What a Classical r-Matrix Really Is

From MaRDI portal

Publication:4498269

Jump to:navigation, search

DOI10.2991/jnmp.1999.6.4.5zbMath1015.17015arXivmath/9910188OpenAlexW3103645568MaRDI QIDQ4498269

Boris A. Kupershmidt

Publication date: 15 August 2000

Published in: Journal of Nonlinear Mathematical Physics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/9910188

zbMATH Keywords

quantum Yang-Baxter equation classical Yang-Baxter equation classical \(r\)-matrix \(\mathcal O\)-operators 2-cocycles linear Poisson brackets symplectic models crossed Lie algebras Clebsch representations differential Lie algebras quadratic Poisson brackets symplectic doubles

Mathematics Subject Classification ID

Quantum groups (quantized enveloping algebras) and related deformations (17B37) Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests (37J35) Poisson manifolds; Poisson groupoids and algebroids (53D17) Poisson algebras (17B63) Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory (37-02)

Related Items

Homotopy relative Rota-Baxter lie algebras, triangular 𝐿_{∞}-bialgebras and higher derived brackets ⋮ Octo-bialgebras ⋮ The L∞-deformations of associative Rota–Baxter algebras and homotopy Rota–Baxter operators ⋮ Conformal r-matrix-Nijenhuis structures, symplectic-Nijenhuis structures, and ON-structures ⋮ Cohomology and deformations of weighted Rota–Baxter operators ⋮ O-operators and related structures on Leibniz algebras ⋮ Anti-flexible bialgebras ⋮ Cohomologies and relative Rota-Baxter-Nijenhuis structures of 3-LieRep pairs ⋮ Representations and cohomologies of relative Rota-Baxter Lie algebras and applications ⋮ On Hom-prealternative bialgebras ⋮ Left-symmetric algebras from linear functions ⋮ Cohomology and deformations of \(\mathcal{O}\)-operators on Hom-associative algebras ⋮ On Real Simple Left-symmetric Algebras ⋮ R-matrix, Lax pair, and multiparameter decompositions of Lie algebras ⋮ Special identities for the pre-Jordan product in the free dendriform algebra ⋮ Admissible Poisson bialgebras ⋮ Coherent categorical structures for Lie bialgebras, Manin triples, classical \(r\)-matrices and pre-Lie algebras ⋮ Deformation of Kupershmidt operators and Kupershmidt-Nijenhuis structures of a Malcev algebra ⋮ Deformations and cohomologies of relative Rota-Baxter operators on Lie algebroids and Koszul-Vinberg structures ⋮ O-Operators on Lie ∞-algebras with respect to Lie ∞-actions ⋮ \(\mathcal{O}\)-operators and Nijenhuis operators of associative conformal algebras ⋮ Anti-pre-Lie algebras, Novikov algebras and commutative 2-cocycles on Lie algebras ⋮ Relative Rota-Baxter operators and symplectic structures on Lie-Yamaguti algebras ⋮ Deformations and homotopy theory of relative Rota-Baxter Lie algebras ⋮ Noncommutative Poisson bialgebras ⋮ Deformations and their controlling cohomologies of \(\mathcal{O}\)-operators ⋮ Kupershmidt-(dual-)Nijenhuis structures on a Lie algebra with a representation ⋮ An operator approach to the rational solutions of the classical Yang-Baxter equation ⋮ \(F\)-manifold algebras and deformation quantization via pre-Lie algebras ⋮ Homotopy Rota-Baxter operators and post-Lie algebras ⋮ Non-abelian extensions of Rota-Baxter Lie algebras and inducibility of automorphisms ⋮ A survey on deformations, cohomologies and homotopies of relative Rota–Baxter Lie algebras ⋮ Twisting theory, relative Rota-Baxter type operators and \(L_\infty\)-algebras on Lie conformal algebras ⋮ On Hom-F-manifold algebras and quantization ⋮ Bimodules over relative Rota-Baxter algebras and cohomologies ⋮ Relative Rota–Baxter Leibniz algebras, their characterization and cohomology ⋮ Operated groups, differential groups and Rota-Baxter groups with an emphasis on the free objects ⋮ Super Kupershmidt operators and Kupershmidt-Nijenhuis structures on Malcev superalgebras ⋮ L ∞ -structures and cohomology theory of compatible O-operators and compatible dendriform algebras ⋮ Post-Hopf algebras, relative