Lie super-bialgebra structures on super-Virasoro algebra
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 \(H^1(L, L\otimes L)=0\).
scientific article; zbMATH DE number 5646349
scientific article