Nil graded self-similar algebras

Abstract

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

of the truncated polynomial ring$K[t_i,i\in \mathbb{N}|t_i^p=0,i\in \mathbb{N}]$ in countably many variables. The associative algebra $\mathbf{A}$ generated by $v_1,v_2$ is equipped with a natural $\mathbb{Z}\oplus \mathbb{Z}$-gradation. In this paper we show that for p, which is not representable as $p=m^2+m+1$, $m\in \mathbb{Z}$, the algebra $\mathbf{A}$ 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 $\mathbf{A}$ 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 ${\mathbb{Z}}^p$-graded, graded nil, and are sums of two locally nilpotent subalgebras.