About the Cohomology of the Lie Superalgebra of Vector Fields on ℝ^n|n

Abstract

In this article, we compute the first space of cohomology of Vect (ℝ ^n|n ), the Lie superalgebra of vector fields on the supermanifold ℝ ^n|n with coefficients in 𝒻 (ℝ ^n|n ), the space of smooth functions on ℝ ^n|n . We give a super analog of the cohomologies of vector fields that where studied for instance by Fuchs [2]. This work allows us to classify the deformations of the action of Vect(ℝ ^n|n ) on 𝒻 (ℝ ^n|n ).

Key Words:

• Cohomology space
• Frölicher–Nijenhuis bracket
• Richardson–Nijenhuis bracket
• The Lie superalgebra of vector fields on the super manifold ℝ ^n|n

2000 Mathematics Subject Classification:

• 17B56
• 53D55

ACKNOWLEDGMENTS

I would like to thank C. Roger and P. Lecomte for their suggestions and remarks.

Notes

Communicated by L. Small.