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Abstracts
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| Operads associated to
weak bialgebras |
| BAO, Yan-Hong |
| Anhui University, China |
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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. |
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Homotopy embedding
tensors and
associated Getzler
trees |
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CHEN, Zhuo |
| Tsinghua University, China |
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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.
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Unary binary qudratic/cubic Operadic Compatible
Structures |
| GAO, Xing |
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Lanzhou University, China |
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| 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. |
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Exact Hochschild extensions and deformed Calabi-Yau
completions |
| HAN, Yang |
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Academy of Mathematics and Systems Science, CAS, China |
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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. |
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Nonsymmetric operads and various structures over
cohomology theories |
| LYU, Wei-Guo |
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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..
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Homotopy Rota-Baxter operators, homotopy O-operators and
homotopy post-Lie algebras |
| SHENG, Yun-He |
| Jilin University, China |
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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. |
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2-dimensional topological field theories and invariants
of Calabi-Yau categories |
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TU,
Jun-Wu
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ShanghaiTech University, China |
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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. |
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| A proof of the Brown-Goodearl
conjecture for module-finite weak Hopf algebras |
| WON, Robert J. |
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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.) |
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Derivations between algebraic operads |
| XU, Yong-Jun |
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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. |
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Hopf algebra structures on weak compositions and signed
permutations |
| YU, Hou-Yi |
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Southwest University, China |
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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.
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Universal Enveloping Algebras and Poincaré-Birkhoff-Witt
Theorem
for Involutive Hom-Lie Algebras |
| ZHENG, Shang-Hua |
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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. |
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The
Structure of connected (graded) Hopf algebras |
| ZHOU, Gui-Song |
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Ningbo University, China |
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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. |
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