Planar trees, free nonassociative algebras, invariants, and elliptic integrals
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 descri...
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| Дата: | 2018 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Lugansk National Taras Shevchenko University
2018
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/806 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| id |
admjournalluguniveduua-article-806 |
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admjournalluguniveduua-article-8062018-03-22T09:39:19Z Planar trees, free nonassociative algebras, invariants, and elliptic integrals Drensky, Vesselin Holtkamp, Ralf 17A50, 17A36, 17A42, 15A72, 33E05 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 elliptic integrals. 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. Lugansk National Taras Shevchenko University 2018-03-22 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/806 Algebra and Discrete Mathematics; Vol 7, No 2 (2008) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/806/336 Copyright (c) 2018 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| baseUrl_str |
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| datestamp_date |
2018-03-22T09:39:19Z |
| collection |
OJS |
| language |
English |
| topic |
17A50 17A36 17A42 15A72 33E05 |
| spellingShingle |
17A50 17A36 17A42 15A72 33E05 Drensky, Vesselin Holtkamp, Ralf Planar trees, free nonassociative algebras, invariants, and elliptic integrals |
| topic_facet |
17A50 17A36 17A42 15A72 33E05 |
| format |
Article |
| author |
Drensky, Vesselin Holtkamp, Ralf |
| author_facet |
Drensky, Vesselin Holtkamp, Ralf |
| author_sort |
Drensky, Vesselin |
| title |
Planar trees, free nonassociative algebras, invariants, and elliptic integrals |
| title_short |
Planar trees, free nonassociative algebras, invariants, and elliptic integrals |
| title_full |
Planar trees, free nonassociative algebras, invariants, and elliptic integrals |
| title_fullStr |
Planar trees, free nonassociative algebras, invariants, and elliptic integrals |
| title_full_unstemmed |
Planar trees, free nonassociative algebras, invariants, and elliptic integrals |
| title_sort |
planar trees, free nonassociative algebras, invariants, and elliptic integrals |
| description |
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 elliptic integrals. 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. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/806 |
| work_keys_str_mv |
AT drenskyvesselin planartreesfreenonassociativealgebrasinvariantsandellipticintegrals AT holtkampralf planartreesfreenonassociativealgebrasinvariantsandellipticintegrals |
| first_indexed |
2025-12-02T15:37:07Z |
| last_indexed |
2025-12-02T15:37:07Z |
| _version_ |
1850411401038790657 |