Reynolds $n$-Lie algebras and NS-$n$-Lie algebras

Abstract

In this paper, first we introduce the notion of a Reynolds operator on an $n$-Lie algebra and illustrate the relationship between Reynolds operators and derivations on an $n$-Lie algebra. We give the cohomology theory of Reynolds operators on an $n$-Lie algebra and study infinitesimal deformations of Reynolds operators using the second cohomology group. Then we introduce the notion of NS-$n$-Lie algebras, which are generalizations of both $n$-Lie algebras and $n$-pre-Lie algebras. We show that an NS-$n$-Lie algebra gives rise to an $n$-Lie algebra together with a representation on itself. Reynolds operators and Nijenhuis operators on an $n$-Lie algebra naturally induce NS-$n$-Lie algebra structures. Finally, we construct Reynolds $(n+1)$-Lie algebras and Reynolds $3$-Lie algebras from Reynolds $n$-Lie algebras and Reynolds commutative associative algebras respectively.

arXiv comment21 pages

References in corpus (8)

• Gauge Symmetry and Supersymmetry of Multiple M2-Branes
• Three-Algebras and N=6 Chern-Simons Gauge Theories
• n-ary algebras: a review with applications
• Nijenhuis operators on $n$-Lie algebras
• Deformations of 3-algebras
• Twisted Rota-Baxter operators and Reynolds operators on Lie algebras and NS-Lie algebras
• Cohomology and Deformations of $n$-Lie algebra morphisms
• Twisted Rota-Baxter operators on 3-Lie algebras and NS-3-Lie algebras