Examples of Self-Iterating Lie Algebras, 2

V. M. Petrogradsky ; I. P. Shestakov

Journal of Lie theory, Tome 19 (2009) no. 4, pp. 697-724.

Résumé

\def\dd{\partial} \def\LL{{\bf L}} \def\Der{\mathop{\rm Der}\nolimits} \def\AA{{\bf A}} \def\Alg{\mathop{\rm Alg}\nolimits} \def\uu{{\bf u}} \def\Lie{\mathop{\rm Lie}\nolimits} \def\N{{\Bbb N}} We study properties of self-iterating Lie algebras in positive characteristic. Let $R=K[t_i| i\in\N]/(t_i^p| i\in\N)$ be the truncated polynomial ring. Let $\partial_i={\dd\over\dd t_i}$, $i\in\N$, denote the respective derivations. Consider the operators \begin{align} v_1 =\dd_1+t_0(\dd_2+t_1(\dd_3+t_2(\dd_4+t_3(\dd_5+t_4(\dd_6+\cdots )))));\\ v_2 =\qquad\quad\;\, \dd_2+t_1(\dd_3+t_2(\dd_4+t_3(\dd_5+t_4(\dd_6+\cdots )))). \end{align} Let $\LL=\Lie_p(v_1,v_2)\subset \Der R$ be the restricted Lie algebra generated by these derivations. We establish the following properties of this algebra in case $p=2,3$. \par a)~$\LL$ has a polynomial growth with Gelfand-Kirillov dimension $\ln p/\ln((1{+}\sqrt 5)/2)$. \par b)~the associative envelope $\AA=\Alg(v_1,v_2)$ of $\LL$ has Gelfand-Kirillov dimension $2\ln p/\ln((1{+}\sqrt 5)/2)$. \par c)~$\LL$ has a nil-$p$-mapping. \par d)~$\LL$, $\AA$ and the augmentation ideal of the restricted enveloping algebra $\uu=u_0(\LL)$ are direct sums of two locally nilpotent subalgebras. The question whether $\uu$ is a nil-algebra remains open. \par e)~the restricted enveloping algebra $u(\LL)$ is of intermediate growth. These properties resemble those of Grigorchuk and Gupta-Sidki groups.