# Nil graded self-similar algebras

### Victor M. Petrogradsky

Ulyanovsk State University, Russian Federation### Ivan P. Shestakov

Universidade de São Paulo, Brazil### Efim Zelmanov

University of California, San Diego, United States

## Abstract

In [19], [24] we introduced a family of self-similar nil Lie algebras $L$ over fields of prime characteristic $p>0$ whose properties resemble those of Grigorchuk and Gupta–Sidki groups. The Lie algebra $L$ is generated by two derivations

of the truncated polynomial ring$K[t_{i},i∈N∣t_{i}=0,i∈N]$ in countably many variables. The associative algebra $A$ generated by $v_{1},v_{2}$ 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 $A$ 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 $p=m_{2}+m+1$ 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.

## Cite this article

Victor M. Petrogradsky, Ivan P. Shestakov, Efim Zelmanov, Nil graded self-similar algebras. Groups Geom. Dyn. 4 (2010), no. 4, pp. 873–900

DOI 10.4171/GGD/112