International Conference
on
Operads
and Universal Algebra
July 4 – July 9, 2010, Nankai University, Tianjin, China
Schedule of the Conference
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Monday July 5 |
Tuesday July 6 |
Wednesday July 7 |
Thursday July 8 |
Friday July 9 |
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8:30-9am |
Opening |
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9-10am |
Y.Chen |
P.L.Curien |
R.Holtkamp |
M.Ronco |
Y.Zhang |
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A.Makhlouf |
D.Liu |
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10:10-11:10am |
C.Xi |
S. Wang |
L.Kong |
H.Li |
Y.Z.Huang |
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11:10-11:30am |
Break |
Break |
Break |
Break |
Break |
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11:30-12:30 |
V.Dotsenko |
Y.Guiraud |
M.Livernet |
M.Mendez |
B.Vallette |
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12:30-2:00pm |
Lunch |
Lunch |
Lunch |
Lunch |
Lunch |
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2:00-2:45pm |
J.Hirsch |
A.Frabetti |
Discussions |
E.Burgunder |
End of Conference |
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2:50-3:35pm |
W.L.Yang |
P. Malbos |
X.Ni |
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3:40-4:25pm |
C.Vespa |
J.Milles |
Y.Fregier |
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4:25-5:00pm |
Break |
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5:00-5:30pm |
Y.S. Chen |
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5:30-6:00pm |
C.Selvaraj |
Titles and Abstracts
Emily Burgunder (Université
Paul Sabatier,
Title: Operads and Kontsevich graph
complexes
Abstract: On the one hand, Kontsevich showed that the
Lie homology of symplectic vector fields can be
computed via a certain graph homology.
This has been extended to operads via the work
of Conant and Vogtmann. On the other hand, Loday, Quillen and Tsygan have proved that the Lie
homology of the Lie matrices can be computed via cyclic homology, which can
then be reinterpreted as graph homology. A similar result due to Procesi and Loday has been proven in the orthogonal case. I
extend these results to the operadic case by
providing functor from operads
to Lie algebras with a group action of one of the groups : sp, o, or sl and relate
the Lie homology of these algebras with a graph complex. A generalisation
to Leibniz homology gives symmetric graphs.
Yongshan Chen (
Title: Groebner-Shirshov bases for Lie
algebras over a commutative algebra
Abstract: Here we present the Composition-Diamond lemma for a free Lie algebra
over a polynomial algebra. It provides a Gr\"{o}bner-Shirshov
bases theory for Lie algebras over a associative commutative algebra. As
applications we give another proofs that classical examples given by A.I. Shirshov (1953) and P. Cartier (1958) of Lie algebras (over
commutative algebras over $GF(2)$) are not embeddable
into associative algebras. We give a proof that the example suggested by P.M. Cohn (1963) of a Lie algebra over a commutative algebra over a
field of characteristic 2 or 3 is not
embeddable into associative algebra. For
characteristic $p>3$ it is an open problem. We find one series of one-relator Lie algebras over a commutative algebra that are embeddable into associative algebras. Finally we prove
that any countably-generated Lie algebra over a
commutative algebra $K$ is embeddable into a two-generated Lie algebra over
$K$.
Yuqun Chen (South China Normal University, China/ Sobolev
Institute of Mathematics, Russia)
Title: Some new results on Groebner-Shirshov bases for universal linear algebras
Abstract: In this survey, we report some recent results on Groebner-Shirshov bases. The following results are included. New
Composition-Diamond lemmas for: right-symmetric
algebras; associative Omega-algebras; tensor product of free associative
algebras; Lie algebras over a
commutative algebra; Dialgebras; Rota-Baxter algebras; differential
algebras; modules over an algebra;
categories; L-algebras. Groebner- Shirshov
bases for: free inverse semigroups; free Lie algebras
as anti-commutative algebras; someone-relator groups;
HNN extensions of groups; Chinese monoids; Braid
groups in Adyan-Thurston generators; The Onsager Lie
algebra (it is due to E. Poroshenko); Classical
simple Lie algebras relative to any order of simple roots (it is due to A. Koryukin). Some applications: Embedding of algebras into
simple algebras and two-generated algebras; Embedding dendriform
algebra into its universal enveloping Rota-Baxter algebra; Schreier
extensions of groups and algebras; PBW type theorems and normal forms theorems.
