Algebra of formal vector fields on the line and Buchstaber’s conjecture

Abstract

We consider the Lie algebra L 1 of formal vector fields on the line which vanish at the origin together with their first derivatives. V. M. Buchstaber and A. V. Shokurov showed that the universal enveloping algebra U(L 1) is isomorphic to the Landweber-Novikov algebra S tensored with the reals. The cohomology H*(L 1) = H*(U(L 1)) was originally calculated by L. V. Goncharova. It follows from her computations that the multiplication in the cohomology H*(L 1) is trivial. Buchstaber conjectured that the cohomology H*(L 1) is generated with respect to nontrivial Massey products by one-dimensional cocycles. B. L. Feigin, D. B. Fuchs, and V. S. Retakh found a representation for additive generators of H*(L 1) in the desired form, but the Massey products indicated by them later proved to contain the zero element. In the present paper, we prove that H*(L 1) is recurrently generated with respect to nontrivial Massey products by two one-dimensional cocycles in H 1(L 1).