Rota-Baxter operators on cocommutative Hopf algebras

Abstract

We generalize the notion of a Rota-Baxter operator on groups and the notion of a Rota-Baxter operator of weight 1 on Lie algebras and define and study the notion of a Rota-Baxter operator on a cocommutative Hopf algebra $H$. If $H=F[G]$ is the group algebra of a group $G$ or $H=U(\mathfrak{g})$ the universal enveloping algebra of a Lie algebra $\mathfrak{g}$, then we prove that Rota-Baxter operators on $H$ are in one to one correspondence with corresponding Rota-Baxter operators on groups or Lie algebras.

arXiv comment14 pages

References in corpus (6)

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• Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras
• Integration and geometrization of Rota-Baxter Lie algebras
• On the Lie enveloping algebra of a post-Lie algebra
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Cited by in corpus (9)

• Integration and geometrization of Rota-Baxter Lie algebras
• Rota-Baxter groups, skew left braces, and the Yang-Baxter equation
• Rota-Baxter operators on groups
• Factorizable Lie bialgebras, quadratic Rota-Baxter Lie algebras and Rota-Baxter Lie bialgebras
• Lie theory and cohomology of relative Rota-Baxter operators
• Rota-Baxter operators on Turaev's Hopf group (co)algebras I: Basic definitions and related algebraic 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
• On the sub-adjacent Hopf algebra of the universal enveloping algebra of a post-Lie algebra