2019 Conference on Operad Theory and Related Topics  

Welcome to participate in 2019 Conference on Operad Theory and Related Topics !  




Operads associated to weak bialgebras
BAO, Yan-Hong
Anhui University, China
Abstract:  In this talk, we provide several constructions of plain operads by weak bialgebras. For a 1-cocommutative weak bialgebra B, the category of algebras over the plain operad associated to B is equivalent to the category of left B-module algebras. Furthermore, a cohomology theory for 1-cocommutative weak bialgebras is obtained. This is joint work with Xiao-Wei Xu, Yu Ye, James J. Zhang and Zhi-Bing Zhao.
Homotopy embedding tensors and associated Getzler trees
CHEN, Zhuo
Tsinghua University, China

Abstract: The embedding tensor appears in the gauging procedure of supergravity theories. Kotov and Strobl prove that there exists a correspondence between embedding tensors and Leibniz algebras, and show that the associated tensor hierarchy only depends on the corresponding Leibniz algebra. In this talk, we consider a homotopic version of embedding tensors in the context of dg geometry and derive SH(strongly homotopy) Leibniz algebras from homotopic embedding tensors. The construction of SH Leibniz structures is based on the operad of weighted Getzler trees. Several interesting examples will be discussed. This is joint work with M. Xiang, W. Jiang and T. Zhang.

Unary binary qudratic/cubic Operadic Compatible Structures
GAO, Xing
Lanzhou University, China
Abstract: Strohmayer characterized linear and total compatibilities of binary quadratic operads. In this talk, we introduce linear, matching and total compatibilities of unary binary qudratic/cubic operads. As applications, we obtain these three  compatibilities on top of differential algebras and Rota-Baxter algebras.  This is a joint work with Li Guo and Huhu Zhang.
Exact Hochschild extensions and deformed Calabi-Yau completions
HAN, Yang
Academy of Mathematics and Systems Science, CAS, China
Abstract: We introduce the Hochschild extensions of dg algebras, which are A-infinite algebras. We show that exact Hochschild extensions are symmetric Hochschild extensions, more precisely, every exact Hochschild extension of a finite dimensional complete typical dg algebra is a symmetric A-infinite algebra. Moreover, we prove that the Koszul dual of trivial extension is Calabi-Yau completion and the Koszul dual of exact Hochschild extension is deformed Calabi-Yau completion, more precisely, the Koszul dual of the trivial extension of a finite dimensional complete dg algebra is the Calabi-Yau completion of its Koszul dual, and the Koszul dual of an exact Hochschild extension of a finite dimensional complete typical dg algebra is the deformed Calabi-Yau completion of its Koszul dual. This talk is based on the preprint arXiv:1909.02200v1 [math.RA]. It is a joint work with Xin LIU and Kai WANG.
Nonsymmetric operads and various structures over cohomology theories
LYU, Wei-Guo
University of Science and Technology of China, China
Abstract: We investigate the relationship between nonsymmetric operads and various structures over cohomology theories. L. Menichi proves in 2004 that the cohomology of the complex given by a nonsymmetric cyclic operad with multiplication is a Batalin-Vilkovisky algebra, and we provide a chain level version of this result by showing that the cohomological complex of a nonsymmetric cyclic operad with multiplication is a Quesney algebra. We define cyclic opposite operad modules with pairing and show that the existence of such cyclic opposite operad module with pairing implies that the operad is a cyclic operad and hence its cohomological complex is a Quesney algebra, continuing a line of research beginning by N. Kowalzig et al..


Homotopy Rota-Baxter operators, homotopy O-operators and homotopy post-Lie algebras
Jilin University, China
Abstract: Rota-Baxter operators, O-operators on Lie algebras and their interconnected pre-Lie and post-Lie algebras are important algebraic structures with broad applications. This paper introduces the notions of a homotopy Rota-Baxter operator and a homotopy O-operator on a symmetric graded Lie algebra. Their characterization by Maurer-Cartan elements of suitable differential graded Lie algebras is provided. Through the action of a homotopy O-operator on a symmetric graded Lie algebra, we arrive at the notion of an operator homotopy post-Lie algebra, together with its characterization in terms of Maurer-Cartan elements. A cohomology theory of post-Lie algebras is established, with an application to 2-term skeletal operator homotopy post-Lie algebras.
2-dimensional topological field theories and invariants of Calabi-Yau categories

