Lie bialgebras of complex type and associated Poisson Lie groups

Abstract

a b s t r a c tIn this work we study a particular class of Lie bialgebras arising from Hermitian structures on Lie algebras such that the metric is ad-invariant. We will refer to them as Lie bialgebras of complex type. These give rise to Poisson Lie groups G whose corresponding duals G * are complex Lie groups. We also prove that a Hermitian structure on g with ad-invariant metric induces a structure of the same type on the double Lie algebra Dg = g ⊕ g * , with respect to the canonical ad-invariant metric of neutral signature on Dg. We show how to construct a 2n-dimensional Lie bialgebra of complex type starting with one of dimension 2(n − 2), n ≥ 2. This allows us to determine all solvable Lie algebras of dimension ≤6 admitting a Hermitian structure with ad-invariant metric. We present some examples in dimensions 4 and 6, including two one-parameter families, where we identify the Lie-Poisson structures on the associated simply connected Lie groups, obtaining also their symplectic foliations.

On pseudo-Hermitian quadratic nilpotent Lie algebras

“…In [2,Th. 24] it is seen that, up to dimension 6, every quadratic Lie algebra (g, ϕ) with sig(ϕ) = (2r, 2s) admits a ϕ-skewsymmetric complex structure J and, therefore, (g, J, ϕ) is pseudo-Hermitian quadratic.…”

“…(1) Notice that the concept of pHQ-double extension defined above is different from the construction that Andrada, Barberis and Ovando give in [2,Th. 21].…”

“…The Lie algebra L is isomorphic to W 3 ⊕ R with the notation of [7] and it is the Lie algebra denoted (0, 0, 0, 12, 14, 24) in the list of 6-dimensional nilpotent Lie algebras admitting complex structures given by Salamon in [13]. In [2], it is denoted by L 3 (1, 2) × R.…”

“…A pseudo-Hermitian quadratic Lie algebra is a triple (g, J, ϕ) where g is a Lie algebra, J is a complex structure on g and ϕ is a non-degenerate ad-invariant bilinear form on g which is compatible with J in the sense that ϕ(Jx, Jy) = ϕ(x, y) for all x, y ∈ g. Geometrically, g is the Lie algebra of a Lie group G endowed with a left-invariant complex structure and a bi-invariant pseudo-Hermitian metric. Further, it was seen in [2] that the complex structure on such Lie algebras defines a classical r-matrix so that they give rise to Lie bialgebras of complex type and, as a consequence, to certain Poisson Lie groups whose duals are complex Lie groups.…”

Extending invariant complex structures

“…They say that the conditions of the Proposition hold in this situation if and only if on the Hermitian Lie algebra (g, J, g) the complex structure J is bi-invariant and the inner product g is adinvariant. However as proved in [6] this is possible only for an abelian Lie algebra g.…”