Quasi-trace function method to cohomologies, symplectic structures and relative Rota-Baxter operators on n-Lie algebras

The aim of this paper is to introduce the notion of quasi-trace functions on n-Lie algebras. We show that the kernel of a linear function on an n-Lie algebra is an ideal (resp. a subalgebra) if and only if the linear function is a trace function (resp. a quasi-trace function). Similar to trace functions, we get that quasi-trace functions can also induce (n+1)-Lie algebras. Some structural properties on induced (n+1)-Lie algebras are obtained. Moreover, we give two sufficient and necessary conditions for characterizing quasi-trace functions in terms of Leibniz algebras and universal enveloping algebras associated with n-Lie algebras. Based on this observation, we construct a representation of an induced (n+1)-Lie algebra. Then we compare the cohomology of n-Lie algebras with those of induced (n+1)-Lie algebras arising from quasi-trace functions. Finally, we establish the relation between symplectic structures and relative Rota-Baxter operators on n-Lie algebras and those structures on the (n+1)-Lie algebras induced by n-Lie algebras with quasi-trace functions.