Lie bialgebras of complex type and associated Poisson Lie groups

A. ANDRADA, M.L. BARBERIS, G. OVANDO

JOURNAL OF GEOMETRY AND PHYSICS

Elsevier

Año: 2008 vol. 58 p. 1310 - 1328

0393-0440

In this work we study a particular class of Lie bialgebras arisingfrom Hermitian structures on Lie algebras such that the metric is ad-invariant. Wewill refer to them as Lie bialgebras  of complex type. These giverise to Poisson Lie groups $G$ whose corresponding duals $G^*$ arecomplex Lie groups. We also prove that a Hermitian structure on$ggo$ with ad-invariant metric induces a structure of the same typeon the double Lie algebra ${mathcalD}mathfrak{g}=mathfrak{g}oplusmathfrak{g}^*$, with respect tothe canonical ad-invariant metric of neutral signature on ${mathcalD}mathfrak{g}$. We show how to construct a $2n$-dimensional Liebialgebra of complex type starting with one of dimension $2(n-2), ;ngeq 2$. This allows us to determine all solvable Lie algebras ofdimension $leq 6$ admitting a Hermitian structure with ad-invariantmetric. We exhibit some examples in dimension $4$ and $6$, includingtwo one-parameter families, where we identify the Lie-Poissonstructures on the associated simply connected Lie groups, obtainingalso their symplectic foliations.