Integration and geometrization of Rota-Baxter Lie algebras

Abstract

This paper first introduces the notion of a Rota-Baxter operator (of weight $1$) on a Lie group so that its differentiation gives a Rota-Baxter operator on the corresponding Lie algebra. Direct products of Lie groups, including the decompositions of Iwasawa and Langlands, carry natural Rota-Baxter operators. Formal inverse of the Rota-Baxter operator on a Lie group is precisely the crossed homomorphism on the Lie group, whose tangent map is the differential operator of weight $1$ on a Lie algebra. A factorization theorem of Rota-Baxter Lie groups is proved, deriving directly on the Lie group level, the well-known global factorization theorems of Semenov-Tian-Shansky in his study of integrable systems. As geometrization, the notions of Rota-Baxter Lie algebroids and Rota-Baxter Lie groupoids are introduced, with the former a differentiation of the latter. Further, a Rota-Baxter Lie algebroid naturally gives rise to a post-Lie algebroid, generalizing the well-known fact for Rota-Baxter Lie algebras and post-Lie algebras. It is shown that the geometrization of a Rota-Baxter Lie algebra or a Rota-Baxter Lie group can be realized by its action on a manifold. Examples and applications are provided for these new notions.

24 pages. Section 3 revised. Comments welcome

References in corpus (2)

• Rota-Baxter operators on cocommutative Hopf algebras
• Rota-Baxter operators on groups

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• Rota-Baxter operators on groups
• Factorizable Lie bialgebras, quadratic Rota-Baxter Lie algebras and Rota-Baxter Lie bialgebras
• Skew braces from Rota--Baxter operators: a cohomological characterisation and some examples
• Rota-Baxter operators on Clifford semigroups and the Yang-Baxter equation
• Cohomology and deformations of weighted Rota-Baxter operators
• Extensions and automorphisms of Rota-Baxter groups
• Relative Rota-Baxter groups and skew left braces
• Lie theory and cohomology of relative Rota-Baxter operators
• Deformations and homotopy theory of Rota-Baxter algebras of any weight
• Schur multiplier and Schur covers of relative Rota-Baxter groups
• Rota--Baxter operators and skew left brace structures over Heisenberg Group
• Rota's program on algebraic operators, rewriting systems and Gröbner-Shirshov bases
• On the integration of relative Rota-Baxter Lie algebras
• Cohomology and Extensions of Relative Rota-Baxter groups
• Deformations and Cohomologies of Relative Rota-Baxter Operators on Lie Algebroids and Koszul-Vinberg Structures
• Banach Poisson-Lie groups, Lax equations and the AKS theorem in infinite dimensions
• Post-Hopf algebroids, post-Lie-Rinehart algebras and geometric numerical integration
• Kupershmidt-Nijenhuis structures on pre-Malcev algebras
• 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
• 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