Higher structures, quantum groups and genus zero modular operad

Abstract

In my Montreal lecture notes of 1988, it was suggested that the theory of linear quantum groups can be presented in the framework of the category of {\\it quadratic algebras} (imagined as algebras of functions on quantum linear spaces ), and quadratic algebras of their inner (co)homomorphisms. \nSoon it was understood (E. Getzler and J. Jones, V. Ginzburg, M. Kapranov, M. Kontsevich, M. Markl, B. Vallette et al.) that the class of {\\it quadratic operads} can be introduced and the main theorems about quadratic algebras can be generalised to the level of such operads, if their components are {\\it linear spaces} (or objects of more general monoidal categories.) \nWhen quantum cohomology entered the scene, it turned out that the basic tree level (genus zero) (co)operad of quantum cohomology not only is {\\it quadratic} one, but its {\\it components are themselves quadratic algebras.} \nIn this short note, I am studying the interaction of quadratic algebras structure with operadic structure in the context of enriched category formalism due to G. M. Kelly et al.