Pierre-Louis Curien (CNRS and
Title: On some applications of operadic
ideas and techniques in the field of the semantics of programming languages
Abstract: Operadic and
abstract homotopic ideas (or variants
thereof) and techniques have been
recently of use in the modelling of the semantics
of programming (rewriting, type theory,
concurrency theory). In the talk, we
shall try to give a feel for one or two such applications.
Vladimir Dotsenko (Dublin Institute for
Advanced Studies and
Title: Operadic homological algebra via shuffle operads
Abstract: In this talk, I shall explain how to use shuffle operads
to answer questions of homological algebra for usual (symmetric) operads. The central result is an explicit free resolution
for a shuffle operad with a known Groebner
basis, which leads to formulas for homology classes analogous to the ones
discovered by Anick in the case of associative algebras.
This result has some immediate applications, including the Koszulness
of operads built from Koszul
quadratic algebras, and the computation of the bar homology of the BV operad (a result that was recently announced by G.Drummond-Cole and B.Vallette).
Another application of this machinery is to prove results combinatorics
of consecutive pattern avoidance. I shall explain some of these results in
details, and give hints about proofs of the remaining ones. The talk is based
on joint works with A.Khoroshkin
Alessandra Frabetti (University of Lyon, France)
Title: Combinatorial Hopf Algebras from
renormalization
Abstract: According to Connes and Kreimer,
renormalization in perturbative quantum field theory
contains a combinatorial procedure which is efficiently described by Hopf algebras. Connes-Kreimer Hopf algebra for the scalar theory \phi^3 gives rise to the
toy model of right-sided combinatorial Hopf algebras
introduced by Loday and Ronco. I describe here three more examples of
right-sided combinatorial Hopf algebras which appear
as renormalization Hopf algebras for other field
theories. It is a joint work with Christian Brouder
and Frederic Menous.
Yael Frégier (
Title: Layer
cake and homotopy representations :
formal geometry approach
Abstract: Representations
up to homotopy of Lie algebras have attracted
recently much attention. On the other hand J. Baez has introduced a way to
build a homotopy Lie algebra
out of a Lie algebra and an n-cocycle. We show in
this work a common framework enabling to generalize both notions (replacing Lie
algebras by homotopy Lie algebras) and extend them
for other types of algebras (commutative and associative). The main tool is the
language of homological vector fields on products of formal manifolds. This is
a joint work with John Baez.
Philippe Malbos (INRIA and Université de Lyon, France)
Title: Higher-dimensional rewriting strategies and acyclic
polygraphs
Abstract: We introduce two
finiteness conditions for small categories. The homotopical
condition FDT(p) is satisfied by categories that admit
a finite presentation by an acyclic polygraph (i.e., by generators, relations,
relations between relations, etc.). The homological condition FP(p) characterizes categories that admit a finite
projective resolution. We give a way to build a free resolution from an acyclic
polygraph, yielding the result FDT(p) => FP(p+2),
and we use rewriting techniques to build an acyclic polygraph from any
convergent presentation. The proofs are based on the notion of
higher-dimensional rewriting strategy, defined as a coherent choice of
representatives in every equivalence class.
Shoufeng
Wang
(Southwest University, China)
Title:
A survey of a topic
on universal algebra: Characterizing and generalizing regular languages by semifilter-congruences
Abstract: The theory of congruences is one of the most
key parts of the theory of universal algebra. In particular, the theory of congruences on free semigroups
which are 2-algebras is crucial in the study of combinatorial semigroups. The present survey is concerning the important
role of the theory of congruences on free semigroups in characterizing and generalizing regular
languages. It is well known that regular languages are important in theoretical
computer science. Recall that a language is regular if it can be accepted by a
finite states automaton. Also, regular languages have a lot of remarkable
algebraic properties. In particular, regular languages can be characterized and
classified by their syntactic (left, right) congruences.
Since regular languages can be characterized by their corresponding syntactic
(left, right) congruences, we may investigate and
generalize regular languages by means of generalizing syntactic (left, right) congruences. This idea is realized firstly by Prodinger who explored a generalization model of syntactic
right congruences by using semifilters.
