

Abstract: In this work, which is a joint work with A. Tsurkov, we compute the quotient group $\mathfrak{A}_{n}/\mathfrak{Y}_{n}$ of all automorphisms $\mathfrak{A}%_{n}$ of the category $\Theta _{n}^{0}$ of all finitely generated free nilpotent algebras of degree $n$ over a field $\Bbbk $, by the normal subgroup $\mathfrak{Y}_{n}$ of all inner automorphisms of this category. We call $\mathfrak{A}_{n}/\mathfrak{Y}_{n}$ the group of outer automorphisms of  $\Theta _{n}^{0}$. In the universal algebraic geometry setting, this group is very important, because it measures the possible difference between geometric and automorphic equivalences in the variety $\Theta _{n}$ of all nilpotent algebras of degree $n$. We prove that $\mathfrak{A}_{n}/\mathfrak{Y% }_{n}\cong \Bbbk ^{\ast }\rtimes Aut\Bbbk $, where $\Bbbk ^{\ast }$ is the multiplicative group of all invertibles elements of $\Bbbk $ and $\operatorname{Aut}\Bbbk $ is the group of all automorphisms of $\Bbbk $. It is interesting that if we consider the category $\Theta ^{0}$ of all finitely generated algebras over a field $\Bbbk $, the group $\mathfrak{A}$ of all automorphisms of this category and the group $\mathfrak{Y}$ of all inner automorphisms of this category, then also $\mathfrak{A}/\mathfrak{Y}\cong \Bbbk ^{\ast }\rtimes \operatorname{Aut}\Bbbk $.

Alan Guimarães (UFRN)

Títle:  A characterization of the natural Z2-grading of the Grassmann algebra

 Abstract: Let F be any field of characteristic different from two and let E be the Grassmann algebra of an infinite dimensional F-vector space L. In this talk, we will present a condition for a Z2-grading on E to behave as the natural Z2-grading Ecan. More specifically, our aim is to prove the validity of a weak version of a conjecture presented in [1]. The conjecture poses that every Z2-grading on E has at least one non-zero homogeneous element of L. As a consequence, we obtain a characterization of Ecan by means of its Z2-graded polynomial identities. Furthermore we construct a Z2-grading on E that refutes the conjecture in general.

References: 

[1] GUIMARÃES, A. ; KOSHLUKOV, P. Automorphisms and superalgebra structures on the grassmann algebra. Journal of Pure and Applied Algebra,  2023. 

 [2] GUIMARÃES, A. ; FIDELES, C. ; GOMES, A. B.; GRISHKOV. A characterization of the natural grading of the Grassmann algebra and its non-homogeneous Z2-gradings. Linear Algebra and its Applications , 2024.    

Arkady Tsurkov (UFRN)

Title: Categories and functors of universal algebraic geometry. Automorphic equivalence of algebras.

Universal algebraic geometry allows considering of algebraic geometry of every universal algebra. When two algebras have same algebraic geometry? We must consider the categories of algebraic closed sets of these algebras to answer this question. The complete coincidence of these categories gives us a concept of the geometric equivalence of algebras. Some type of isomorphisms of these categories gives us a concept of the automorphic equivalence of algebras. This concept has been considered since Plotkin. We will give by language of category theory one more elegant definition of this concept and recall some theorems related to this concept.

References: 

[1] B. Plotkin, Algebras with the same algebraic geometry. 

[2] Proceedings of the Steklov Institute of Mathematics}, MIAN, Vol. 242 (2003).

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