Mathematics – Rings and Algebras
Scientific paper
2007-10-02
Algebra Discrete Math. 2008, no. 2, 1-41
Mathematics
Rings and Algebras
Revised version, after suggestions of the anonymous referee, 39 pages
Scientific paper
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 Nielsen-Schreier 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.
Drensky Vesselin
Holtkamp Ralf
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