Odd and even derivations, transposed Poisson superalgebra and 3-Lie superalgebra.
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| Title: | Odd and even derivations, transposed Poisson superalgebra and 3-Lie superalgebra. |
|---|---|
| Alternate Title: | Paaritud ja paarisderivatsioonid, transponeeritud Poissoni superalgebra ja 3-Lie superalgebra. |
| Authors: | Abramov, Viktor1, Sovetnikov, Nikolai1 nikolai.sovetnikov@ut.ee |
| Source: | Proceedings of the Estonian Academy of Sciences. 2025, Vol. 74 Issue 4, p487-499. 13p. |
| Subjects: | Superalgebras, Commutative algebra, Lie superalgebras, Jordan algebras |
| Abstract (English): | One important example of a transposed Poisson algebra can be constructed by means of a commutative algebra and its derivation. This approach can be extended to superalgebras; that is, one can construct a transposed Poisson superalgebra given a commutative superalgebra and its even derivation. In this paper, we show that including odd derivations in the framework of this approach requires introducing a new notion. It is a super vector space with two operations that satisfy the compatibility condition of a transposed Poisson superalgebra. The first operation is determined by a left supermodule over a commutative superalgebra and the second is a Jordan bracket. Then it is proved that the super vector space generated by an odd derivation of a commutative superalgebra satisfies all the requirements of the introduced notion. We also show how to construct a 3-Lie superalgebra if we are given a transposed Poisson superalgebra and its even derivation. [ABSTRACT FROM AUTHOR] |
| Abstract (Estonian): | Ühe olulise näite transponeeritud Poissoni algebrast saab konstrueerida kommutatiivse algebra ja selle derivatsiooni abil. Seda lähenemist saab laiendada superalgebratele, st konstrueerida transponeeritud Poissoni superalgebra, kui on antud kommutatiivne superalgebra ja selle paarisderivatsioon. Artiklis näitame, et paaritute derivatsioonide kaasamine selle lähenemisviisi raamistikku eeldab uue struktuuri defineerimist. See on supervektorruum kahe tehtega, mis rahuldavad transponeeritud Poissoni superalgebra kooskõlatingimust. Esimese tehte määrab vasakpoolne supermoodul üle kommutatiivse superalgebra ja teine on Jordani sulg. Seejärel tõestatakse, et kommutatiivse superalgebra paaritu derivatsiooni tekitatud supervektorruumi korral on kõik nimetatud struktuuri tingimused täidetud. Samuti näidatakse, kuidas konstrueerida 3-Lie-superalgebrat, kui on antud transponeeritud Poissoni superalgebra ja selle paarisderivatsioon. [ABSTRACT FROM AUTHOR] |
| Copyright of Proceedings of the Estonian Academy of Sciences is the property of Teaduste Akadeemia Kirjastus and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Links: – Type: pdflink Text: Availability: 0 |
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| Header | DbId: egs DbLabel: Engineering Source An: 189725020 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Odd and even derivations, transposed Poisson superalgebra and 3-Lie superalgebra. – Name: TitleAlt Label: Alternate Title Group: TiAlt Data: Paaritud ja paarisderivatsioonid, transponeeritud Poissoni superalgebra ja 3-Lie superalgebra. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Abramov%2C+Viktor%22">Abramov, Viktor</searchLink><relatesTo>1</relatesTo><br /><searchLink fieldCode="AR" term="%22Sovetnikov%2C+Nikolai%22">Sovetnikov, Nikolai</searchLink><relatesTo>1</relatesTo><i> nikolai.sovetnikov@ut.ee</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Proceedings+of+the+Estonian+Academy+of+Sciences%22">Proceedings of the Estonian Academy of Sciences</searchLink>. 2025, Vol. 74 Issue 4, p487-499. 13p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Superalgebras%22">Superalgebras</searchLink><br /><searchLink fieldCode="DE" term="%22Commutative+algebra%22">Commutative algebra</searchLink><br /><searchLink fieldCode="DE" term="%22Lie+superalgebras%22">Lie superalgebras</searchLink><br /><searchLink fieldCode="DE" term="%22Jordan+algebras%22">Jordan algebras</searchLink> – Name: Abstract Label: Abstract (English) Group: Ab Data: One important example of a transposed Poisson algebra can be constructed by means of a commutative algebra and its derivation. This approach can be extended to superalgebras; that is, one can construct a transposed Poisson superalgebra given a commutative superalgebra and its even derivation. In this paper, we show that including odd derivations in the framework of this approach requires introducing a new notion. It is a super vector space with two operations that satisfy the compatibility condition of a transposed Poisson superalgebra. The first operation is determined by a left supermodule over a commutative superalgebra and the second is a Jordan bracket. Then it is proved that the super vector space generated by an odd derivation of a commutative superalgebra satisfies all the requirements of the introduced notion. We also show how to construct a 3-Lie superalgebra if we are given a transposed Poisson superalgebra and its even derivation. [ABSTRACT FROM AUTHOR] – Name: Abstract Label: Abstract (Estonian) Group: Ab Data: Ühe olulise näite transponeeritud Poissoni algebrast saab konstrueerida kommutatiivse algebra ja selle derivatsiooni abil. Seda lähenemist saab laiendada superalgebratele, st konstrueerida transponeeritud Poissoni superalgebra, kui on antud kommutatiivne superalgebra ja selle paarisderivatsioon. Artiklis näitame, et paaritute derivatsioonide kaasamine selle lähenemisviisi raamistikku eeldab uue struktuuri defineerimist. See on supervektorruum kahe tehtega, mis rahuldavad transponeeritud Poissoni superalgebra kooskõlatingimust. Esimese tehte määrab vasakpoolne supermoodul üle kommutatiivse superalgebra ja teine on Jordani sulg. Seejärel tõestatakse, et kommutatiivse superalgebra paaritu derivatsiooni tekitatud supervektorruumi korral on kõik nimetatud struktuuri tingimused täidetud. Samuti näidatakse, kuidas konstrueerida 3-Lie-superalgebrat, kui on antud transponeeritud Poissoni superalgebra ja selle paarisderivatsioon. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Proceedings of the Estonian Academy of Sciences is the property of Teaduste Akadeemia Kirjastus and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.3176/proc.2025.4.02 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 13 StartPage: 487 Subjects: – SubjectFull: Superalgebras Type: general – SubjectFull: Commutative algebra Type: general – SubjectFull: Lie superalgebras Type: general – SubjectFull: Jordan algebras Type: general Titles: – TitleFull: Odd and even derivations, transposed Poisson superalgebra and 3-Lie superalgebra. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Abramov, Viktor – PersonEntity: Name: NameFull: Sovetnikov, Nikolai IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 10 Text: 2025 Type: published Y: 2025 Identifiers: – Type: issn-print Value: 17366046 Numbering: – Type: volume Value: 74 – Type: issue Value: 4 Titles: – TitleFull: Proceedings of the Estonian Academy of Sciences Type: main |
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