# Poisson–Lie sigma models on Drinfel’d double

## Jan Vysoký and Ladislav Hlavatý

Address: Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Břehová 7, 115 19 Prague 1, Czech Republic

E-mail:

hlavaty@fjfi.cvut.cz

vysous@gmail.com

Abstract: Poisson sigma models represent an interesting use of Poisson manifolds for the construction of a classical field theory. Their definition in the language of fibre bundles is shown and the corresponding field equations are derived using a coordinate independent variational principle.
The elegant form of equations of motion for so called Poisson-Lie groups is derived.
Construction of the Poisson-Lie group corresponding to a given Lie bialgebra is widely known only for coboundary Lie bialgebras. Using the adjoint representation of Lie group and Drinfel’d double we show that Poisson-Lie group can be constructed for general Lie bialgebra.

AMSclassification: primary 53D17; secondary 70G45.

Keywords: Poisson sigma models, Poisson manifolds, Poisson-Lie groups, bundle maps.

DOI: 10.5817/AM2012-5-423