Generalizations of the classical Yang-Baxter equation and O-operators

Abstract

Tensor solutions ($r$-matrices) of the classical Yang-Baxter equation (CYBE) in a Lie algebra, obtained as the classical limit of the $R$-matrix solution of the quantum Yang-Baxter equation (QYBE), is an important structure appearing in different areas such as integrable systems, symplectic geometry, quantum groups and quantum field theory. Further study of CYBE led to its interpretation as certain operators, giving rise to the concept of O-operators. In [3], the O-operators were in turn interpreted as tensor solutions of CYBE by enlarging the Lie algebra. The purpose of this paper is to extend this study to a more general class of operators that were recently introduced [4] in the study of Lax pairs in integrable systems. Relationship between $\calo$-operators, relative differential operators and Rota-Baxter operators are also discussed.

17 pages

References in corpus (2)

• Loday-type Algebras and the Rota-Baxter Relation
• A Unified Algebraic Approach to Classical Yang-Baxter Equation

Cited by in corpus (13)

• Rota-Baxter operators on $sl(2,C)$ and solutions of the classical Yang-Baxter equation
• 3-Lie-Rinehart Algebras
• Rota-Baxter operators on Witt and Virasoro algebras
• Rota-Baxter multiplicative 3-ary Hom-Nambu-Lie algebras
• Kupershmidt-(dual-)Nijenhuis structures on a Lie algebra with a representation
• Deformations and cohomology theory of Rota-Baxter $3$-Lie algebras of arbitrary weights
• $3$-BiHom-Lie superalgebras induced by BiHom-Lie superalgebras
• Homogeneous conformal averaging operators on semisimple Lie algebras
• Cross-domain attribute representation based on convolutional neural network
• Rota-Baxter Operators on pre-Lie superalgebras and beyond
• Operator forms of nonhomogeneous associative classical Yang-Baxter equation
• On $\mathcal{O}$-operators on modules over Lie algebras
• Lagrangian and orthogonal splittings, quasitriangular Lie bialgebras and almost complex product structures