Cohomologies and Deformations of Lie Superalgebras with Superderivations

Abstract

In this paper, we introduce the concept of a LiesDer pair (i.e., a Lie superalgebra with a superderivation). We define a representation of a LiesDer pair and study its corresponding cohomology. We prove that the infinitesimals of two equivalent 1-parameter formal deformations of a LiesDer pair are in the same cohomology class. We show that a LiesDer pair is rigid if the second cohomology group is trivial, and a deformation of order n is extensible if and only if its obstruction class is trivial. We use the second cohomology group with coefficients in the trivial representation to classify central extensions of the LiesDer pair. Finally, we prove that a pair of superderivations is extensible if and only if its obstruction class is trivial.

Keywords

Lie superalgebras / superderivations / representations / cohomologies / deformations / central extensions / 17B10 / 17B40 / 17B56

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Peking University