Quasi-triangular and factorizable antisymmetric infinitesimal bialgebras. (English) Zbl 1529.16035
Working with finite-dimensional vector spaces and algebras over a field \({\mathbb K}\), the authors of this paper introduce the notions of of quasitriangular antisymmetric infinitesimal bialgebra and factorizable antisymmetric infinitesimal bialgebra as an special classes of coboundary antisymmetric infinitesimal bialgebras. In Proposition 2.3 they prove that a factorizable antisymmetric infinitesimal bialgebra leads to a factorization of the underlying associative algebra and in Theorem 2.3 they obtain that the associative double of an antisymmetric infinitesimal bialgebra naturally has a factorizable antisymmetric infinitesimal bialgebra structure. Next they introduce the notion of symmetric Rota-Baxter Frobenius algebras and they show that there is a one-to-one correspondence between factorizable antisymmetric infinitesimal bialgebras and symmetric Rota-Baxter Frobenius algebras (see Theorem 4.6). Finally, they also show in Theorem 4.13 that a symmetric Rota-Baxter Frobenius algebra can give rise to an isomorphism from the regular representation to the coregular representation of a Rota-Baxter associative algebra.
Reviewer: Ramón González Rodríguez (Vigo)
Keywords:
quasi-triangular antisymmetric; infinitesimal bialgebras; factorizable antisymmetric; infinitesimal bialgebras; symmetric Rota-Baxter Frobenius algebraReferences:
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