By applying these kinds of generalized syntactic right congruences,
Prodinger defined and investigated some classes of
generalized regular languages. Following Prodinger,
several authors, such as Guo, Yuqi, Wang, Shuiting, Li, Lian and Zhang, Shuhua, are also
devoted to this topic. Recently, the authors continued to pay attention to this
topic and obtained some new results. In this paper, we shall survey the results
obtained in this line. Moreover, some problems are proposed.
Joseph Hirsh (City
Title: Homotopy-coherent morphisms of
algebraic structures
Abstract: The study of
differential-graded algebraic structures up to quasi-isomorphism is aided by
understanding "homotopy-coherent morphisms", or "infinity morphisms."
We will develop this notion in terms of infinity-categorical limits, and use
this particular presentation to develop a natural obstruction theory for the
problem of extending algebra maps on homology to infinity morphisms
on the chain level.
Ralf Holtkamp (
Title: On sub-operads of ComMag
Abstract: The operad ComMag
of commutative magmatic algebras, with generating series $1-\sqrt{1-2t}$, allows a
factorization (or splitting) of the form ComMag = Com
$\circ$ P in a similar way, in which the operad Ass can be factorized as
Com $\circ$ Lie. Here P is an operad with generating series $(2 \sqrt{1-2t} + 2t - 1)^{-1}.$ We study the
operad P and related sub-operads
of ComMag.
Yi-Zhi Huang (Rutgers University-Piscataway, USA)
Title:
Operads in two-dimensional conformal field theory
Abstract: Many algebraic structures in two-dimensional conformal field theory
can be formulated and studied as algebras over operads.
Such formulations and studies have many applications. For example, in a joint
work with Zhao, I proved a conjecture of Lian-Zuckerman
on topological vertex operator algebras and homotopy
Gerstenhaber algebras. In this talk, I will discuss the operadic
formulations and studies of these algebraic structures, including vertex
operator algebras, intertwining operator algebras, topological vertex operator
algebras, superconformal vertex operator algebras,
open string vertex operator algebras and full field algebras. I will also
discuss the applications of these formulations and studies.
Liang Kong (Tsinghua University, China)
Title: On the classification of
rational open-closed conformal field theories
Abstract: I will start with a modified version of Segal's
definition of conformal field theory, and discuss its relation to vertex
operator algebra. Then I will show how to use the representation theory of
vertex operator algebra to give a classification of rational conformal field
theory. Using this classification, we can easily prove the Holographic
principle and duality-defect relation in this context.
Haisheng Li (Rutgers University-Camden,
Title:
Formal groups and vertex algebras
Abstract: For every 1-dimensional formal group F, we study vertex F-algebras and
their $\phi$-coordinated modules where $\phi$ is what we call an associate of
F, and we give a canonical isomorphism between the category of vertex
F-algebras and the category of ordinary vertex algebras. Meanwhile, for every
formal group we completely determine its associates. We also study
$\phi$-coordinated modules for vertex operator algebras with $\phi(x,z)=xe^{z}$
(a particular associate of the additive formal group) and we prove certain
results closely related to a result of Zhu on change-of-coordinate and to a
result of Lepowsky on X-operators.
Dong Liu: (Huzhou Teachers College, China)
Title: Alternative
dialgebras and Leibniz algebras.
Abstract: In
this talk, we introduce the definition of alternative dialgebras
and some relations between alternative dialgebras and
Leibniz algebras.
Muriel Livernet (
Title: A tor interpretation
of E_n-homology (joint work with Birgit Richter).
Abstract: We prove that
the E_n homology of a commutative algebra can be
computed via functor homology whose category source
is the category of batanin n-level trees. E_n-homology is defined as the homology theory associated
to an e_n operad. Benoit Fresse proved that for commutative algebras this homology
corresponds to the n-th iterated bar complex of the
commutative algebra. We prove that the homology of the latter complex has a tor
interpretation. We'll explain also how this approach can be useful to
obstruction theory.
Abdenacer Makhlouf
(
Title:An
overview of Hom-algebras, Cohomologies
and Deformations.
