Lie bialgebras arising from Hermitian structures on Lie algebras

ADRIAN ANDRADA, MARIA LAURA BARBERIS, GABRIELA OVANDO

Madrid

Congreso; International congress of mathematicians; 2006

n this work we study a particular class of Lie bialgebras arising from Hermitianstructures on Lie algebras such that the metric is ad-invariant. We will refer to themas Lie bialgebras of complex type. These give rise to Poisson Lie groups G whosecorresponding duals G are complex Lie groups. We also prove that a Hermitianstructure on g with ad-invariant metric induces a structure of the same type on thedouble Lie algebra Dg = g⊕g*, with respect to the canonical ad-invariant metric ofneutral signature on Dg. We show how to construct a 2n-dimensional Lie bialgebraof complex type starting with one of dimension 2(n − 2), n ≥ 2. This allows us todetermine all solvable Lie algebras of dimension ≤ 6 admitting a Hermitian structurewith ad-invariant metric.We exhibit some examples in dimensions 4 and 6, includingtwo one-parameter families, where we identify the Lie-Poisson structures on theassociated simply connected Lie groups, obtaining also their symplectic foliations.