Bicharacters, braids and Jacobi identity

Abstract

For an abelian group G we consider braiding in a category of G-graded modules $M^{kG}$ given by a bicharacter \chi on G. For $(G,\chi)$-bialgebra A in $M^{kG}$ an analog of Lie bracket is defined. This bracket is determined by a linear map $E\in\End(A)$ and n-ary operations $\Omega^{n}_{E}$ on A. Our result states that if $E(1)=0,E^{2}=0$ and $\Omega^{3}_{E}=0$ then a braided Jacobi identity holds and the linear map E is a braided derivation of a braided Lie algebra.