Planar trees, free nonassociative algebras, invariants, and elliptic integrals
Abstract
We consider absolutely free nonassociative algebras and, more generally, absolutely free algebras with (maybe infinitely) many multilinear operations. Such 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. These algebras satisfy the NielsenSchreier property and their subalgebras are also free. 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2007
 arXiv:
 arXiv:0710.0493
 Bibcode:
 2007arXiv0710.0493D
 Keywords:

 Mathematics  Rings and Algebras;
 Mathematics  Combinatorics;
 17A50 (Primary);
 05A15;
 05C05;
 17A36;
 17A42;
 15A72;
 33E05 (Secondary)
 EPrint:
 Revised version, after suggestions of the anonymous referee, 39 pages