Rota-Baxter operators on Turaev's Hopf group (co)algebras I: Basic definitions and related algebraic structures

We find a natural compatible condition between the Rota-Baxter operator and Turaev's (Hopf) group-(co)algebras, which leads to the concept of Rota-Baxter Turaev's (Hopf) group-(co)algebra. Two characterizations of Rota-Baxter Turaev's group-algebras (abbr. T-algebras) are obtained: one by Atkinson factorization and the other by T-quasi-idempotent elements. The relations among some related Turaev's group algebraic structures (such as (tri)dendriform T-algebras, Zinbiel T-algebras, pre-Lie T-algebras, Lie T-algebras) are discussed, and some concrete examples from the algebras of dimensions 2,3 and 4 are given. At last we prove that Rota-Baxter Poisson T-algebras can produce pre-Poisson T-algebras and Poisson T-algebras can be obtained from pre-Poisson T-algebras.