Lie bialgebra structures on the Schrödinger–Virasoro Lie algebra

Abstract

Abstract. In this paper we investigate Lie bialgebra structures on the Schrödinger-Virasoro algebra L. Surprisingly, we find out an interesting fact that not all Lie bialgebra structures on the Schrödinger-Virasoro algebra are triangular coboundary, which is different from the related known results of some Lie algebras related to the Virasoro algebra.

Schrödinger-Virasoro type Lie bialgebra: a twisted case

“…Lie superbialgebra structures on the Ramond N = 2 super Virasoro algebra were considered and determined in [26]. Recently, the Lie bialgebra structures on the original Schrödinger-Virasoro algebra were considered by [6], which are different from that given in [15]. In this paper, we shall investigate Lie bialgebra structures on L. Compared to the case considered in [6], it is more complicated according to the different basis and brackets between them and interesting based on the analysis presented above.…”

Quantization of Schrödinger-Virasoro Lie algebra

“…(17) Proof The first equation of (16) is obvious by Lemma 4 (ii) and since M k commutes with e. From equations (4), (11), Lemmas 3 and 4 (ii), one has…”

“…Since then, a great deal of attention has been paid to the study of the quantization of Lie bialgebras as well as Lie bialgebra structures of some Lie algebras (e.g., [5,6,9,11]). The following result can be found in several sources (e.g., [20]).…”

“…Equation (3) implies that Δ r is an inner derivation of L. For the Schrödinger-Virasoro Lie algebra L defined in (1), it is showed in [11] that a Lie bialgebra (L , [·, ·], Δ) is triangular coboundary if and only if Δ is an inner derivation, which is determined by the classical Yang-Baxter r-matrix r. From Proposition 1, we notice that the classical Yang-Baxter r-matrix r is uniquely expressed as the antisymmetric tensor of two distinguished elements a, b up to nonzero scalars satisfying [a, b] = kb (k = 0). In fact, for a given r-matrix in Proposition 1, we may take two distinguished elements of the form h := k −1 a and e := b such that [h, e] = e with 0 = k ∈ F.…”

“…Recently, the Lie bialgebra structures on the Schrödinger-Virasoro Lie algebra L were discussed in [11], which turned out to be not all coboundary triangular (for definition, see [7, p. 28]). In the present paper, we use the general quantization method by Drinfel'd twists (cf.…”