Investigation of the $n$-th powers of the Fibonacci and Lucas $Q$-matrices via Lor...
Homological and combinatorial methods in the theory of Lie Algebras and groups
| Author(s): |
Total Authors: 2
|
| Affiliation: | [1] Ulyanovsk State Univ. Fac Math
[2] Univ Sao Paulo. Inst Math & Estat
Total Affiliations: 2
|
| Document type: | Journal article |
| Source: | JOURNAL OF LIE THEORY; v. 23, n. 2, p. 407-431, 2013. |
| Web of Science Citations: | 1 |
| Abstract | |
Let R = K{[}t(i)vertical bar i >= 0]/(t(i)(p)vertical bar i >= 0) be the truncated polynomial ring, where K is a field of characteristic 2. Let partial derivative(i) = partial derivative/partial derivative(ti), i >= 1, 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(v(1), v(2)) and L = Lie(p)(v(1), v(2)) subset of Der R be the Lie algebra and the restricted Lie algebra generated by these derivations, respectively. These algebras were introduced by the first author and called Fibonacci Lie algebras. It was established that L has polynomial growth and a nil p-mapping. The latter property is a natural analogue of periodicity of Grigorchuk and Gupta-Sidki groups. We also proved that L, the associative algebra generated by these derivations A = Alg(v(1), v(2)) subset of End(R), and the augmentation ideal of the restricted enveloping algebra u(0)(L) are direct sums of two locally nilpotent subalgebras. The goal of the present paper is to study Fibonacci Lie algebras in more details. We give a clear basis for the algebras L and L. We find functional equations and recurrence formulas for generating functions of L and G, also we find explicit formulas for these functions. We determine the center, terms of the lower central series, values of regular growth functions, and terms of the derived series of G. We observed before that L is not just infinite dimensional. Now we introduce one more restricted Lie algebra G = Lie(p)(partial derivative(1), v(2)) and prove that it is just infinite dimensional. Finally, we formulate open problems. (AU) | |
| FAPESP's process: | 10/50347-9 - Algebras, representations e applications |
| Grantee: | Ivan Chestakov |
| Support Opportunities: | Research Projects - Thematic Grants |
| FAPESP's process: | 05/60337-2 - Lie and Jordan algebras, their representations and generalizations |
| Grantee: | Ivan Chestakov |
| Support Opportunities: | Research Projects - Thematic Grants |