Poster session
Title: Nambu-Poisson structures and compatible pairs of Leibniz algebroids
Authors: Edith Padrón and Diana de las Nieves Sosa
Abstract :
Nambu proposed in 1973 a generalization of Hamiltonian Mechanics substituting the Poisson bracket by an n-bracket of functions on $R^n$. Later, Taktajan in 1994 introduced the notion of a Nambu-Poisson structure in order to give an axiomatic formalism for the n-brackets considered by Nambu. Recently, a strong effort is being made to understand the geometric (local and global) properties of the Nambu-Poisson manifolds. In particular, Ibáñez et al have introduced the notion of a Leibniz algebroid (a generalization of a Lie algebroid and a Leibniz algebra ), showing that each Nambu-Poisson manifold M of order n>2 has associated a structure of this type on $\wedge^n(T^*M)$. Afterwards, Hagiwara has obtained a different Leibniz algebroid structure on the same space without the restriction n>2.
In this poster we study, for a Nambu-Poisson manifold M, compatibility conditions between the canonical Lie algebroid TM and the Leibniz algebroid $\wedge^n(T^*M)$ by Hagiwara in order to introduce the notion of a Nambu-Lie bialgebroid, a generalization of a Lie bialgebroid such that a Nambu-Poisson manifold has associated a canonic Nambu-Lie bialgebroid. In this sense, some partial results have been obtained by Wade. We will compare our results with those given by Wade.
Title: Skinner-Rusk formulation in classical field theory
Authors: A. Echeverría-Enríquez, J. Marín-Solano, M.C. Muñoz-Lecanda, N. Román-Roy
Abstract : The Skinner-Rusk formulation of mechanics is extended to first-order classical field theories, thus giving a unified framework for describing their Lagrangian and Hamiltonian formalisms. This description incorporates all the characteristics of these formalisms, both for the regular and singular cases.
Title:Dirac-Nijenhuis structures
Authors: Jesus Clemente-Gallardo and Joana M. Nunes da Costa
Abstract: The deformation of structures by means of Nijenhuis operators has been proved to be a very useful and interesting method in the context of Poisson and Jacobi manifolds. The deformation of the Lie structure of a Lie algebroid has also been studied, and used for the analysis of those two cases. The natural step forward is to study the deformation of Lie bialgebroids and Dirac manifolds. In this poster we define the concept of Dirac-Nijenhuis structures as a generalizarion of the concept of Poisson-Nijenhuis and Jacobi-Nijenhuis structures.
Title:Gerbes and Parallel Transport
Authors: Marco Mackaay and R. Picken
Abstract: We present an overview of our work on abelian gerbes in their local form, via transition functions and connection forms, and in terms of their holonomy and parallel transport along surfaces.
References:
M. Mackaay. A note on the holonomy of connections in twisted
bundles.
To appear in Cahiers Topologie Géom.
Différentielle Catég., math.DG/0106019, 2001.
M. Mackaay and R. Picken. Holonomy and Parallel Transport for Abelian Gerbes}.
Adv. Math. 170:287--339, 2002.
R. Picken. TQFT's and gerbes.
math.DG/0302065, 2003.
R. Picken. A cohomological description of Abelian bundles and gerbes.
math.DG/0305147, 2003.
Title: Formal and hard Lefschetz symplectic manifolds with no Kahler metrics
Authors: José A. Santisteban
Abstract: We show examples of compact symplectic manifolds which are formal and satisfy the hard Lefschetz property. However, according to the results proved by Campana,
such manifolds do not admit Kahler metrics since their fundamental groups cannot be the fundamental group of any compact Kahler manifold.
References:
F. Campana, Remarques sur les groupes de Kahler nilpotents, Ann. Scient. Ec. Norm. Sup., 28 (1995), 307--316.
P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, Real homotopy theory of Kahler manifolds,
Invent. Math., 29 (1975), 245--274.
