Lie Theory in Mathematical Physics online Workshop
December 26-27, 2020
Husileng Xiao (Harbin Engineering University)
December 26, 9:00 am, 腾讯会议，ID：901 129 375
Title: On the finite W-(super)algebras and their representations
Abstract: In this talk, I will give a brief survey to the constructions and representation of finite W-(super)algebras. First, I will introduce how to construct W-(super)algebras by quantum Hamiltonian reduction and Daboux-Weinstein theorem. Then I will introduce the Schur-Weyl dualities and Soergel functors for finite W-(super)algebras.
Yanpeng Li (University of Toronto)
December 26, 10:00 am, 腾讯会议，ID：901 129 375
Title: Integrable systems from tropical limits
Abstract: In this talk, we use tropical limit and the trick of averaging Moser flows to construct new Gelfand-Zeitlin type integrable systems for unitary Lie algebras. As an application, we find Gelfand-Zeitlin type integral systems for real symplectic Lie algebras. This is a joint work with Xiaomeng Xu.
Jianghua Lu, (University of Hongkong)
December 26, 11:00 am, 腾讯会议，ID：901 129 375
Title: Dirac Geometry and Integration of Poisson Homogeneous Spaces
Abstract: We use tools from Dirac geometry to show that every Poisson homogeneous space of any Poisson Lie group has a symplectic groupoid. This is joint work with Bursztyn and Iglesias-Ponte.
Jiefeng Liu, (Northeast Normal University)
December 27, 9:00 am, 腾讯会议，ID：357 234 945
Title: F-manifold algebras and deformation quantization via pre-Lie algebras
Abstract: The notion of an F-manifold algebra is the underlying algebraic structure of an F-manifold. We introduce the notion of pre-Lie formal deformations of commutative associative algebras and show that F-manifold algebras are the corresponding semi-classical limits. We study pre-Lie infinitesimal deformations and extension of pre-Lie n-deformation to pre-Lie (n+1)-deformation of a commutative associative algebra through the cohomology groups of pre-Lie algebras. We introduce the notions of pre-F-manifold algebras and show that a pre-F-manifold algebra gives rise to an F-manifold algebra through the sub-adjacent associative algebra and the sub-adjacent Lie algebra. We use Rota-Baxter operators, more generally O-operators on F-manifold algebras to construct pre-F-manifold algebras. This is a joint work with Chengming Bai and Yunhe Sheng.
Yuancheng Xie, (Ohio State University)
December 27, 10:00 am, 腾讯会议，ID：357 234 945
Title: A biased introduction to Toda lattice
Abstract: In 1967, Japanese physicist Morikazu Toda proposed an integrable lattice model to desribe motions of a chain of particles with exponential interactions between nearest neighbors. Since then, Toda lattice and its various generalizations become the test ground for various techniques and philosophies in integrable systems and they have been analyzed from the perspectives of analysis, geometry, combinatorics, representation theory etc. In this talk, I will give a surely biased introduction to the classical Toda lattice and discuss some recent progress and problems of its generalizations.
Yunhe Sheng, (Jilin University)
December 27, 11:00 am, 腾讯会议，ID：357 234 945
Title: Leibniz bialgebras, relative Rota-Baxter operators and the classical Leibniz Yang-Baxter equation
Abstract: We introduce the notion of a Leibniz bialgebra and show that matched pairs of Leibniz algebras, Manin triples of Leibniz algebras and Leibniz bialgebras are equivalent. Then we introduce the notion of a (relative) Rota-Baxter operator on a Leibniz algebra and construct the graded Lie algebra that characterizes relative Rota-Baxter operators as Maurer-Cartan elements. By these structures and the twisting theory of twilled Leibniz algebras, we further define the classical Leibniz Yang-Baxter equation, classical Leibniz r-matrices and triangular Leibniz bialgebras. Finally, we construct solutions of the classical Leibniz Yang-Baxter equation using relative Rota-Baxter operators and Leibniz-dendriform algebras.