Abstract: The Hom-Lie
algebras arise naturally in discretizations and
deformations of vector Fields and differential calculus, to describe the
structures on some q-deformations of the Witt and the Virasoro
algebras. Recently, the Hom-type algebras were
intensively investigated. The main feature of Hom-algebras
is that the classical identities are twisted by a homomorphism. The purpose of
my talk is to summarize, present recent developments and provide some key
constructions and examples of Hom-associative, Hom-Leibniz, Hom-Lie, Hom-Poisson and Hom-Hopf
algebraic structures. Then, I will focuss on the cohomology
structures of Hom-associative algebras and Hom-Lie algebras.
Among the relevant formulas for a generalization of Hochschild
cohomology for Hom-associative
algebras and a Chevalley-Eilenberg cohomology for Hom-Lie algebras,
there is a Gerstenhaber bracket on the space of multiplicative multilinear mappings of Hom-associative
algebras and Nijenhuis-Richardson bracket on the
space of multiplicative multilinear mappings of Hom-Lie algebras. Also, I will show that these complexes
are suitable for a formal deformations theory.
Philippe Malbos (
Title: Identities among relations for polygraphic rewriting
Abstract: Fox-Squier theory relates the decidability by
rewriting of the word-problem in a monoid with
homological and homotopical finiteness conditions on
the monoid. I will survey generalisations
of Fox-Squier theory to higher-dimensional
categories. In particular, I will present a homotopical
property on higher categories characterised by the
notion of critical
branching and identities among relations (a notion well-known for group
presentations, generalised here to presentations of
higher categories). Talk based on joint work with Yves Guiraud.
Miguel Mendez (IVIC and Universidad Central de Venezuela, Venezuela)
Title:
A formula for the antipode of the natural Hopf
algebra associated to a set-operad.
Abstract: A
set-operad is a monoid in
the category of combinatorial species with respect to the operation of substitution.
From a left cancellative set-operad,
we can construct a family of partially ordered sets. By defining an order
compatible equivalence relation over the intervals we get a reduced incidence Hopf algebra, that we call the
natural Hopf algebra associated to the set-operad. We obtain a combinatorial formula for its antipode,
generalizing Hayman-Schmitt formula for the Faá di Bruno Hopf algebra. We
reformulate the combinatorial proof of Lagrange inversion formula given by
William Y. C. Chen in terms of our antipode formula for the set-operad of pointed sets. We also present very interesting
connections of our formula with the antipode of Connes and Kreimer
Hopf algebra and some related Hopf
algebras obtained by considering the set-operad of
enriched rooted trees.
Joan Millès (Université de
Nice - Sophia Antipolis, France)
Title:
Curved Koszul duality theory and homotopy
unital associative algebras
Abstract: The classical Koszul duality theory is
defined for augmented associative algebras, operads
or properads. To study non-augmented operads or properads, we show
that a curvature on the bar-construction and in the Maurer-Cartan
equation controls this default of augmentation. We obtain resolutions for the properad encoding unital and counital Frobenius algebras and
for the operad encoding unital
associative algebras. Then we present a definition for homotopy
unital associative algebras, holding good homotopy properties.
Xiang Ni (
Title:
Nonabelian generalized Lax
pairs, O-operators and various successors of
some binary algebraic operads.
Abstract: We generalize the classical
study of (generalized) Lax pairs and the related O-operators and the (modified)
classical Yang-Baxter equation by introducing the concepts of nonabelian generalized Lax pairs, O-operators and the classical
Yang-Baxter equation. We study in this context the nonabelian
generalized r-matrix ansatz and the related double
Lie algebra structures. Relationship between O-operators and the classical Yang-Baxter equation is
established. We prove that an O-operator gives a PostLie
algebra. In the second part of the talk we define and study various successors
of a binary algebraic operad defined by several
generating operations and some relations. The relationships between successors
and O-operators are discussed. We treat some examples and an interesting one
involves the PostLie algebra discussed before. This
talk is based on the joint works with Chengming Bai and Li Guo.
Maria Ronco (University of
Valparaiso, Chile)
Title: Algebra and coalgebra structures on the generalized associahedron
Abstract: For any finite simple graph
G, Carr and Devadoss defined a convex polytope KG, associated to it. We study algebraic
structures on the space spanned by the faces of KG and compare them with
well-known combinatorial Hopf algebras.
