## 李理论系列报告二十七则

2021年4月17日 ID：659 221 582，密码：210417

2021年4月18日 ID：280 594 168，密码：210418

Quantum affine algebras are quantum groups associated with affine Lie algebras. In this talk, I will present the recent joint work with Ming Liu and Alexander Molev on R-matrix presentations of quantum affine algebras, which shows the isomorphism with the Drinfeld realization.

：姚裕丰 教授

We construct a family of non-weight modules over the super-Virasoro algebras and the $N=2$ superconformal algebras. These modules when regarded as modules over the Cartan subalgebra $\mathfrak{h}$ (modulo the center) are free of rank 1 or 2. We obtain a sufficient and necessary condition for such modules to be simple. Moreover, we determine the isomorphism classes of these modules. Finally, we show that these modules constitute a complete classification of $U(\mathfrak{h})$-free modules of rank 1 or 2. A relationship among such non-weight modules over these superalgebras is precisely presented. This is a joint work with Hengyun Yang and Limeng Xia.

：董超平 教授

After briefly recalling the background of Dirac cohomology, I will introduce the recent progresses on the classification of Dirac series.

：胡峻 教授

In this talk we shall give a brief introduction to the Z-graded representations of the BGG category O for any semisimple Lie algebra. We shall illustrate how Kazhdan-Lusztig theory plays a central role in this Z-graded picture.

：陈良云 教授

In this talk, we study some properties of biderivations and $(\sigma,\tau)$-derivations and obtain their relations with centroids, respectively. Moreover, centroids are also considered. This talk is a report on joint work with Yao Ma and Chenrui Yao.

：万金奎 教授

In a joint work with Weiqiang Wang, we established the stability property of the centers of the integral group algebra of the general linear group GL(n,q) over a finite field. We show that the stability property can be extended to the case of affine groups P(n,q), symplectic groups Sp(2n,q) and unitary groups U(n,q^{2}). More precisely, the group algebras of these finite groups admit a filtration defined via the affine reflections, transvections, respectively. We establish that the structure constants of the associated graded algebras are independent of n.

In a previous paper by the authors, we obtain the first example of a finitely freely generated simple $\mathbb{Z}$-graded Lie conformal algebra of linear growth that cannot be embedded into any general Lie conformal algebra. In this talk, we obtain, as a byproduct, another class of such Lie conformal algebras by classifying $\mathbb{Z}$-graded simple Lie conformal algebras containing the Virasoro conformal algebra. These algebras include some Lie conformal algebras of Block type.

：邓少强 教授

In this talk, we present our recent results on normal homogeneous Finsler spaces. We first define the notion of a normal homogeneous Finsler space, using the method of isometric submersion of Finsler metrics. Then we establish a technique to reduce the classification of normal homogeneous Finsler spaces of positive flag curvature to an algebraic problem. We show that a coset space G/H admits a positively curved normal homogeneous Finsler metric if and only if it admits a positively curved normal homogeneous Riemannian metric. We also give a complete description of the coset spaces admitting non-Riemannian positively curved normal homogeneous Finsler spaces.

：安金鹏 教授

One-parameter subgroup actions on homogeneous spaces of Lie groups exhibit rich structures and are related to problems in number theory. In this talk, we will discuss some problems and results concerning divergent forward orbits of such actions and their relations with Diophantine approximation. After reviewing some background, we will report a recent result for which the space is a product of spaces of the form SL(d,R)/SL(d,Z). The latter is a joint work with Lifan Guan, Antoine Marnat and Ronggang Shi.

