Locally Nilpotent Derivations of Free Algebra of Rank Two
In commutative algebra, if δ is a locally nilpotent derivation of the polynomial algebra [x₁, …, xd] over a field of characteristic 0 and w is a nonzero element of the kernel of δ, then Δ=wδ is also a locally nilpotent derivation with the same kernel as δ. In this paper, we prove that the locally n...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2019 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2019
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/210297 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Locally Nilpotent Derivations of Free Algebra of Rank Two / V. Drensky, L. Makar-Limanov // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 31 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862744647852359680 |
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| author | Drensky, V. Makar-Limanov, L. |
| author_facet | Drensky, V. Makar-Limanov, L. |
| citation_txt | Locally Nilpotent Derivations of Free Algebra of Rank Two / V. Drensky, L. Makar-Limanov // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 31 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In commutative algebra, if δ is a locally nilpotent derivation of the polynomial algebra [x₁, …, xd] over a field of characteristic 0 and w is a nonzero element of the kernel of δ, then Δ=wδ is also a locally nilpotent derivation with the same kernel as δ. In this paper, we prove that the locally nilpotent derivation Δ of the free associative algebra ⟨X, Y⟩ is determined up to a multiplicative constant by its kernel. We also show that the kernel of Δ is a free associative algebra and give an explicit set of its free generators.
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| first_indexed | 2025-12-07T21:25:04Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-210297 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:25:04Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Drensky, V. Makar-Limanov, L. 2025-12-05T09:23:54Z 2019 Locally Nilpotent Derivations of Free Algebra of Rank Two / V. Drensky, L. Makar-Limanov // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 31 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 16S10; 16W25; 16W20; 13N15 arXiv: 1909.13262 https://nasplib.isofts.kiev.ua/handle/123456789/210297 https://doi.org/10.3842/SIGMA.2019.091 In commutative algebra, if δ is a locally nilpotent derivation of the polynomial algebra [x₁, …, xd] over a field of characteristic 0 and w is a nonzero element of the kernel of δ, then Δ=wδ is also a locally nilpotent derivation with the same kernel as δ. In this paper, we prove that the locally nilpotent derivation Δ of the free associative algebra ⟨X, Y⟩ is determined up to a multiplicative constant by its kernel. We also show that the kernel of Δ is a free associative algebra and give an explicit set of its free generators. The authors are grateful to the Beijing International Center for Mathematical Research for its warm hospitality during their visit when this work was started. The second author is grateful to the Max-Planck-Institut f¨ur Mathematik in Bonn, where he was a visitor when this project was finished. While working on this project, he was also supported by a FAPESP grant awarded by the State of São Paulo, Brazil. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Locally Nilpotent Derivations of Free Algebra of Rank Two Article published earlier |
| spellingShingle | Locally Nilpotent Derivations of Free Algebra of Rank Two Drensky, V. Makar-Limanov, L. |
| title | Locally Nilpotent Derivations of Free Algebra of Rank Two |
| title_full | Locally Nilpotent Derivations of Free Algebra of Rank Two |
| title_fullStr | Locally Nilpotent Derivations of Free Algebra of Rank Two |
| title_full_unstemmed | Locally Nilpotent Derivations of Free Algebra of Rank Two |
| title_short | Locally Nilpotent Derivations of Free Algebra of Rank Two |
| title_sort | locally nilpotent derivations of free algebra of rank two |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210297 |
| work_keys_str_mv | AT drenskyv locallynilpotentderivationsoffreealgebraofranktwo AT makarlimanovl locallynilpotentderivationsoffreealgebraofranktwo |