Weak commutativity and nilpotency

Abstract

We continue the analysis of the weak commutativity construction for Lie algebras. This is the Lie algebra χ(g) generated by two isomorphic copies g and gψ of a fixed Lie algebra, subject to the relations [x,xψ]=0 for all x∈g. In this article we study the ideal L=L(g) generated by x−xψ for all x∈g. We obtain an (infinite) presentation for L as a Lie algebra, and we show that in general it cannot be reduced to a finite one. With this in hand, we study the question of nilpotency. We show that if g is nilpotent of class c, then χ(g) is nilpotent of class at most c+2, and this bound can improved to c+1 if g is 2-generated or if c is odd. We also obtain concrete descriptions of L(g) (and thus of χ(g)) if g is free nilpotent of class 2 or 3. Finally, using methods of Gröbner-Shirshov bases we show that the abelian ideal R(g)=[g,[L,gψ]] is infinite-dimensional if g is free of rank at least 3.