Adjoint cohomology of graded Lie algebras of maximal class

TL;DR

This paper explicitly computes the adjoint cohomology of two specific infinite-dimensional graded Lie algebras of maximal class, revealing a near-isomorphism with a tensor product involving scalar cohomology.

Contribution

It provides explicit calculations of the adjoint cohomology for two N-graded Lie algebras of maximal class, extending previous work on related algebras.

Findings

H*(m_j,m_j) is almost isomorphic to m_j tensor H^*(m_j)

Explicit cohomology computations for m_0 and m_2

Connections to known cohomology of the Witt algebra

Abstract

We compute explicitly the adjoint cohomology of two N-graded Lie algebras of maximal class (infinite dimensional filiform Lie algebras) m_0 and m_2. It is known that up to an isomorphism there are only three N-graded Lie algebras of the maximal class. The third algebra from this list is the "positive" part L_1 of the Witt (or Virasoro) algebra and its adjoint cohomology was computed earlier by Feigin and Fukhs. We show that the total space H*(m_j,m_j) is "almost" isomorphic to the completed tensor product of the algebra m_j by scalar cohomology space H^*(m_j), j=0,2.