Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras

Abstract

We generalize the classical study of (generalized) Lax pairs and the related $O$-operators and the (modified) classical Yang-Baxter equation by introducing the concepts of nonabelian generalized Lax pairs, extended $\calo$-operators and the extended classical Yang-Baxter equation. We study in this context the nonabelian generalized $r$-matrix ansatz and the related double Lie algebra structures. Relationship between extended $O$-operators and the extended classical Yang-Baxter equation is established, especially for self-dual Lie algebras. This relationship allows us to obtain explicit description of the Manin triples for a new class of Lie bialgebras. Furthermore, we show that a natural structure of PostLie algebra is behind $O$-operators and fits in a setup of triple Lie algebra that produces self-dual nonabelian generalized Lax pairs.

38 pages

References in corpus (2)

• Loday-type Algebras and the Rota-Baxter Relation
• Infinitesimal bialgebras, pre-Lie and dendriform algebras

Cited by in corpus (59)

• Splitting of operations, Manin products and Rota-Baxter operators
• What is a post-Lie algebra and why is it useful in geometric integration
• Integration and geometrization of Rota-Baxter Lie algebras
• Rota-Baxter groups, skew left braces, and the Yang-Baxter equation
• Post-groups, (Lie-)Butcher groups and the Yang-Baxter equation
• Generalizations of the classical Yang-Baxter equation and O-operators
• Rota-Baxter operators on cocommutative Hopf algebras
• Factorizable Lie bialgebras, quadratic Rota-Baxter Lie algebras and Rota-Baxter Lie bialgebras
• Rota-Baxter operators on $sl(2,C)$ and solutions of the classical Yang-Baxter equation
• Post-Lie algebras and factorization theorems
• Post-Lie Algebras and Isospectral Flows
• Cohomology and deformations of weighted Rota-Baxter operators
• Rota-Baxter operators on Witt and Virasoro algebras
• Post-symmetric braces and integration of post-Lie algebras
• Totally compatible associative and Lie dialgebras, tridendriform algebras and PostLie algebras
• $3$-post-Lie algebras and relative Rota-Baxter operators of nonzero weight on $3$-Lie algebras
• Rota-Baxter type operators, rewriting systems and Gröbner-Shirshov bases
• Deformations and cohomology theory of Rota-Baxter $3$-Lie algebras of arbitrary weights
• Kupershmidt-(dual-)Nijenhuis structures on a Lie algebra with a representation
• Rota-Baxter multiplicative 3-ary Hom-Nambu-Lie algebras
• Rota-Baxter operators on the polynomial algebras, integration and averaging operators
• Cacti and filtered distributive laws
• Hopf algebra of multi-decorated rooted forests, free matching Rota-Baxter algebras and Gröbner-Shirshov bases
• Polylogarithms and multiple zeta values from free Rota-Baxter algebras
• Splitting of operads and Rota-Baxter operators on operads
• Cohomology of weighted Rota-Baxter Lie algebras and Rota-Baxter paired operators
• Representations and Modules of Rota-Baxter Algebras
• Lie theory and cohomology of relative Rota-Baxter operators
• PostLie algebra structures on the Lie algebra sl(2,C)
• Combinatorial Hopf algebra for interconnected nonlinear systems
• Quasi-triangular, factorizable Leibniz bialgebras and relative Rota-Baxter operators
• O-operators on associative algebras and dendriform algebras
• Post-Lie Algebras, Factorization Theorems and Isospectral-Flows
• Matching Rota-Baxter algebras, matching dendriform algebras and matching pre-Lie algebras
• The tridendriform structure of a Magnus expansion
• Universal enveloping Lie Rota-Baxter algebra of preLie and post-Lie algebras
• Functorial PBW theorems for post-Lie algebras
• Homotopy Rota-Baxter operators, homotopy $\mathcal{O}$-operators and homotopy post-Lie algebras
• Rota's program on algebraic operators, rewriting systems and Gröbner-Shirshov bases
• Commutative matching Rota-Baxter operators, shuffle products with decorations and matching Zinbiel algebras
• On the integration of relative Rota-Baxter Lie algebras
• Affine actions on Lie groups and post-Lie algebra structures
• Commutative post-Lie algebra structures on Kac--Moody algebras
• Operator forms of nonhomogeneous associative classical Yang-Baxter equation
• Manin triples and quasitriangular structures of Hom-Poisson bialgebras
• On the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra
• Rota-Baxter operators on crossed modules of Lie groups and categorical solutions of the Yang-Baxter equation
• The category and operad of matching dialgebras
• Post-Hopf algebroids, post-Lie-Rinehart algebras and geometric numerical integration
• Equivariant Nijenhuis Lie Algebras: Extensions to Classical Lie-Theoretic Structures
• An associative analogy of Lie H-pseudobialgebra
• Polarization and deformations of generalized dendriform algebras
• Domain transfer convolutional attribute embedding
• Commutative modified Rota-Baxter algebras, shuffle products and Hopf algebras
• Post-Lie algebra structures on the Witt algebra
• Classification of monomial Rota-Baxter operators on k[x]
• Lagrangian and orthogonal splittings, quasitriangular Lie bialgebras and almost complex product 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