Rota–Baxter operators on Witt and Virasoro algebras

Abstract

Abstract. The homogeneous Rota-Baxter operators on the Witt and Virasoro algebras are classified. As applications, the induced solutions of the classical Yang-Baxter equation and the induced pre-Lie and PostLie algebra structures are obtained.

Post-Lie Algebra Structures on the Witt Algebra

“…Proposition 2.5 tells us that, up to isomorphism, there are 3 types of Lie algebras induced by graded post-Lie algebras from Proposition 1.3. We note that this fact does not agree with Proposition 6.4 in [11]. In fact, it is easy to check that every Lie algebra of Proposition 6.4 in [11] has missed the…”

“…We note that this fact does not agree with Proposition 6.4 in [11]. In fact, it is easy to check that every Lie algebra of Proposition 6.4 in [11] has missed the…”

“…Munthe-Kaas [14], post-Lie algebras also arise naturally from the differential geometry of homogeneous spaces and Klein geometries, topics that are closely related to Cartans method of moving frames. In addition, post-Lie algebras also turned up in relations with Lie groups [5,14], classical Yang-Baxter equation [1], Hopf algebra and classical r-matrices [10] and Rota-Baxter operators [11].…”

“…It is natural to consider the infinite-dimensional case. In particular, the authors [11] obtained a class of induced post-Lie algebra structures on the Witt algebra by use of the homogeneous Rota-Baxter operators. As a matter of fact, the Rota-Baxter operators were originally defined on associative algebras by G. Baxter to solve an analytic formula in probability [2] and then developed by the Rota school [16].…”

“…The Rota-Baxter operators on the Witt algebra W can be given a class of solutions of the classical Yang-Baxter equation (CYBE) on W ⋉ ad * W * . The details can be found in [11], which discuss the homogeneous case. Along the same lines, we can also consider the non-homogeneous case by use of Theorem 5.4.…”