2020/01/19 by Luis Augusto de Mendonça, de Mendonça, Luis Augusto
Mathematics · #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA
paper · pdf · doi:10.48550/arxiv.2001.06903
18 pages
arxiv created 2020/01/19 · arxiv updated 2020/01/22
We continue the analysis of the weak commutativity construction for Lie algebras. This is the Lie algebra χ(\mathfrakg) generated by two isomorphic copies \mathfrakg and \mathfrakgψ of a fixed Lie algebra, subject to the relations [x,xψ]=0 for all x ∈ \mathfrakg. In this article we study the ideal L =L(\mathfrakg) generated by x-xψ for all x ∈ \mathfrakg. 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 \mathfrakg is nilpotent of class c, then χ(\mathfrakg) is nilpotent of class at most c+2, and this bound can improved to c+1 if \mathfrakg is 2-generated or if c is odd. We also obtain concrete descriptions of L(\mathfrakg) (and thus of χ(\mathfrakg)) if \mathfrakg is free nilpotent of class 2 or 3. Finally, using methods of Gröbner-Shirshov bases we show that the abelian ideal R(\mathfrakg) = [\mathfrakg, [L, \mathfrakgψ]] is infinite-dimensional if \mathfrakg is free of rank at least 3.