On the algebra generated by $\overline{\mu}, \overline{\partial}, \partial, \mu$

TL;DR

This paper characterizes the algebra generated by specific differential operators on almost complex manifolds, showing it is a universal enveloping algebra and analyzing its cohomology structure.

Contribution

It identifies the algebra as a universal enveloping algebra of a graded Lie algebra and computes its cohomology with respect to various differentials.

Findings

The algebra is the universal enveloping algebra of a certain graded Lie algebra.

The cohomology of the graded Lie algebra is explicitly determined.

The structure of the algebra generated by the operators is fully characterized.

Abstract

In this note, we determine the structure of the associative algebra generated by the differential operators $\overline{\mu}, \overline{\partial}, \partial, \mu$ that act on complex-valued differential forms of almost complex manifolds. This is done by showing it is the universal enveloping algebra of the graded Lie algebra generated by these operators and determining the structure of the corresponding graded Lie algebra. We then determine the cohomology of this graded Lie algebra with respect to its canonical inner differential $[d,-]$, as well as its cohomology with respect to all its inner differentials.