：生云鹤 教授

Rota-Baxter operators were originally defined on a commutative associative algebra by Rota. Then it was defined on Lie algebras as the operator form of the classical Yang-Baxter equation. Kupershmidt introduced a more general notion called O-operator (later called relative Rota-Baxter operator) for arbitrary representation. Rota-Baxter operators have fruitful applications in mathematical physics. We determine the L-infty-algebra that characterizes relative Rota-Baxter Lie algebras as Maurer-Cartan elements. As applications, first we determine the L-infty-algebra that controls deformations of a relative Rota-Baxter Lie algebra and show that it is an extension of the dg Lie algebra controlling deformations of the underlying Lie algebra and representation by the dg Lie algebra controlling deformations of the relative Rota-Baxter operator. Then we define the cohomology of relative Rota-Baxter Lie algebras and relate it to their infinitesimal deformations. In particular the cohomolgoy of Rota-Baxter Lie algebras and triangular Lie bialgebras are given. Finally we introduce the notion of homotopy relative Rota-Baxter operators and show that the underlying structure is pre-Lie-infinity algebras. This talk is based on joint works with Chenming Bai, Li Guo, Andrey Lazarev and Rong Tang.

：陈小伍 教授

The folding of root lattices is classical in Lie theory. Following Gabriel and Geiss-Leclerc-Schoerer, the root lattices are categorified by the module categories of some finite dimensional quiver algebras. We categorify the folding projective, namely obtain an ecact functor between the module categoes whose $K_{0}$-shadow coincides with the folding projection. The construction uses EI categories of Cartan type. This is joint with Ren Wang.

For a commutative algebra $A$ over $\mathbb{C}$, let $\mathfrak{g}=\text{Der}(A)$. A module over the smash product $A\# U(\mathfrak{g})$ is called a jet $\mathfrak{g}$-module, where $U(\mathfrak{g})$ is the universal enveloping algebra of $\mathfrak{g}$. In this talk, we talk about jet modules when $A=\mathbb{C}[t_1^{\pm 1},t_2]$. We show that  $A\#U(\mathfrak{g})\cong\mathcal{D}\otimes U(L)$, where $\mathcal{D}$ is the Weyl algebra $\mathbb{C}[t_1^{\pm 1},t_2, \frac{\partial}{\partial t_1},\frac{\partial}{\partial t_2}]$, and $L$ is a Lie subalgebra of  $A\# U(\mathfrak{g})$ called the jet Lie algebra corresponding to $\mathfrak{g}$. Using a Lie algebra isomorphism $\theta:L \rightarrow \mathfrak{m}_{1,0}\Delta$,  where $\mathfrak{m}_{1,0}\Delta$ is the subalgebra of vector fields vanishing at the point $(1,0)$, we show that any irreducible finite dimensional  $L$-module is isomorphic to an irreducible $\gl_2$-module. As an application, we give tensor product realizations of irreducible jet modules over $\mathfrak{g}$ with uniformly bounded weight spaces.

：袁腊梅 副教授

In this talk, we will give a classification of finitely irreducible conformal modules over a class of Lie conformal algebras W(b), and classify extensions of conformal modules over them. This is based on a series of joint work with Henan Wu and Kaijing Ling.

：包益欣 助理教授

In this talk, we briefly introduce the concept of locally Nash groups. We review the algebraization method established by Hrushovski and Pillay. Based on this techinique, we give a complete classification of connected commutative locally Nash groups.

：刘东 教授

In this talk, we introduce some progresses on the representations over the Ovisenko-Roger Lie algebras, which are an abelian extension of the Virasoro algebra by the tensor density modules. We classify all simple weight modules with finite dimensional weight spaces, and determine simple restricted modules generalizing highest weight modules and Whittaker modules. As a corollary, we classify all simple generalized Verma modules. It is based on some joint researches with Yufeng Pei, Limeng Xia and Kaiming Zhao.

：陈洪佳 教授

In this talk, I will first give an introduction of Nilsson's results about the $\mathcal{U}(\mathfrak{h})$-free modules over $\mathfrak{sl}_2(\mathbb{C})$ and an explicit formula for taking tensor product with a finite dimensional simple module. Then I will  show some recent computation about the tensor product. This talk is based on the paper by J. Nilsson and on some discussions with X. Guo, N. Jing and Y. Ma.

：罗栗 副教授

In this talk, we will provide a geometric realization of Howe duality via flag varieties of classical type. In particular, our (U_q(gl_m), U_q(gl_n))-duality coincides with the one established by Zhang Ruibin via quantum coordinate algebras.

