A Class of Infinite-Dimensional Lie Bialgebras Containing the Virasoro Algebra

Abstract

In this note, we show explicitly how to obtain the structure of a Lie bialgebra on the Virasoro algebra (with or without a central extension), on the Witt algebra, and on many other Lie algebras. Previously, V. G. Drinfel′d (in a fundamental paper (1983, Soviet Math. Dokl. 27, No. 1, 68-71)), introduced the notion of what later (1987, in "Proceedings, International Congress of Mathematicians, 1986, Berkeley, CA," pp. 798-820, Amer. Math. Soc., Providence, RI) came to be known as a triangular, coboundary Lie bialgebra G associated to a solution r ∈ G ⊗ G of the Classical Yang-Baxter Equation; and together with A. A. Belavin he showed (1982, Functional Anal. Appl. 16, No. 3, 159-180) that all invertible solutions r of that equation could be obtained as inverses of 2-cocycles. E. Witten (1988, Comm. Math. Phys. 114, 1-53) considered the case of such invertible r for diff( S 1), the Lie algebra of smooth vector fields on the circle. E. Beggs and S. Majid (1990, Ann. Inst. H. Poincaré Phys. Theor. 53, No. 1, 15 34) considered the invertible case further (in a topological framework) and found solutions for the complexification of diff( S 1) but not for the real Lie algebra diff( S 1). In this note, we exhibit a Lie bialgebra structure corresponding to a non-invertible solution r. The method by which we obtain such a structure works for both the real and the complex case of diff( S 1), the Virasoro algebra (with or without a central extension), the Witt algebra, and many other Lie algebras. We present an explicit expression for the diagonal in all these cases. In fact, we show, more generally, how to obtain the structure of a triangular, coboundary Lie bialgebra on any Lie algebra containing linearly independent elements a and b satisfying [ a, b] = k · b for some non-zero k ∈ K. Depending on the particular Lie algebra under consideration, the flexibility provided by allowing k to vary enables us to obtain different Lie bialgebra structures (some locally finite, others not) on the same underlying Lie algebra.