TU, Jun-Wu

ShanghaiTech University, China
Abstract: In this talk, I shall report some recent progress on defining enumerative invariants of Calabi-Yau categories through 2d topological field theories. We discuss its applications in algebraic geometry, symplectic geometry and mirror symmetry. The talk is based on joint works with Kevin Costello and Andrei Caldararu.
A proof of the Brown-Goodearl conjecture for module-finite weak Hopf algebras
WON, Robert J.
University of Washington, USA
Abstract: Brown and Goodearl conjectured that any noetherian Hopf algebra should have finite injective dimension. This conjecture is known to be true in certain cases, in particular for affine polynomial identity Hopf algebras. Weak Hopf algebras are an important generalization of Hopf algebras. Just as for Hopf algebras, the category of modules over a weak Hopf algebra has a monoidal structure, and this has important consequences for homological properties of the algebra. We study the extension of the Brown-Goodearl conjecture to the case of weak Hopf algebras, and show that a weak Hopf algebra which is finite over an affine center has finite injective dimension and is a direct sum of AS Gorenstein algebras. (Joint with Daniel Rogalski and James Zhang.)
Derivations between algebraic operads
XU, Yong-Jun
Qufu Normal University, China
Abstract: In a recent work, Bao and his coauthors studied various invariants such as cohomology groups, derivations, automorphisms of algebraic operads. In my talk, I will report some results about homomorphisms, derivations and the first cohomology groups between some algebraic operads corresponding to some algebra structures. Moreover, I will introduce the relationship between derivations and universal deformations of algebraic operads. This is a joint work with James Zhang.
Hopf algebra structures on weak compositions and signed permutations
YU, Hou-Yi
Southwest University, China
Abstract: Motivated by a question of Rota relating symmetric functions to Rota-Baxter algebras, quasi-symmetric functions parameterized by compositions were generalized to those parameterized by weak compositions, yielding the Hopf algebra of weak quasi-symmetric functions which has the Hopf algebra of quasi-symmetric functions as both a Hopf subalgebra and Hopf quotient algebra. In order to give a combinatorial interpretation and a simple multiplication rule for the fundamental weak quasi-symmetric functions, we construct a Hopf algebra from signed permutations which naturally contains the Malvenuto-Reutenauer Hopf algebra from permutations as a Hopf subalgebra. Furthermore, we obtain a commutative diagram of these Hopf algebras revealing a relationship among compositions, weak compositions, permutations and signed permutations. This is a joint work with Li Guo and Jean-Yves Thibon.
             Universal Enveloping Algebras and Poincaré-Birkhoff-Witt Theorem
for Involutive Hom-Lie Algebras
ZHENG, Shang-Hua
Jiangxi Normal University, China
Abstract: Hom-type algebras, in particular Hom-Lie algebras, have attracted quite much attention in recent years. A Hom-Lie algebra is called involutive if its Hom map is multiplicative and involutive.
In this talk, we obtain an explicit construction of the free involutive Hom-associative algebra on a Hom-module. We then apply this construction to obtain the universal enveloping algebra of an involutive Hom-Lie algebra. Finally, we generalize the well-known Poincaré-Birkhoff-Witt theorem for enveloping algebras of Lie algebras to involutive Hom-Lie algebras.
The Structure of connected (graded) Hopf algebras
ZHOU, Gui-Song
Ningbo University, China
Abstract: In this talk, we will present a structure theorem for connected graded Hopf algebras over a field of characteristic 0 by claiming the existence of a  family of homogeneous generators and a total order on the index set  that satisfy some excellent conditions. The approach to the structure theorem is based on the combinatorial properties  of Lyndon words and the standard bracketing on words. As an immediate consequence of the structure theorem, it turns out that connected graded Hopf algebras of finite Gelfand-Kirillov dimension over a field of characteristic 0 are all iterated Hopf Ore extensions of the base field. This is a Joint work with Di-Ming Lu and Yuan Shen.