：胡乃红 教授

In this short talk, I'll propose a conjecture on the modular finite-dimensional pointed Hopf algebras, together with some possible evidences from our own side.

：夏利猛 教授

In this talk, I will briefly introduce some results on (twisted)Whittaker modules of several quantum groups.

：陈智奇 教授

In this talk, we focus on pseudo-Riemannian Lie groups with non-Killing conformal vector fields. In the beginning, we show that such Lie groups are solvable for Lorentz case and type (n,2), and then give non-solvable examples for type (p,q) with p,q bigger than 3, finally we give the classification for Lorentz case.

：郭向前 教授

We construct a class of module over the affine algebra ${\hat\sl}_2$ from the degree two Weyl algebra ${\mathcal K}_2$. The irreducibility and isomorphisms among these algebras are determined.

：刘公祥 教授

In this talk, we will give the finite duals of some Hopf algebras of GK-1. From this, we can construct some Hopf parings naturally. Some questions are posed.

：夏春光 副教授

We classify finite irreducible conformal modules over a class of infinite Lie conformal algebras of Block type and their super generalizations. In particular, we prove that a finite irreducible conformal module admits a nontrivial extension of a finite conformal module over a Virasoro or Neveu-Schwarz conformal subalgebra if and only if p=-1. Extension problem for conformal modules over nonsuper case is also completely solved. This talk is based on the joint work with Yucai Su and Lamei Yuan.

：张洋 博士

For any finite Coxeter group there is a noncrossing partition (NCP) lattice comprising elements between the identity and a fixed Coxeter element. In analogy with the Orlik-Solomon algebra, I will define a finite dimensional Z-graded algebra, called noncrossing algebra, associated to any NCP lattice in two different ways. Each graded component of this algebra is isomorphic to the direct sum of all homology of order complexes of lower intervals of a fixed rank. In terms of this algebra, I will give an explicit finite chain complex of free abelian groups whose homology is the integral homology of the Milnor fibre of the corresponding Coxeter arrangement. This permits us to calculate the homology of Milnor fibres computationally. The actions of both W and the monodromy can be partly described by our chain complex. Time permitting, I will talk about connections with the integral homology of Artin groups and pure Artin groups.

：曹彬涛 副教授

We establish an explicit bijection between the sets of singular solutions of the (super) KZ equations associated to the Lie superalgebra, of infinite rank, of type $\mf{a, b, c, d}$ and to the corresponding Lie algebra. As a consequence, the singular solutions of the super KZ equations associated to the classical Lie superalgebra, of finite rank, of type $\mf{a, b, c, d}$ for the tensor product of certain parabolic Verma modules (resp., irreducible modules) are obtained from the singular solutions of the KZ equations for the tensor product of the corresponding parabolic Verma modules (resp., irreducible modules) over the corresponding Lie algebra of sufficiently large rank, and vice versa. The analogous results for some special kinds of trigonometric (super) KZ equations are obtained. This is a joint work with Ngau Lam.

：王浩 博士

For $\mathcal {S}$-local vertex operator algebra $V\# H$, where $H$ is the group algebra of a finite subgroup of $\Aut(V)$, Frobenius reciprocity theorem are investigated. We also give an explicit construction and classification of admissible $V\# H$-modules from admissible $V$-modules. We also give a complete set of irreducible inequivalent admissible $V\# H$-modules.

：贺艳 博士

Let $\fg=\sl_{n+1}, \cW_n^+$ or $\cW_n$, $\cD=\oplus_{i=1}^n\bC\frac{\ptl}{\ptl t_i}$. We classify the full subcategory $\fD(\fg)$ of $\fg$-$\Mod$ consisting of modules which are free of rank $1$ when restricted to $U(\cD)$. The irreducibility of modules in $\fD(\fg)$ is determined. Using modules in $\fD(\sl_{n+1})$, we construct classes of modules which are free of finite rank and infinite rank when restricted to the Cartan subalgebra.