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 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.

arXiv commentRevised version, after suggestions of the anonymous referee, 39 pages

References in corpus (2)

• Constants of Weitzenböck derivations and invariants of unipotent transformations acting on relatively free algebras
• Partial magmatic bialgebras

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• Graded algebras with prescribed Hilbert series
• Groebner-Shirshov Bases for Associative Algebras with Multiple Operators and Free Rota-Baxter Algebras
• Gröbner--Shirshov bases for Vinberg--Koszul--Gerstenhaber right-symmetric algebras