Trivial central extensions of Lie bialgebras

Marco A Farinati,Alejandra Patricia Jancsa

doi:10.1016/j.jalgebra.2013.05.011

Abstract

From a Lie algebra g satisfying Z(g)=0 and Λ2(g)g=0 (in particular, for g semisimple) we describe explicitly all Lie bialgebra structures on extensions of the form L=g×K in terms of Lie bialgebra structures on g (not necessarily factorizable nor quasi-triangular) and its biderivations, for any field K of characteristic different form 2, 3. If moreover, [g,g]=g, then we describe also all Lie bialgebra structures on extensions L=g×Kn. In interesting cases we characterize the Lie algebra of biderivations.

Highlights

Introduction and preliminariesRecall [D1, D2] that a Lie bialgebra over a field K is a triple (g, [−, −], δ) where (g, [−, −]) is a Lie algebra over K and δ : g → Λ2g is such that δ : g → Λ2g satisfies co-Jacobi identity, namely Alt((δ ⊗ Id) ◦ δ) = 0, δ : g → Λ2g is a 1-cocycle in the Chevalley-Eilenberg complex of the Lie algebra (g, [−, −]) with coefficients in Λ2g.In the finite dimensional case, δ : g → Λ2g satisfies co-Jacobi identity if and only if the bracket defined by δ∗ : Λ2g∗ → g∗ satisfies Jacobi identity
Our point of view is the following: From a Lie algebra g over a field K with char K = 0 satisfying Z(g) = 0 and Λ2(g)g = 0 we describe explicitly all the Lie bialgebra structures on extensions of the form L = g × K in terms of Lie bialgebra structures on g and its biderivations
If [g, g] = g, we describe all the Lie bialgebra structures on extensions L = g × Kd for any d

Summary

Introduction and preliminaries

Recall [D1, D2] that a Lie bialgebra over a field K is a triple (g, [−, −], δ) where (g, [−, −]) is a Lie algebra over K and δ : g → Λ2g is such that. Coboundary Lie. bialgebras are denoted by (g, r), r is in general not unique. There exists a non-degenerate, symmetric, invariant, bilinear form on g with corresponding Casimir element Ω, a Cartan subalgebra h, a system of simple roots ∆, a BD-triple (Γ1, Γ2, τ ) and continuous parameter r0 ∈ h ⊗ h such that δ(x) = adx(r) for all x ∈ g, with r given by r = r0 +. This characterization includes the reductive factorizable case, but we obtain all Lie bialgebra structures on L = g×Kd that restrict to a given Lie bialgebra structure on g, which include non-factorizable and even non-coboundary ones The latter were not considered in previous works. 3. If dim V = 1 and g is semisimple, H1(L, ΛL) = 0, in particular, every Lie bialgebra structure on L is coboundary. If (g, δ) is a R-Lie bialgebra, it is coboundary if and only if its complexification is coboundary

Biderivations

Main Construction for Trivial Abelian Extensions