Cohomologies of a Lie algebra with a derivation and applications

By Rong Tang, Yael Fregier, Y. Sheng. Journal of Algebra, 2019

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Abstract The main object of study of this paper is the notion of a Lie Der pair, i.e. a Lie algebra with a derivation. We introduce the concept of a representation of a Lie Der pair and study the corresponding cohomologies. We show that a Lie Der pair is rigid if the second cohomology group is trivial, and a deformation of order n is extensible if its obstruction class, which is defined to be an element is the third cohomology group, is trivial. We classify central extensions of Lie Der pairs using the second cohomology group with the coefficient in the trivial representation. For a pair of derivations, we define its obstruction class and show that it is extensible if and only if the obstruction class is trivial. Finally, we classify Lie2 Der pairs using the third cohomology group of a Lie Der pair.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/54266f0bde78d471db35c6418fbc6836751d4d44