Weak commutativity and nilpotency

[摘要] We continue the analysis of the weak commutativity construction for Lie algebras. This is the Lie algebra chi(g) generated by two isomorphic copies g and g(psi) of a fixed Lie algebra, subject to the relations [x, x(psi)] = 0 for all x is an element of g. In this article we study the ideal L = L(g) generated by x - x(psi) for all x is an element of 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 chi(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 chi(g)) if gis free nilpotent of class 2 or 3. Finally, using methods of Grobner-Shirshov bases we show that the abelian ideal R(g) =[g, [ L, g(psi)]] is infinite-dimensional if gis free of rank at least 3. (C) 2020 Elsevier Inc. All rights reserved.

[发布日期] 2020-12-15

[关键词] Lie algebras;Cohomology;Nilpotency;Finite presentability;Grobner-Shirshov bases