What a classical r-matrix really is

Abstract

The notion of classical $r$-matrix is re-examined, and a definition suitable to differential (-difference) Lie algebras, -- where the standard definitions are shown to be deficient, -- is proposed, the notion of an $\mathcal{O}$-operator. This notion has all the natural properties one would expect form it, but lacks those which are artifacts of finite-dimensional isomorpisms such as not true in differential generality relation $\mbox{End}\, (V) \simeq V^* \otimes V$ for a vector space $V$. Examples considered include a quadratic Poisson bracket on the dual space to a Lie algebra; generalized symplectic-quadratic models of such brackets (aka Clebsch representations); and Drinfel'd's 2-cocycle interpretation of nondegenate classical $r$-matrices.

References in corpus (3)

• On the nature of the Virasoro algebra
• Quantum differential forms
• Coadjoint Poisson actions of Poisson-Lie groups

Cited by in corpus (49)

• Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras
• A Unified Algebraic Approach to Classical Yang-Baxter Equation
• Left-symmetric Bialgebras and An Analogue of the Classical Yang-Baxter Equation
• Integration and geometrization of Rota-Baxter Lie algebras
• Some results on L-dendriform algebras
• From Rota-Baxter Algebras to Pre-Lie Algebras
• Deformations and homotopy theory of relative Rota-Baxter Lie algebras
• Nijenhuis operators on $n$-Lie algebras
• Anti-pre-Lie algebras, Novikov algebras and commutative 2-cocycles on Lie algebras
• Twisted Rota-Baxter operators and Reynolds operators on Lie algebras and NS-Lie algebras
• Generalizations of the classical Yang-Baxter equation and O-operators
• $\mathcal{O}$-Operators on Hom-Lie algebras
• Classification of Graded Left-symmetric Algebra Structures on Witt and Virasoro Algebras
• Factorizable Lie bialgebras, quadratic Rota-Baxter Lie algebras and Rota-Baxter Lie bialgebras
• Purely Hom-Lie bialgebras
• Bijective 1-cocycles and classification of 3-dimensional Left-symmetric Algebras
• Infinite-dimensional Lie bialgebras via affinization of Novikov bialgebras and Koszul duality
• Representations and cohomologies of relative Rota-Baxter Lie algebras and applications
• A bialgebra theory for transposed Poisson algebras via anti-pre-Lie bialgebras and anti-pre-Lie-Poisson bialgebras
• Conformal classical Yang-Baxter equation, $S$-equation and $\mathcal{O}$-operators
• Left-symmetric algebroids
• Deformations of Lie 2-algebras
• Cohomology and deformations of weighted Rota-Baxter operators
• Hom-Big Brackets: Theory and Applications
• The $L_{\infty }$-deformations of associative Rota-Baxter algebras and homotopy Rota-Baxter operators
• Twisted Rota-Baxter operators on 3-Lie algebras and NS-3-Lie algebras
• $3$-post-Lie algebras and relative Rota-Baxter operators of nonzero weight on $3$-Lie algebras
• Anti-flexible bialgebras
• Kupershmidt-(dual-)Nijenhuis structures on a Lie algebra with a representation
• Special identities for the pre-Jordan product in the free dendriform algebra
• Infinite-dimensional Lie bialgebras via affinization of perm bialgebras and pre-Lie bialgebras
• Kupershmidt operators on Hom-Malcev algebras and their deformation
• A survey on deformations, cohomologies and homotopies of relative Rota-Baxter Lie algebras
• Conformal $r$-matrix-Nijenhuis structures, symplectic-Nijenhuis structures and $\mathcal{O}N$-structures
• $\mathcal{O}$-operators on Lie $\infty$-algebras with respect to Lie $\infty$-actions
• $\mathcal{O}$-operators and related structures on Leibniz algebras
• Cohomology and relative Rota-Baxter-Nijenhuis structures on LieYRep pairs
• Coherent categorical structures for Lie bialgebras, Manin triples, classical $r$-matrices and pre-Lie algebras
• Some Non-Abelian Phase Spaces in Low Dimensions
• On the integration of relative Rota-Baxter Lie algebras
• Deformations and Cohomologies of Relative Rota-Baxter Operators on Lie Algebroids and Koszul-Vinberg Structures
• Deformations and homotopy of Rota-Baxter operators and $\mathcal{O}$-operators on Lie algebras
• An operator approach to the rational solutions of the classical Yang-Baxter equation
• Equivariant Nijenhuis Lie Algebras: Extensions to Classical Lie-Theoretic Structures
• Relative Rota-Baxter operators of weight 0 on groups, pre-groups, braces, the Yang-Baxter equation and $T$-structures
• Formal integration of complete Rota-Baxter Lie algebras
• Kupershmidt-Nijenhuis structures on pre-Malcev algebras
• Rota-Baxter operators on crossed modules of Lie groups and categorical solutions of the Yang-Baxter equation
• Weighted $\mathcal{O}$-operators on Hom-Lie triple systems