Examples of Self-Iterating Lie Algebras, 2

Author(s)

Petrogradsky, V. M. ^[1] ; Shestakov, I. P. ^[2]

Affiliation

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

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

• Document type: Journal article
• Source: JOURNAL OF LIE THEORY; v. 19, n. 4, p. 697-724, 2009.
• Web of Science Citations: 8

Abstract

We study properties of self-iterating Lie algebras in positive characteristic. Let R = K{[}t(i)vertical bar i is an element of N]/(t(i)(p)vertical bar i is an element of N) be the truncated polynomial ring. Let partial derivative(i) = partial derivative/partial derivative t(i), i is an element of N, denote the respective derivations. Consider the operators v(1) = partial derivative(1) + t(0)(partial derivative(2) + t(1)(partial derivative(3) + t(2)(partial derivative(4) + t(3)(partial derivative(5) + t(4)(partial derivative(6) + ...))))); v(2) = partial derivative(2) + t(1)(partial derivative(3) + t(2)(partial derivative(4) + t(3)(partial derivative(5) + t(4)(partial derivative(6) + ...)))). Let L = Lie(p)(v(1), v(2)) subset of Der R be the restricted Lie algebra generated by these derivations. We establish the following properties of this algebra in case p = 2, 3. a) L has a polynomial growth with Gelfand-Kirillov dimension lnp/ln((1+root 5)/2). b) the associative envelope A = Alg(v(1), v(2)) of L has Gelfand-Kirillov dimension 2 lnp/ln((1+root 5)/2). c) L has a nil-p-mapping. d) L, A and the augmentation ideal of the restricted enveloping algebra u = u(0)(L) are direct sums of two locally nilpotent subalgebras. The question whether u is a nil-algebra remains open. e) the restricted enveloping algebra u(L) is of intermediate growth. These properties resemble those of Grigorchuk and Gupta-Sidki groups. (AU)

• FAPESP's process: 05/60337-2 - Lie and Jordan algebras, their representations and generalizations
• Grantee: Ivan Chestakov
• Support Opportunities: Research Projects - Thematic Grants
• FAPESP's process: 05/58376-0 - Victor Petrogradsky | Ulyanovsk State University - Russia
• Grantee: Ivan Chestakov
• Support Opportunities: Research Grants - Visiting Researcher Grant - International