{"title":"Affine Nijenhuis Operators and Hochschild Cohomology of Trusses","authors":"Tomasz Brzezi'nski, J. Papworth","doi":"10.3842/SIGMA.2023.056","DOIUrl":null,"url":null,"abstract":"The classical Hochschild cohomology theory of rings is extended to abelian heaps with distributing multiplication or trusses. This cohomology is then employed to give necessary and sufficient conditions for a Nijenhuis product on a truss (defined by the extension of the Nijenhuis product on an associative ring introduced by Carinena, Grabowski and Marmo in [Internat. J. Modern Phys. A 15 (2000), 4797-4810, arXiv:math-ph/0610011]) to be associative. The definition of Nijenhuis product and operators on trusses is then linearised to the case of affine spaces with compatible associative multiplications or associative affgebras. It is shown that this construction leads to compatible Lie brackets on an affine space.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2023-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Symmetry Integrability and Geometry-Methods and Applications","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.3842/SIGMA.2023.056","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 2
Abstract
The classical Hochschild cohomology theory of rings is extended to abelian heaps with distributing multiplication or trusses. This cohomology is then employed to give necessary and sufficient conditions for a Nijenhuis product on a truss (defined by the extension of the Nijenhuis product on an associative ring introduced by Carinena, Grabowski and Marmo in [Internat. J. Modern Phys. A 15 (2000), 4797-4810, arXiv:math-ph/0610011]) to be associative. The definition of Nijenhuis product and operators on trusses is then linearised to the case of affine spaces with compatible associative multiplications or associative affgebras. It is shown that this construction leads to compatible Lie brackets on an affine space.
期刊介绍:
Scope
Geometrical methods in mathematical physics
Lie theory and differential equations
Classical and quantum integrable systems
Algebraic methods in dynamical systems and chaos
Exactly and quasi-exactly solvable models
Lie groups and algebras, representation theory
Orthogonal polynomials and special functions
Integrable probability and stochastic processes
Quantum algebras, quantum groups and their representations
Symplectic, Poisson and noncommutative geometry
Algebraic geometry and its applications
Quantum field theories and string/gauge theories
Statistical physics and condensed matter physics
Quantum gravity and cosmology.