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


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

of the truncated polynomial ring in countably many variables. The associative algebra generated by is equipped with a natural -gradation. In this paper we show that for p, which is not representable as , , 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 the algebra 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 -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