A class of infinite-dimensional Lie bialgebras containing the Virasoro algebra
Let \(L\) be a Lie algebra, with a 2-dimensional subalgebra with basis \(a\), \(b\) so that \((a,b)= kb\), \(k\) a non-zero scalar. Then the author notes that \(r= a\wedge b= a\otimes b-b\otimes a\) satisfies the classical Yang- Baxter equation, and thus the coboundary \(\Delta_ r: L\to L\wedge L\), with the adjoint action on \(L\wedge L\), gives \(L\) the structure of a (triangular coboundary) Lie bialgebra. Examples are given for Witt and Virasoro algebras, where, if \(x_ n= x^{n+1} d/dx\), \(r_ n= x_ 0\wedge x_ n\) yields such a Lie bialgebra structure. Reviewer's note: The reviewer also noted these structures on the Witt and Virasoro algebras [J. Pure Appl. Algebra 87, 301-312 (1993; Zbl 0786.17015)]. He obtained the uniqueness of these structures at characteristic 0 for the one-sided Witt algebra, i.e., distinct \(n\) yield non-isomorphic Lie coalgebra (bialgebra) structures. He also discussed the locally finite part of these examples at arbitrary characteristic. The author states that if \(k\) is allowed to vary in \([a,b]= kb\), one can obtain different Lie bialgebra structures (some locally finite, some not). This is not so since the Lie bialgebra structure depends only on the 2-dimensional subalgebra. In the example with \(r_ n= x_ 0\wedge x_ n\), varying \(n\) does change the locally finite part.
scientific article; zbMATH DE number 703897
scientific article