Cohomologie de Chevalley des graphes ascendants

TL;DR

This paper computes the cohomology of the Lie algebra of ascending tensors on bfcrf6d with respect to the adjoint action, using aerial Kontsevitch graphs, revealing it is generated by products of odd wheels.

Contribution

It provides the first explicit computation of the cohomology for this specific Lie algebra of ascending tensors using graph-based methods.

Findings

Cohomology is generated by products of odd wheels.

The cohomology is freely generated by these wheel products.

The results extend previous work on vectorial graphs.

Abstract

The space $T_{poly}(\mathbb R^d)$ of all tensor fields on $\mathbb R^d$, equipped with the Schouten bracket is a Lie algebra. The subspace of ascending tensors is a Lie subalgebra of $T_{poly}(\mathbb R^d)$. In this paper, we compute the cohomology of the adjoint representations of this algebra (in itself and $T_{poly}(\mathbb R^d)$), when we restrict ourselves to cochains defined by aerial Kontsevitch's graphs like in our previous work (Pacific J of Math, vol 229, no 2, (2007) 257-292). As in the vectorial graphs case, the cohomology is freely generated by all the products of odd wheels.