Lie super-bialgebra structures on super-Virasoro algebra
type: Item
instance of: scholarly article
title: Lie super-bialgebra structures on super-Virasoro algebra
zbMATH Open document ID: 1250.17029
author: Hengyun Yang
published in: Frontiers of Mathematics in China
publication date: 2009-12-14T00:00:00Z
review text: S-H. Ng and the reviewer showed that all Lie bialgebra structures on the complex Virasoro algebra \(V\) with central charge \(C\), are triangular coboundary, i.e., the Lie coalgebra structure is given by a one-coboundary determined by a skew-symmetric element of \(V\otimes V\) satisfying the classical Yang-Baxter equation [\textit{S.-H. Ng} and \textit{E. J. Taft}, J. Pure Appl. Algebra 151, No. 1, 67--88 (2000; Zbl 0971.17008)]. Since then, this result has been proved for many variants of the Witt and Virasoro algebras. The paper under review treats a super-Virasoro algebra \(L\), whose even part is \(V\) with the usual basis \(\{L_i\mid i\) an integer \(\}\), and whose odd part has basis \(\{G_i\mid i\) half an odd integer \(\}\), with multiplication given by \([L_i, G_j]=(j-i/2)G_{(i+j)}\), \([G_i, G_j]=2L_{(i+j)}\) for \(i+j\) not zero, and \([G_i, G_{-i}]=2 L_0-(\tfrac 13(i^2-\tfrac i4))C\). The notions of Lie super-bialgebra, coboundary and triangular are similar to those for Lie bialgebras, but with some sign changes due to the grading. The author proves that all Lie super-bialgebra structures on \(L\) are triangular coboundary. As usual, the main ingredient of the proof is that \(H^1(L, L\otimes L)=0\).
reviewed by: Earl J. Taft
Mathematics Subject Classification ID: 17B62
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 17B68
Mathematics Subject Classification ID: 17B70
zbMATH DE Number: 5646349
zbMATH Keywords: Super-Virasoro algebra
zbMATH Keywords: Lie super-bialgebra
MaRDI profile type: Publication
full work available at URL: s11464-009-0012-x
OpenAlex ID: W2058580700
Wikidata QID: Q115377963
cites work: A class of infinite-dimensional Lie bialgebras containing the Virasoro algebra
cites work: Classification of the Lie bialgebra structures on the Witt and Virasoro algebras
cites work: The structure of the dual Lie coalgebra of the Witt algebra
cites work: Harish–Chandra modules of the intermediate series over the high rank Virasoro algebras and high rank super-Virasoro algebras
cites work: Classification of harish-chandra modules over the super-virasoro algebras
cites work: Witt and Virasoro algebras as Lie bialgebras
cites work: Lie bialgebras of generalized Virasoro-like type
cites work: Lie Bialgebras of Generalized WITT Type, II
DOI: 10.1007/S11464-009-0012-X
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description: scientific article; zbMATH DE number 5646349