|
Algebra and Discrete Mathematics, 2008, Issue 2, Pages 1–41
(Mi adm156)
|
|
|
|
This article is cited in 8 scientific papers (total in 8 papers)
RESEARCH ARTICLE
Planar trees, free nonassociative algebras, invariants, and elliptic integrals
Vesselin Drenskya, Ralf Holtkampb a Institute of Mathematics and Informatics,
Bulgarian Academy of Sciences, 1113 Sofia,
Bulgaria
b Fakultät für Mathematik, Ruhr-Universität,
44780 Bochum, Germany
Abstract:
We consider absolutely free algebras with (maybe infinitely) many multilinear operations. Such multioperator algebras were introduced by Kurosh in 1960. Multioperator algebras satisfy the Nielsen–Schreier property and subalgebras of free algebras are also free. Free multioperator algebras are described in terms of labeled reduced planar rooted trees. This allows to apply combinatorial techniques to study their Hilbert series and the asymptotics of their coefficients. Then, over a field of characteristic 0, we investigate the subalgebras of invariants under the action of a linear group, their sets of free generators and their Hilbert series. It has turned out that, except in the trivial cases, the algebra of invariants is never finitely generated. In important partial cases the Hilbert series of the algebras of invariants and the generating functions of their sets of free generators are expressed in terms of elliptic integrals.
Received: 03.01.2008 Revised: 15.07.2008
Citation:
Vesselin Drensky, Ralf Holtkamp, “Planar trees, free nonassociative algebras, invariants, and elliptic integrals”, Algebra Discrete Math., 2008, no. 2, 1–41
Linking options:
https://www.mathnet.ru/eng/adm156 https://www.mathnet.ru/eng/adm/y2008/i2/p1
|
Statistics & downloads: |
Abstract page: | 244 | Full-text PDF : | 92 | First page: | 1 |
|