C. Selvaraj (
Title: Topological Conditions of 2-primal Rings
Abstract: A 2-primal ring is one in
which the prime radical is exactly the set of nilpotent elements. A ring is clean
provided every element is the sum of a unit and an idempotent. Keith Nicholson
introduced clean rings in 1977 and proved: Every clean ring is an exchange
ring. Conversely, every exchange ring in which all idempotents
are central is clean. In this paper, we investigate some of the relationships
among ring-theoretic properties and topological conditions, such as a 2-primal
weakly exchange ring and its prime spectrum Spec(R). We also prove that
2-primal exchange rings are clean and show that cleanness is equivalent to the
prime spectrum being strongly zero dimensional (every open neighborhood of a
closed set contains a clopen neighborhood).
Bruno Vallette (University of Nice, France, and
Title:
Homotopy Batalin-Vilkovisky algebras
Abstract: First, I will give two explicit cofibrant resolutions of the operad
encoding Batalin-Vilkovisky algebras, the first one
coming from the Koszul duality theory and the second
being the minimal model. The later one is related to the moduli
space of genus 0 curves. This allows one to study the homotopy
properties of Batalin-Vilkovisky algebras. We will
give applications on Topological Conformal Field Theories, Vertex algebras and
double loop spaces. Finally, we will extend Kontsevich
formality theorem to the homotopy BV case, which
includes the divergence operator.
Christine Vespa (
Title: Quadratic
functors on pointed categories
Abstract: Let
C be a pointed algebraic theory (i. e. a category
having a null object and with an object E such that any object of C is a finite
sum of copies of E). We establish a functorial
equivalence between quadratic functors from C to abelian groups and certain minimal algebraic data called
"quadratic C-modules". This result generalizes works of Baues and Pirashvili obtained for
C the category of finitely generated free groups. In this case, "quadratic
C-modules" are equivalent to abelian square
groups.
(This is a joint
work with Manfred Hartl.)
Changchang Xi (Beijing Normal University,
China)
Title:
Auslander-Yoneda algebras and derived equivalences.
Abstract: Let ${\mathbb N}_0$ be the additive monoid of all natural numbers
including zero. Recall that a subset $\Phi$ of ${\mathbb N}_0$ is admissible if
(1) $\Phi$ contains zero, and (2) for any $d\in \Phi$ with a decomposition $d=i+j+k$, $i,j,k\in \Phi$, there
holds that $i+j\in \Phi$ implies $j+k\in
\Phi$. Let $\Phi$ be an admissible subset of ${\mathbb
N}_0$. Given an ${\mathbb
N}_0$-graded $k$-algebra $\Lambda=\bigoplus_{i\ge 0}^{\infty}\Lambda_i$, where $k$ is a commutative ring and each $\Lambda_i$ is a $k$-module,
we define a $\Phi$-Auslander-Yoneda algebra
$A(\Phi):=\oplus_{i\in
\Phi} \Lambda_i$ based on $\Lambda$, where the
multiplication in $A(\Phi)$ is: for $a_i\in \Lambda_i$ and $b_j\in \Lambda_j$ with $i,j\in \Phi$, we
define $a_i\cdot b_j = a_ib_j$ if $i+j\in \Phi$, and zero otherwise. Our aim is to construct
derived equivalences between algebras of this form. In particular, we show
that, for a self-injective algebra $A$ and an $A$-module $X$, there is a
derived equivalence between the $\Phi$-Auslander-Yoneda
algebras of $A\oplus X$ and $A\oplus
\Omega(X)$ for any admissible subset $\Phi$ of ${\mathbb N}_0$, where $\Omega$ is the Heller loop operator of
$A$. Consequently, if two representation-finite, self-injective algebras are
derived-equivalent, then so are their Auslander
algebras.
Wen-Li Yang (
Title: Differential operator realizations of (super)algebras and free-field representations of corresponding
current algebras
Abstract: Based on the particular orderings introduced for the positive roots of
infinite-dimensional Lie (super)algebras, we construct
the explicit differential operator representations of the algebras and the
explicit free-field realizations of corresponding current algebras at an
arbitrary level.
Yong Zhang (
Title: Variations of
the dendriform quadri-algebra
Abstract: The dendriform quadri-algebra is a
regular opeard with four binary operations introduced
by Aguiar and Loday. It can be written as the operadic
product of two copies of the dendriform dialgebra. We consider variations of the dendriform quadri-algebra and
study the relationship between these quadri-algebras
and their relationship with Rota-Baxter algebras.