Cohomology ring of 𝑛-Lie algebras

Abstract

Natural graded Lie brackets on the space of cochains of 𝑛-Leibniz and 𝑛 -Lie algebras are introduced. It turns out that these brackets agree under the natural embedding introduced by Gautheron. Moreover, 𝑛 -Leibniz and 𝑛 -Lie algebras turn to be canonical structures for these brackets in a similar way in which associative algebras (respectively, Lie algebras) are canonical structures for the Gerstenhaber bracket (respectively, Nijenhuis-Richardson bracket). This allows to define the corresponding cohomology operators and graded Lie algebra structures on the cohomology spaces in an uniform simple way by means of square zero elements.