Nil graded self-similar algebras

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.