# 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

v1 = ∂1 + t0p − 1(∂2 + t1p − 1(∂3 + t2p − 1(∂4 + t3p − 1(∂5 + t4p − 1(∂6 + ··· ))))),

v2 = ∂2 + t1p − 1(∂3 + t2p − 1(∂4 + t3p − 1(∂5 + t4p − 1(∂6 + ··· ))))

of the truncated polynomial ring K[ti, i ∈ ℕ | tip = 0, i ∈ ℕ] in countably many variables. The associative algebra **A** generated by v1, v2 is equipped with a natural ℤ ⊕ ℤ-gradation. In this paper we show that for p, which is not representable as p = m2 + 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 = m2 + 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 ℤp-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