Rota-Baxter operators on Turaev’s Hopf group (co)algebras. I: Basic definitions and related algebraic structures. (English) Zbl 1484.17027
A theory of Rota-Baxter for Turaev (co-)algebras is developed. A Turaev algebra is an associative algebra which is graded by a semigroup \(\Omega\) and a Turaev Rota-Baxter algebra is a Turaev associative algebra with a homogeneous Rota-Baxter operators. This includes Rota-Baxter algebras and Rota-Baxter pairs.
These Rota-Baxter T-algebras are characterised into two different ways, the first one using Atkinson factorization and the second one using idempotents. Examples of dimension 2, 3, and 4 are given. Classical relations between Rota-Baxter algebras and (tri)-dendriform, pre-Lie, Lie, zinbiel algebras are extended to the T-context, as well as results on pre-Poisson and Poisson algebras.
These Rota-Baxter T-algebras are characterised into two different ways, the first one using Atkinson factorization and the second one using idempotents. Examples of dimension 2, 3, and 4 are given. Classical relations between Rota-Baxter algebras and (tri)-dendriform, pre-Lie, Lie, zinbiel algebras are extended to the T-context, as well as results on pre-Poisson and Poisson algebras.
Reviewer: Loïc Foissy (Calais)
MSC:
| 17B38 | Yang-Baxter equations and Rota-Baxter operators |
| 16T05 | Hopf algebras and their applications |
| 16T10 | Bialgebras |
| 17B63 | Poisson algebras |
Keywords:
Rota-Baxter (co,bi)algebras; (Tri)dendriform algebras; Zinbiel algebras; (Pre-)Lie algebras; Rota-Baxter Poisson algebras; Rota-Baxter Hopf algebrasReferences:
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