Nil graded self-similar algebras

Autor(es)

Petrogradsky, Victor M. ^[1] ; Shestakovand, Ivan P. ^[2] ; Zelmanov, Efim ^[3]

Afiliação do(s) autor(es)

^[1] Ulyanovsk State Univ, Fac Math & Comp Sci, Ulyanovsk 432970 - Russia

^[2] Univ Sao Paulo, Inst Math & Estat, BR-05315970 Sao Paulo - Brazil

^[3] Univ Calif San Diego, Dept Math, La Jolla, CA 92093 - USA

• Tipo de documento: Artigo Científico
• Fonte: Groups Geometry and Dynamics; v. 4, n. 4, p. 873-900, 2010.
• Citações Web of Science: 8

Resumo

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 v(1) = partial derivative(1) + t(0)(p-1) (partial derivative(2) + t(1)(p-1) (partial derivative(3) + t(2)(p-1) (partial derivative(4) + t(3)(p-1) (partial derivative(5) + t(4)(p-1) (partial derivative(6) + ...))))), v(2) = partial derivative(2) + t(1)(p-1) (partial derivative(3) + t(2)(p-1) (partial derivative(4) + t(3)(p-1) (partial derivative(5) + t(4)(p-1) (partial derivative(6) + ...)))) of the truncated polynomial ring K{[}t(i), i is an element of N vertical bar t(j)(p) =0, i is an element of N] in countably many variables. The associative algebra A generated by v(1), v(2) is equipped with a natural Z circle plus Z-gradation. In this paper we show that for p, which is not representable as p = m(2) + m + 1, m is an element of Z, the algebra A is graded nil and can be represented as a sum of two locally nilpotent subalgebras. L. Bartholdi {[}3] andYa. 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. (AU)

• Processo FAPESP: 05/60337-2 - Álgebras de Lie e de Jordan, suas representações e generalizações
• Beneficiário: Ivan Chestakov
• Modalidade de apoio: Auxílio à Pesquisa - Temático
• Processo FAPESP: 05/58376-0 - Victor Petrogradsky | Ulyanovsk State University - Rússia
• Beneficiário: Ivan Chestakov
• Modalidade de apoio: Auxílio à Pesquisa - Pesquisador Visitante - Internacional