On Properties of the Fibonacci Restricted Lie Algebra

V. M. Petrogradsky ; I. P. Shestakov

Journal of Lie theory, Tome 23 (2013) no. 2, pp. 407-431.

Résumé

\def\dd{\partial} \def\LLL{{\cal L}} \def\LL{{\Bbb L}} \def\AA{{\Bbb A}} \def\GG{{\Bbb G}} \def\Lie{\mathop{\rm Lie}\nolimits} \def\Der{\mathop{\rm Der}\nolimits} \def\Alg{\mathop{\rm Alg}\nolimits} \def\End{\mathop{\rm End}\nolimits} Let $R=K[t_i| i\ge 0]/(t_i^p| i\ge 0)$ be the truncated polynomial ring, where $K$ is a field of characteristic 2. Let $\partial_i={\dd\over \dd t_i}$, $i\ge 1$, denote the respective derivations. Consider the operators $$ v_1 = \dd_1+t_0(\dd_2+t_1(\dd_3+t_2(\dd_4+t_3(\dd_5+t_4(\dd_6+\cdots ))))); $$ $$ v_2 = \dd_2+t_1(\dd_3+t_2(\dd_4+t_3(\dd_5+t_4(\dd_6+\cdots ))))\ . $$ Let $\LLL=\Lie(v_1,v_2)$ and $\LL=\Lie_p(v_1,v_2)\subset \Der R$ be the Lie algebra and the restricted Lie algebra generated by these derivations, respectively. These algebras were introduced by the first author and called Fibonacci Lie algebras. It was established that $\LL$ has polynomial growth and a nil $p$-mapping. The latter property is a natural analogue of periodicity of Grigorchuk and Gupta-Sidki groups. We also proved that $\LL$, the associative algebra generated by these derivations $\AA=\Alg(v_1,v_2)\subset \End(R)$, and the augmentation ideal of the restricted enveloping algebra $u_0(\LL)$ are direct sums of two locally nilpotent subalgebras. \endgraf The goal of the present paper is to study Fibonacci Lie algebras in more details. We give a clear basis for the algebras $\LL$ and $\LLL$. We find functional equations and recurrence formulas for generating functions of $\LL$ and $\LLL$, also we find explicit formulas for these functions. We determine the center, terms of the lower central series, values of regular growth functions, and terms of the derived series of $\LLL$. We observed before that $\LL$ is not just infinite dimensional. Now we introduce one more restricted Lie algebra $\GG=\Lie_p(\dd_1,v_2)$ and prove that it is just infinite dimensional. Finally, we formulate open problems.

Détail

Classification : 16P90, 16S32, 16N40, 17B65, 17B66, 17B50, 17B70
Mots-clés : Growth, self-similar algebras, nil-algebras, graded algebras, restricted Lie algebras, Lie algebras of differential operators, Fibonacci numbers