Classical and variational Poisson cohomology
We prove that, for a Poisson vertex algebra V , the canonical injective homomorphism of the variational cohomology of V to its classical cohomology is an isomorphism, provided that V , viewed as a differential algebra, is an algebra of differential polynomials in finitely many differential variables...
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| Veröffentlicht in: | Japanese journal of mathematics 2021-11, Vol.16 (2), p.203-246 |
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| container_title | Japanese journal of mathematics |
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| creator | Bakalov, Bojko De Sole, Alberto Heluani, Reimundo Kac, Victor G. Vignoli, Veronica |
| description | We prove that, for a Poisson vertex algebra
V
, the canonical injective homomorphism of the variational cohomology of
V
to its classical cohomology is an isomorphism, provided that
V
, viewed as a differential algebra, is an algebra of differential polynomials in finitely many differential variables. This theorem is one of the key ingredients in the computation of vertex algebra cohomology. For its proof, we introduce the sesquilinear Hochschild and Harrison cohomology complexes and prove a vanishing theorem for the symmetric sesquilinear Harrison cohomology of the algebra of differential polynomials in finitely many differential variables. |
| doi_str_mv | 10.1007/s11537-021-2109-2 |
| format | Article |
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V
, the canonical injective homomorphism of the variational cohomology of
V
to its classical cohomology is an isomorphism, provided that
V
, viewed as a differential algebra, is an algebra of differential polynomials in finitely many differential variables. This theorem is one of the key ingredients in the computation of vertex algebra cohomology. For its proof, we introduce the sesquilinear Hochschild and Harrison cohomology complexes and prove a vanishing theorem for the symmetric sesquilinear Harrison cohomology of the algebra of differential polynomials in finitely many differential variables.</description><identifier>ISSN: 0289-2316</identifier><identifier>EISSN: 1861-3624</identifier><identifier>DOI: 10.1007/s11537-021-2109-2</identifier><language>eng</language><publisher>Tokyo: Springer Japan</publisher><subject>Algebra ; History of Mathematical Sciences ; Homology ; Homomorphisms ; Isomorphism ; Mathematics ; Mathematics and Statistics ; Original Paper ; Polynomials ; Theorems</subject><ispartof>Japanese journal of mathematics, 2021-11, Vol.16 (2), p.203-246</ispartof><rights>The Mathematical Society of Japan and Springer Japan KK, part of Springer Nature 2021</rights><rights>The Mathematical Society of Japan and Springer Japan KK, part of Springer Nature 2021.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c382t-e958a84ea837a33c39080554d7b41d89554c88fd311e3749a881100b85bfe89f3</citedby><cites>FETCH-LOGICAL-c382t-e958a84ea837a33c39080554d7b41d89554c88fd311e3749a881100b85bfe89f3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s11537-021-2109-2$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s11537-021-2109-2$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27922,27923,41486,42555,51317</link.rule.ids></links><search><creatorcontrib>Bakalov, Bojko</creatorcontrib><creatorcontrib>De Sole, Alberto</creatorcontrib><creatorcontrib>Heluani, Reimundo</creatorcontrib><creatorcontrib>Kac, Victor G.</creatorcontrib><creatorcontrib>Vignoli, Veronica</creatorcontrib><title>Classical and variational Poisson cohomology</title><title>Japanese journal of mathematics</title><addtitle>Jpn. J. Math</addtitle><description>We prove that, for a Poisson vertex algebra
V
, the canonical injective homomorphism of the variational cohomology of
V
to its classical cohomology is an isomorphism, provided that
V
