Deformations and extensions of Lie–Yamaguti algebras

Abstract

The deformation and extension theory of Lie–Yamaguti algebras is studied. We prove that a 1-parameter infinitesimal deformation of a Lie–Yamaguti algebra corresponds to a Lie–Yamaguti algebra of deformation type and a (2,3)-cocycle of with coefficients in the adjoint representation. The notion of Nijenhuis operators for Lie–Yamaguti algebra is introduced to describe trivial deformations. We also prove that equivalence classes of abelian extensions of Lie–Yamaguti algebras are in one-to-one correspondence to elements of the (2,3)-cohomology group.