Classification of the Lie bialgebra structures on the Witt and Virasoro algebras

Lie bialgebra structures on the Schrödinger–Virasoro Lie algebra

“…(ii) Let L be a Lie algebra and Proof (i) If r satisfies MYBE, by Lemma 2.1, c(r) ∈ FM ⊗3 0 . As in [16], one has c(r) = 0. The reverse statement is obvious.…”

“…Since then, many papers on Lie bialgebras appeared, e.g., [3, 11, 13-16, 18, 19, 22, 23]. Witt type Lie bialgebras introduced in [19] were classified in [16], whose generalized cases were considered in [18,22]. Lie bialgebra structures on generalized Virasoro-like and Block Lie algebras were investigated in [23,11].…”

“…As a matter of fact, investigating Lie bialgebras and quantilizations is a complicated problem. In this paper we shall determine Lie bialgebra structures on the Schrödinger-Virasoro algebra L. It is known that every Lie bialgebra structure on the Lie algebras considered in [11,16,18,23] is triangle coboundary. Surprisingly, we find out an interesting fact that not all Lie bialgebra structures on the Schrödinger-Virasoro algebra are triangular coboundary.…”

“…This is one of the reasons why it is difficult to classify all Lie bialgebra structures (in case when all Lie bialgebra structures are triangular coboundary, the classification of Lie bialgebra structures is equivalent to solving all solutions of CYBE (cf. Lemma 2.2), which is not done even for the Virasoro algebra [16]). §2.…”

“…[16]), Proposition 2.3 nevertheless provides us some rule to decide when r ∈ Im(1 − τ ) is not a solution to CYBE. Indeed, Proposition 2.3 classifies all possible highest components r p of r for which r ∈ Im(1 − τ ) and c(r) = 0.…”