Factorizable Lie bialgebras, quadratic Rota-Baxter Lie algebras and Rota-Baxter Lie bialgebras
arXiv:2112.07902 · doi:10.1007/s00220-022-04501-y
Abstract
In this paper, first we introduce the notion of quadratic Rota-Baxter Lie algebras of arbitrary weight, and show that there is a one-to-one correspondence between factorizable Lie bialgebras and quadratic Rota-Baxter Lie algebras of nonzero weight. Then we introduce the notions of matched pairs, bialgebras and Manin triples of Rota-Baxter Lie algebras of arbitrary weight, and show that Rota-Baxter Lie bialgebras, Manin triples of Rota-Baxter Lie algebras and certain matched pairs of Rota-Baxter Lie algebras are equivalent. The coadjoint representations and quadratic Rota-Baxter Lie algebras play important roles in the whole study. Finally we generalize some results to the Lie group context. In particular, we show that there is a one-to-one correspondence between factorizable Poisson Lie groups and quadratic Rota-Baxter Lie groups.
arXiv comment26 pages, comments are welcome
References in corpus (16)
- Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem
- Splitting of operations, Manin products and Rota-Baxter operators
- What a classical r-matrix really is
- Geometry and classification of solutions of the Classical Dynamical Yang-Baxter Equation
- Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras
- A Unified Algebraic Approach to Classical Yang-Baxter Equation
- Spitzer's Identity and the Algebraic Birkhoff Decomposition in pQFT
- A noncommutative Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogoliubov's recursion
- Quantum Analogy of Poisson Geometry, Related Dendriform Algebras and Rota-Baxter Operators
- Integration and geometrization of Rota-Baxter Lie algebras
- Rota-Baxter groups, skew left braces, and the Yang-Baxter equation
- Simple finite-dimensional double algebras
- Rota-Baxter operators on cocommutative Hopf algebras
- Weak composition quasi-symmetric functions, Rota-Baxter algebras and Hopf algebras
- Rota-Baxter operators on groups
- Representations and cohomologies of relative Rota-Baxter Lie algebras and applications