Compatible \(\mathcal{O}\)-operators on bimodules over associative algebras
Motivated by the study of associative analogues of Poisson structures, \textit{K. Uchino} [Lett. Math. Phys. 85, No. 2--3, 91--109 (2008; Zbl 1243.17002)] introduced the notion of a generalized Rota-Baxter operator on a bimodule over an associative algebra. In the Lie algebra context, such a structure was introduced by \textit{B. A. Kupershmidt} [J. Nonlinear Math. Phys. 6, No. 4, 448--488 (1999; Zbl 1015.17015)] and called an \(\mathcal{O}\)-operator. The authors introduce compatibility conditions between an \(\mathcal{O}\)-operator and define the notion of an \(\mathcal{O}N\)-structure on a bimodule over an associative algebra, show that it gives rise to a hierarchy of \(\mathcal{O}\)-operators and that a solution of the strong Maurer-Cartan equation [\textit{K. Uchino}, J. Noncommut. Geom. 4, No. 3, 349--379 (2010; Zbl 1248.16027)] on the associative twilled algebra associated to an \(\mathcal{O}\)-operator gives rise to a pair of \(\mathcal{O}N\)-structures which are naturally in duality. Furthermore, \(PN\)-, \(\Omega N\)-, \(P\Omega\)-structures on associative algebras are defined and then their relations studied.
scientific article; zbMATH DE number 7073740
scientific article