Trivial central extensions of Lie bialgebras

Abstract

From a Lie algebra g satisfying Z(g) = 0 and Λ 2 (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 with char K = 0. If moreover, [g, g] = g, then we describe also all Lie bialgebra structures on extensions L = g × K n . In interesting cases we characterize the Lie algebra of biderivations.

Rota-Baxter operators and non-skew-symmetric solutions of the classical Yang–Baxter equation on quadratic Lie algebras

“…It is known that if g is a simple complex finite-dimensional Lie algebra, then any structure of a Lie bialgebra is either triangular or factorizable, that is, is induced by a special type Rota-Baxter operator of a weight λ ∈ C. If g is an almost simple real finite-dimensional Lie algebra (that is, the complexification of g is a complex simple Lie algebra), then in [2] it was noted that a Lie bialgebra structure on g is either triangular, factorizable, or almost-factorizable (a similar result for reductive Lie algebras, that are extensions of finite-dimensional almost-simple real Lie algebras, follows from [12]).…”

“…Theorem 1. Let g be a finite-dimensional Lie algebra over a field F and ω be a non-degenerate bilinear invariant form on g. Let r = i a i ⊗ b i ∈ g ⊗ g and R r be the corresponding map defined as in (12). Then (1) r is a solution of CYBE with g-invariant symmetric part r + τ (r) if and only if the map R r is a Rota-Baxter operator of a weight µ ∈ Cent(g) satisfying (15) R r + R * r + µ = 0, where R * is the adjoint to R with respect to the form ω map.…”

“…Proposition 3. Let (g, ω) be an arbitrary finite-dimentional quadratic Lie algebra, and let R : g → g be a linear operator satisfying R + R * + µ = 0 for some µ ∈ Cent(g) and r ∈ g ⊗ g be the corresponding tensor defined by (12). Then the multiplication in the dual algebra of the coalgebra (g, δ r ) is given by…”

Contact Lie systems: theory and applications

“…Moreover, every real non-abelian three-dimensional Lie algebra is isomorphic to (E, [•, •]), where E is a three-dimensional vector space and the Lie bracket is given on a canonical basis {e 1 , e 2 , e 3 } of E by one of the cases in table 1 (see e.g. [36,38]).…”

Extending Structures for Lie Conformal Algebras

“…Another method to construct Lie bialgebraic structures is through extensions of a Lie bialgebra by another Lie bialgebra. Such extensions were investigated in [4,5,8,15] and so on. In this paper, we investigate a more general problem as follows.…”