M. Fernandez, V. Muñoz, J. Santisteban, Cohomologically Kahler Manifolds with no Kahler Metrics, preprint 2003.
Title: Helicoidal maximal surfaces in Minkowski space
Authors: José Antonio Pastor González
Abstract: We investigate helicoidal surfaces in Minkowski space. A purely geometric method is used for classifying the helicoidal maximal surfaces. This description consists on exposing helicoidal maximal surfaces as solutions to certain adequate Björling problems, and shows that a maximal surface is helicoidal if and only if it lies in the associate family of a catenoid. We shall also give characteristic properties of the catenoids and helicoids in the Minkowski space.
Title:A classification theorem of invariant star products in
the setting of Etingof-Kazhdan quantization
Authors: Carlos Moreno and Joana Teles
Abstract :
Given a Drinfeld associator $\Phi$ over the field $\mathbb{C}$ we
consider the corresponding Etingof-Kazh\-dan quantization of any
non-degenerate triangular Lie bialgebra over $\mathbb{C}$ $(\A,
[,]_{\A}$, $ \varepsilon_{\A}=d_c r_1)$. The element $\tilde{J}$
in $\CU\A^{\otimes^2}[[\hh]]$, which appears in this quantization,
is an invariant star product (ISP) on $(\A, [,]_{\A},
\varepsilon_{\A}=d_c r_1)$. After \textit{twisting} the usual
triangular Hopf algebra on $\CU\A[[\hh]]$ via $\tilde{J}^{-1}$ we
obtain a triangular Hopf QUE algebra structure on $\CU\A[[\hh]]$,
denoted by $A_{\A[[\hh]], \tilde{J}^{-1}}$, which is a
quantization of the given Lie bialgebra. We obtain the following
results: a) replacing $r_1$ by a non-degenerate formal solution of
the CYBE $r_{\hh} = r_1 + \sum_{k \geq 2} r_k \hh^{k-1} \in (\A
\wedge \A)[[\hh]]$ we obtain a quantization $A_{\A[[\hh]],
\tilde{J}_{r_{\hh}}^{-1}}$ of $\A = (\A, [,]_{\A},
\varepsilon_{\A}= d_c r_1)$; {\it b)} given any triangular Hopf
QUE algebra of the type $A_{\A[[\hh]], F^{-1}}$ which is a
quantization of $\A$, we prove that there exists $r_{\hh}= r_1 +
r_2 \hh + \dots \in (\A \wedge \A)[[\hh]]$ such that the ISP $F$
and $\tilde{J}_{r_{\hh}}$ are equivalent, this implies that the
quantizations $A_{\A[[\hh]], F^{-1}}$ and
$A_{\A[[\hh]],\tilde{J}_{r_{\hh}}^{-1}}$ are isomorphic; {\it c)}
corresponding to two Drinfeld associators $\Phi$ and $\Phi'$ the
Hopf QUE algebra $A_{\A[[\hh]], F^{-1}}$ determines $r_{\hh},
r'_{\hh}$ such that $A_{\A[[\hh]], F^{-1}}$, $A_{\A[[\hh]],
(\tilde{J}_{r_{\hh}}^{\Phi})^{-1}}$ and $A_{\A[[\hh]],
(\tilde{J}_{r'_{\hh}}^{\Phi'})^{-1}}$ are isomorphic.
Using Knizhnik-Zamolodchikov associator we compute the first
terms of the invariant star product $\tilde{J}$ and we prove that
it coincides with the one obtained by Drinfeld up to the term in
$\hh^2$ and, in two examples of dimension 2 and 4 that we
considered, up to the term in $\hh^3$.
We prove a classification theorem for ISP on a Lie group endowed
with an invariant symplectic structure, $(G,\beta_1)$, obtained
using Etingof-Kazhdan quantization method. This theorem can be
stated exactly in the same way as the one in Drinfeld's setting:
two ISP $\tilde{J}_{r_{\hh}}$ and $\tilde{J}_{r_{\hh}'}$ are
equivalent if and only if the non-degenerate $2$-cocycles
$\beta_{\hh}= \mu_{r_{\hh}}(r_{\hh})$ and $\beta'_{\hh}=
\mu_{r'_{\hh}}(r'_{\hh})$ are in the same formal cohomological
class.