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- W3002748485 abstract "We continue the analysis of the weak commutativity construction for Lie algebras. This is the Lie algebra $chi(mathfrak{g})$ generated by two isomorphic copies $mathfrak{g}$ and $mathfrak{g}^{psi}$ of a fixed Lie algebra, subject to the relations $[x,x^{psi}]=0$ for all $x in mathfrak{g}$. In this article we study the ideal $L =L(mathfrak{g})$ generated by $x-x^{psi}$ for all $x in mathfrak{g}$. 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 $mathfrak{g}$ is nilpotent of class $c$, then $chi(mathfrak{g})$ is nilpotent of class at most $c+2$, and this bound can improved to $c+1$ if $mathfrak{g}$ is $2$-generated or if $c$ is odd. We also obtain concrete descriptions of $L(mathfrak{g})$ (and thus of $chi(mathfrak{g})$) if $mathfrak{g}$ is free nilpotent of class $2$ or $3$. Finally, using methods of Grobner-Shirshov bases we show that the abelian ideal $R(mathfrak{g}) = [mathfrak{g}, [L, mathfrak{g}^{psi}]]$ is infinite-dimensional if $mathfrak{g}$ is free of rank at least $3$." @default.
- W3002748485 created "2020-01-30" @default.
- W3002748485 creator A5034188406 @default.
- W3002748485 date "2020-01-19" @default.
- W3002748485 modified "2023-09-27" @default.
- W3002748485 title "Weak commutativity and nilpotency" @default.
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