Rota-Baxter operators and solutions to the Yang-Baxter equation ⋮ Cohomologies and deformations of O-operators on Lie triple systems ⋮ Cohomologies of relative Rota-Baxter Lie algebras with derivations and applications ⋮ Operator identities on Lie algebras, rewriting systems and Gröbner-Shirshov bases ⋮ Infinite-dimensional Lie bialgebras via affinization of Novikov bialgebras and Koszul duality ⋮ A left-symmetric algebraic approach to left invariant flat (pseudo-)metrics on Lie groups ⋮ Kupershmidt operators on Hom-Malcev algebras and their deformation ⋮ Lie theory and cohomology of relative Rota–Baxter operators ⋮ Cohomology and deformations of generalized Reynolds operators on Leibniz algebras ⋮ Quasi-triangular pre-Lie bialgebras, factorizable pre-Lie bialgebras and Rota-Baxter pre-Lie algebras ⋮ Homotopical algebra and higher structures. Abstracts from the workshop held September 19--25, 2021 (hybrid meeting) ⋮ Cohomology and relative Rota-Baxter-Nijenhuis structures on \(\mathsf{LieYRep}\) pairs ⋮ Lie \(n\)-algebras and cohomologies of relative Rota-Baxter operators on \(n\)-Lie algebras ⋮ LEFT-SYMMETRIC BIALGEBRAS AND AN ANALOGUE OF THE CLASSICAL YANG–BAXTER EQUATION ⋮ Leibniz bialgebras, relative Rota-Baxter operators, and the classical Leibniz Yang-Baxter equation ⋮ Factorizable Lie bialgebras, quadratic Rota-Baxter Lie algebras and Rota-Baxter Lie bialgebras ⋮ On parakähler Hom-Lie algebras and Hom-left-symmetric bialgebras ⋮ Splitting of operations for alternative and Malcev structures ⋮ Unnamed Item ⋮ Lie 3-algebras and deformations of relative Rota-Baxter operators on 3-Lie algebras ⋮ Unnamed Item ⋮ On a class of Lie groups with a left invariant flat pseudo-metric ⋮ Deformations of Lie 2-algebras ⋮ 𝒪-Operators of Loday Algebras and Analogues of the Classical Yang–Baxter Equation ⋮ Poisson bialgebras ⋮ Crossed morphisms, integration of post-Lie algebras and the post-Lie Magnus expansion ⋮ Twisted Rota–Baxter operators and Reynolds operators on Lie algebras and NS-Lie algebras ⋮ Conformal classical Yang-Baxter equation, \(S\)-equation and \({\mathcal{O}}\)-operators ⋮ Purely Hom-Lie bialgebras ⋮ Some non-abelian phase spaces in low dimensions ⋮ Compatible \(\mathcal{O}\)-operators on bimodules over associative algebras ⋮ Unnamed Item ⋮ J-dendriform algebras ⋮ Yang-Baxter equations on matrix dialgebras with a bar unit ⋮ OPERATOR FORMS OF THE CLASSICAL YANG–BAXTER EQUATION IN LIE SUPERALGEBRAS ⋮ Some results on L-dendriform algebras ⋮ Left-symmetric superalgebra structures on the N = 2 superconformal algebras ⋮ Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras ⋮ The Riccati and Ermakov-Pinney hierarchies ⋮ Integration and geometrization of Rota-Baxter Lie algebras ⋮ Bijective 1-Cocycles and Classification of 3-Dimensional Left-Symmetric Algebras ⋮ An example of a simple double Lie algebra ⋮ CLASSIFICATION OF GRADED LEFT-SYMMETRIC ALGEBRAIC STRUCTURES ON WITT AND VIRASORO ALGEBRAS ⋮ O -operators on hom-Lie algebras ⋮ Left-symmetric algebroids ⋮ Categorification of Pre-Lie Algebras and Solutions of 2-graded Classical Yang-Baxter Equations ⋮ Rota-Baxter operators on pre-Lie superalgebras ⋮ Cohomologies of \textsf{PoiMod} pairs and compatible structures on Poisson algebras ⋮ A FURTHER STUDY ON NON-ABELIAN PHASE SPACES: LEFT-SYMMETRIC ALGEBRAIC APPROACH AND RELATED GEOMETRY ⋮ Kupershmidt-(dual-)Nijenhuis structure on the alternative algebra with a representation ⋮ RELATIVE ROTA–BAXTER OPERATORS AND TRIDENDRIFORM ALGEBRAS ⋮ Left-symmetric conformal algebras and vertex algebras ⋮ 3-post-Lie algebras and relative Rota-Baxter operators of nonzero weight on 3-Lie algebras ⋮ Pre-Lie analogues of Poisson-Nijenhuis structures and Maurer–Cartan equations ⋮ Generalizations of the classical Yang-Baxter equation and ${\mathcal O}$O-operators ⋮ Hom-big brackets: theory and applications

Cites Work

Retrieved from "https://portal.mardi4nfdi.de/w/index.php?title=Publication:4498269&oldid=18586389"