2021/12/21 by Bai, Chengming, Guo, Li, Ma, Tianshui · 5 citations
#16T05 #16T10 #16T25 #16W99 #17A30 #17B38 #17B62 #57R56 #81$60 #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2112.10928
Rota-Baxter operators and bialgebras go hand in hand in their applications, such as in the Connes-Kreimer approach to renormalization and the operator approach to the classical Yang-Baxter equation. We establish a bialgebra structure that is compatible with the Rota-Baxter operator, called the Rota-Baxter antisymmetric infinitesimal (ASI) bialgebra. This bialgebra is characterized by generalizations of matched pairs of algebras and double constructions of Frobenius algebras to the context of Rota-Baxter algebras. The study of the coboundary case leads to an enrichment of the associative Yang-Baxter equation (AYBE) to Rota-Baxter algebras. Antisymmetric solutions of the equation are used to construct Rota-Baxter ASI bialgebras. The notions of an O-operator on a Rota-Baxter algebra and a Rota-Baxter dendriform algebra are also introduced to produce solutions of the AYBE in Rota-Baxter algebras and thus to provide Rota-Baxter ASI bialgebras. An unexpected byproduct is that a Rota-Baxter ASI bialgebra of weight zero gives rise to a quadri-bialgebra instead of bialgebra constructions for the dendriform algebra.