Poisson d'Avril 2012

Institute: Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, China

Time: May 6–9 (Sun-Wed), 2012
Organizers: Melchior Grützmann; Chen, Zhuo (Tsinghua University)
Accomodation: Zhenghe Binguan-Hotel on campus (see here for details).

Aaron LaudaUniversity of Southern California5-8†
Bai, ChengmingNankai University6-8
Chen, ZhuoTsinghua University5-8
Liu, ZhangjuPeking University6-8
Mathieu StiénonPennsylvania State University5-9
Melchior GrützmannNorthwestern Polytechnical Universityalways
Sheng, YunheJilin University5-8
Xu, PingPennsylvania State University5-9
Xu, XiaomengPeking University5-8
Yannick VoglaireUniversité Catholique de Louvain5-10
Zhong, DeshouChina Youth University for Political Sciences6-9

†participation cancelled due to a family emergency.



8:55Bus to the new campus
10:05-10:55Opening words Bai Chengming Discussion
11:10-12:00Xu Ping Chen Zhuo Discussion
14:30-15:20M. Stiénon M. Grützmann Social activity and discussion
15:30-16:30Liu Zhangju Y. Voglaire
16:55bus back to the city center
18:30Workshop dinner  


Aaron Lauda: Categorification of Quantum Groups (Monday morning and afternoon)

Traditional representation theory of Lie algebras studies actions of the Lie algebra on vector spaces. Categorical representation theory studies actions of Lie algebras on categories, with Lie algebra generators acting by functors, and equations between elements lifting to isomorphisms of functors. Categorified quantum groups govern what kinds of natural transformations one can expect between these functors. All of this structure can be encoded into a simple diagrammatic calculus in the plane. It turns out that this higher structure is useful for constructing equivalences between categories. As an application we will describe Cautis, Kamnitzer and Licata's work using these equivalences to construct knot invariants.

Aaron Lauda: Odd structures arising from categorified quantum groups (Tue morning)

Khovanov homology is a categorification of the Jones polynomial that paved the way for other categorifications of quantum link invariants. The theory of categorified quantum groups provides a representation theoretic explanation of these homological link invariants via the work of Webster and others. Surprisingly, the categorification of the Jones polynomial is not unique. Ozsvath, Rasmussen, and Szabo introduced an "odd" analog of Khovanov homology that also categorifies the Jones polynomial, and the even and odd categorification are not equivalent. In this talk I will explain joint work with Alexander Ellis and Mikhail Khovanov that aims to develop odd analogs of categorified quantum groups to give a representation theoretic explanation of odd Khovanov homology. These odd categorifications lead to surprising new "odd" structures in geometric representation theory including odd analogs of the cohomology of the Grassmannian and Springer varieties.

Bai, Chengming: Classical Yang-Baxter equation and some related structures (Tuesday, 10:05)

In this talk, I will briefly introduce the classical Yang-Baxter equation. In particular, I emphasize the unification of the tensor and the operator forms of the classical Yang-Baxter equation. As applications, certain generalizations of the classical Yang-Baxter equation with motivation from the study of integrable systems and some new algebraic structures with an operadic interpretation are given. Furthermore, some bialgebraic structures are constructed upon the relationship between Lie bialgebras and the classical Yang-Baxter equation.

Chen, Zhuo: From Atiyah Classes to Homotopy Leibniz Algebras (Tuesday, 11:10)

Given a Lie pair (L,A), i.e. a Lie algebroid L together with a Lie subalgebroid A (whose sheaf of smooth sections is noted Γ(A) ), we define the Atiyah class αE of an A-module E as the obstruction to the existence of an A-compatible L-connection on E. We prove that the Atiyah classes αL/A and αE respectively make L/A[-1] and E[-1] into a Lie algebra and a Lie algebra module in the derived category D+(A), where A is the category of coherent sheaves of Γ(A)-modules. This generalizes a result of Kapranov concerning the classical Atiyah class of a holomorphic vector bundle. Moreover, we produce a homotopy Leibniz algebra and a homotopy Leibniz module stemming from the Atiyah classes of L/A and E, and inducing the aforesaid Lie structures in D+(A).
This will be a connecting talk introducing some computation tools after Xu Ping's abstract talk to the same topic.

Liu, Zhangju: On twisted Courant algebroids (Monday, 15:40)

In this talk which is based on joint work with Sheng Yunhe, we give the categorification of Leibniz algebras, which is equivalent to 2-term sh Leibniz algebras. They reveal the algebraic structure of omni-Lie 2-algebras introduced in joint work with C. Zhu as well as twisted Courant algebroids by closed 4-forms introduced by Hansen and Strobl. We also prove that Dirac structures of twisted Courant algebroids give rise to 2-term L-algebras and geometric structures behind them are exactly H-twisted Lie algebroids introduced by Grützmann.

Mathieu Stiénon: Poisson Lie 2-groups (Monday, 14:30)

It is a theorem of Drinfeld that connected, simply connected Poisson–Lie groups are in bijection with Lie bialgebras. We will discuss an analogue for Lie 2-groups.

Melchior Grützmann: Quantization of Symplectic Supermanifolds of degree 2 (Tuesday, 14:30)

In this joint work with J.-P. Michel and Ping Xu we give a quantization formula for what physicists call NP2-manifolds, i.e. N-graded manifolds of maximum coordinate degree 2 with a symplectic structure (also of degree 2). This explains the canonical quantization formula Alekseev and Xu have given for Courant algebroids with a spin-bundle.

Sheng, Yunhe: Non-abelian extensions of Lie 2-algebras

We introduce the notion of derivations of Lie 2-algebras and construct the associated derivation Lie 3-algebra. We prove that isomorphism classes of non-abelian extensions of Lie 2-algebras are classified by equivalence classes of morphisms from a Lie 2-algebra to a derivation Lie 3-algebra.

Xu, Ping: Geometry of Maurer-Cartan elements on complex manifolds (Monday, 11:10)

Maurer-Cartan elements on a complex manifold are extensions of holomorphic Poisson structures. We study the geometry of these structures by investigating their cohomology and homology theory. In particular, we describe a duality on the homology groups, which generalizes the Serre duality of Dolbeault cohomology. This is a joint work with Chen and Stiénon.

Yannick Voglaire: Quantization, midpoints, and reduction in (symplectic) symmetric spaces (Tuesday, 15:30)

In the framework of his quantization by groupoids program, A. Weinstein showed in 1994 strong links between quantization and symplectic geometry, especially symplectic areas of "double triangles". He conjectured that for hermitian symmetric spaces, the phase of an oscillatory integral defining a quantum product at the level of symbols should be equal to the area of a double triangle. The symplectic symmetric spaces constitute the natural framework for this conjecture. I will give an introduction to these spaces and present some structural tools allowing to narrow down the domain of validity of the conjecture, and to build new examples generalizing the flat Moyal-Weyl product and the solvable examples of P. Bieliavsky.

The topics not scheduled were discussed on Sunday and Wednesday afternoon.

This page is maintained by M. Grützmann.