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.

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