Lie Bialgebras of Generalized Virasoro–like Type

Abstract

In two recent papers by the authors, all Lie bialgebra structures on Lie algebras of generalized Witt type are classified. In this paper all Lie bialgebra structures on generalized Virasoro-like algebras are determined. It is proved that all such Lie bialgebras are triangular coboundary.

Lie bialgebra structures on the Schrödinger–Virasoro Lie algebra

“…Proof It can be proved directly by using the similar arguments as those presented in the proof of Lemma 2.2 of [23]. (ii) Let L be a Lie algebra and Proof (i) If r satisfies MYBE, by Lemma 2.1, c(r) ∈ FM ⊗3 0 .…”

“…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]. The Schrödinger-Virasoro Lie algebra [6] was introduced in the context of non-equilibrium statistical physics during the process of investigating the free Schrödinger equations.…”

“…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.…”

Quantization of the q-analog Virasoro-like algebras

“…[15,16]), which is probably why this type of Lie algebras has attracted some attentions in the literature (cf. [10,11,12,13,14,17,18]). …”

Quantization of Hamiltonian-type Lie algebras

“…Lie bialgebra structures of generalized Witt type were classified by , and then quantized by Hu and Wang [2007]. Wu, Song, and Su [2006] defined Lie bialgebras of generalized Virasoro-like type, and the authors quantized these algebras in [Song et al 2008]. Here, we will present the quantization of Hamiltonian-type Lie algebras, whose Lie bialgebra structures were classified by Xin, Song and Su [2007].…”