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Journal of Lie Theory 23 (2013), No. 2, 407--431
Copyright Heldermann Verlag 2013

On Properties of the Fibonacci Restricted Lie Algebra

V. M. Petrogradsky
Faculty of Mathematics, Ulyanovsk State University, Leo Tolstoy 42, Ulyanovsk 432970, Russia

I. P. Shestakov
Instituto de Mathemßtica e EstatÝstica, Universidade de Sa§ Paulo, Caixa postal 66281, CEP 05315-970, Sa§ Paulo, Brazil


\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.

Keywords: Growth, self-similar algebras, nil-algebras, graded algebras, restricted Lie algebras, Lie algebras of differential operators, Fibonacci numbers.

MSC: 16P90, 16S32, 16N40, 17B65, 17B66, 17B50, 17B70

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