Lie Bialgebra Structures on Lie Algebras of Generalized Weyl Type

Abstract

Lie bialgebra structures on Lie algebras of generalized Weyl type are studied. They are shown to be triangular coboundary.

Lie bialgebra structures on the twisted Heisenberg–Virasoro algebra

“…For any 2 elements α, α † ∈ C, one can easily verify that the linear map ζ α,α † : Proof. For any σ ∈ Der(G, G ⊗ G), we can suppose that σ(t) = 0(see [24], or [25]). With [t m , t] = 0 and [t m D, t] = t m+1 , we have σ(t m ), σ(t m D) ∈ G 1 ⊗ G 1 .…”

“…Now we consider Lie bialgebra structures on the Lie algebra of differential operators defined in Section 2. Although such works for the Lie algebra of Weyl type (including the centerless Lie algebra of differential operators) was consider in [24], our calculation is very simple. Clearly C d = C{1} is the center of G. Moreover, we have Lemma 4.3.…”

“…Witt type Lie bialgebras were studied in [15,16,20,17], whose generalized cases were considered in [19,21]. Lie bialgebra structures on some Lie (super)algebras including the Schrödinger-Virasoro Lie algebra, the Lie algebra of Weyl type, the N = 2 superconformal algebra, were investigated case by case in [8], [6], [24], [12], [23], etc.. However, the calculations in these papers are very complicated.…”

Lie bialgebra structures on the extended affine Lie algebra sl2(CqXu1, Li2 2013Journal of Pure and Applied Algebra4000View full textAdd to dashboardCite

“…Lie bialgebra structures on the Witt algebra and the Virasoro algebra were considered and classified in [21,25]. Since then, Lie bialgebra structures as well as their quantizations on some infinitedimensional graded Lie algebras, in particular those containing the Virasoro algebra (or its q-analog) have been extensively studied (e.g., [7,13,17,19,22,23,[25][26][27][28]). We have noticed that, for the above-mentioned infinite-dimensional graded Lie algebras, the Virasoro algebra plays a very crucial role in determining their bialgebra structures because of the fact that the modules of the intermediate series of the Virasoro algebra have relatively simple module structures (see, e.g., [24]).…”

Lie bialgebra structures on generalized loop Schrödinger-Virasoro algebras

“…Lie bialgebra structures on some Lie (super)algebras including generalized Witt type, generalized Virasoro like type and generalized Weyl type Lie algebras, the Schrödinger-Virasoro Lie algebra, the N = 2 superconformal algebra, etc., were constructed (cf. [4], [6], [7], [10], [12], [13], [14], [15] [16]) since the notion was first introduced by Drinfeld in 1983 (cf. [1], [2]) in a connection with quantum groups.…”