# Groups, Geometry, and Dynamics

Full-Text PDF (327 KB) | Metadata | Table of Contents | GGD summary

**Volume 4, Issue 4, 2010, pp. 873–900**

**DOI: 10.4171/GGD/112**

Published online: 2010-10-15

Nil graded self-similar algebras

Victor M. Petrogradsky^{[1]}, Ivan P. Shestakov

^{[2]}and Efim Zelmanov

^{[3]}(1) Ulyanovsk State University, Russian Federation

(2) Universidade de São Paulo, Brazil

(3) University of California, San Diego, United States

In [19], [24] we introduced a family of self-similar nil Lie algebras ** L** over fields
of prime characteristic

`p`> 0

**is generated by two derivations**

`L``v`

_{1}= ∂

_{1}+

`t`

_{0}

^{p − 1}(∂

_{2}+

`t`

_{1}

^{p − 1}(∂

_{3}+

`t`

_{2}

^{p − 1}(∂

_{4}+

`t`

_{3}

^{p − 1}(∂

_{5}+

`t`

_{4}

^{p − 1}(∂

_{6}+ ··· ))))),

`v`

_{2}= ∂

_{2}+

`t`

_{1}

^{p − 1}(∂

_{3}+

`t`

_{2}

^{p − 1}(∂

_{4}+

`t`

_{3}

^{p − 1}(∂

_{5}+

`t`

_{4}

^{p − 1}(∂

_{6}+ ··· ))))

`K`[

`t`

_{i},

`i`∈ ℕ |

`t`

_{i}

^{p}= 0,

`i`∈ ℕ] in countably many variables. The associative algebra

**generated by**

`A``v`

_{1},

`v`

_{2}is equipped with a natural ℤ ⊕ ℤ-gradation. In this paper we show that for

`p`, which is not representable as

`p`=

`m`

^{2}+

`m`+ 1,

`m`∈ ℤ, the algebra

**is graded nil and can be represented as a sum of two locally nilpotent subalgebras. L. Bartholdi [3] and Ya. S. Krylyuk [15] proved that for**

`A``p`=

`m`

^{2}+

`m`+ 1 the algebra

`A`is not graded nil. However, we show that the second family of self-similar Lie algebras introduced in [24] and their associative hulls are always ℤ

^{p}-graded, graded nil, and are sums of two locally nilpotent subalgebras.

*Keywords: *Modular Lie algebras, growth, nil-algebras, self-similar, Gelfand–Kirillov dimension, Lie algebras of vector fields, Grigorchuk group, Gupta–Sidki group

Petrogradsky Victor, Shestakov Ivan, Zelmanov Efim: Nil graded self-similar algebras. *Groups Geom. Dyn.* 4 (2010), 873-900. doi: 10.4171/GGD/112