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Matching Hom-setting of Rota-Baxter algebras, dendriform algebras, and pre-Lie algebras. (English) Zbl 1482.17036

In the “classical” setting, there are well-known relations between associative Rota-Baxter algebras, Lie Rota-Baxter algebras, dendriform algebras and pre-Lie algebras, which can be summed up by a commutative diagram between the four corresponding categories (or between the four corresponding operads). This diagram has been lifted to the setting of hom-objects, where the usual axioms are twisted by a linear map in a convenient way, by Mahklouf and Silvestrov. It has also been lifted to the setting of matching objects, where each product of the classical setting is replaced by a family of products indexed by a given set, in such a way that any linear span of these products gives rise to a classical object.
Here, this diagram is finally lifted to the setting of matching hom-objects, which are introduced in this paper. Matching hom associative, Lie, (tri)dendriform, Rota-Baxter associative, Rota-Baxter Lie, pre-Lie algebras are introduced and it is shown that indeed the four classical functors can be extended here and give a similar commuting diagram. As an example, an associative matching hom algebra has a family of binary product \((\cdot_\alpha)_{\alpha \in \Omega}\) indexed by a given set \(\Omega\), and an endomorphism \(p\), with the axioms \[ (x\cdot_\alpha y)\cdot_\beta p(z)=p(x)\cdot_\alpha(y\cdot_\beta z), \] for any \(\alpha,\beta\in \Omega\).

MSC:

17B38 Yang-Baxter equations and Rota-Baxter operators
17A30 Nonassociative algebras satisfying other identities
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References:

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