, viewed as a differential algebra, is an algebra of differential polynomials in finitely many differential variables. This theorem is one of the key ingredients in the computation of vertex algebra cohomology. For its proof, we introduce the sesquilinear Hochschild and Harrison cohomology complexes and prove a vanishing theorem for the symmetric sesquilinear Harrison cohomology of the algebra of differential polynomials in finitely many differential variables.</description><subject>Algebra</subject><subject>History of Mathematical Sciences</subject><subject>Homology</subject><subject>Homomorphisms</subject><subject>Isomorphism</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Original Paper</subject><subject>Polynomials</subject><subject>Theorems</subject><issn>0289-2316</issn><issn>1861-3624</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNp1kE1LAzEQhoMoWKs_wFvBq9FMZrM7OUrxCwp60HNId7N1y3ZTM63Qf29kBU-e5oP3fZl5hLgEdQNKVbcMYLCSSoPUoKzUR2ICVILEUhfHYqI05SVCeSrOmNdKGa2hnIjree-Zu9r3Mz80sy-fOr_r4pDn19gxx2FWx4-4iX1cHc7FSet7Dhe_dSreH-7f5k9y8fL4PL9byBpJ72SwhjwVwRNWHrFGq0gZUzTVsoCGbG5rorZBgIBVYT0R5CeWZJZtINviVFyNudsUP_eBd24d9ynfxE4bawBLKjGrYFTVKTKn0Lpt6jY-HRwo9wPFjVBchuJ-oDidPXr0cNYOq5D-kv83fQOfJWJQ</recordid><startdate>20211101</startdate><enddate>20211101</enddate><creator>Bakalov, Bojko</creator><creator>De Sole, Alberto</creator><creator>Heluani, Reimundo</creator><creator>Kac, Victor G.</creator><creator>Vignoli, Veronica</creator><general>Springer Japan</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20211101</creationdate><title>Classical and variational Poisson cohomology</title><author>Bakalov, Bojko ; De Sole, Alberto ; Heluani, Reimundo ; Kac, Victor G. ; Vignoli, Veronica</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c382t-e958a84ea837a33c39080554d7b41d89554c88fd311e3749a881100b85bfe89f3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Algebra</topic><topic>History of Mathematical Sciences</topic><topic>Homology</topic><topic>Homomorphisms</topic><topic>Isomorphism</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Original Paper</topic><topic>Polynomials</topic><topic>Theorems</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bakalov, Bojko</creatorcontrib><creatorcontrib>De Sole, Alberto</creatorcontrib><creatorcontrib>Heluani, Reimundo</creatorcontrib><creatorcontrib>Kac, Victor G.</creatorcontrib><creatorcontrib>Vignoli, Veronica</creatorcontrib><collection>CrossRef</collection><jtitle>Japanese journal of mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bakalov, Bojko</au><au>De Sole, Alberto</au><au>Heluani, Reimundo</au><au>Kac, Victor G.</au><au>Vignoli, Veronica</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Classical and variational Poisson cohomology</atitle><jtitle>Japanese journal of mathematics</jtitle><stitle>Jpn. J. Math</stitle><date>2021-11-01</date><risdate>2021</risdate><volume>16</volume><issue>2</issue><spage>203</spage><epage>246</epage><pages>203-246</pages><issn>0289-2316</issn><eissn>1861-3624</eissn><abstract>We prove that, for a Poisson vertex algebra
V
, the canonical injective homomorphism of the variational cohomology of
V
to its classical cohomology is an isomorphism, provided that
V
, viewed as a differential algebra, is an algebra of differential polynomials in finitely many differential variables. This theorem is one of the key ingredients in the computation of vertex algebra cohomology. For its proof, we introduce the sesquilinear Hochschild and Harrison cohomology complexes and prove a vanishing theorem for the symmetric sesquilinear Harrison cohomology of the algebra of differential polynomials in finitely many differential variables.</abstract><cop>Tokyo</cop><pub>Springer Japan</pub><doi>10.1007/s11537-021-2109-2</doi><tpages>44</tpages></addata></record> |
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| ispartof | Japanese journal of mathematics, 2021-11, Vol.16 (2), p.203-246 |
| issn | 0289-2316 1861-3624 |
| language | eng |
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| subjects | Algebra History of Mathematical Sciences Homology Homomorphisms Isomorphism Mathematics Mathematics and Statistics Original Paper Polynomials Theorems |
| title | Classical and variational Poisson cohomology |
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