Compatible Structures of Operads by Polarization, Their Koszul Duality and Manin Products

Abstract Algebraic structures with replicate operations interrelated by various compatibility conditions have long been studied in mathematics and mathematical physics. They are broadly referred as linearly compatible, matching, and totally compatible structures. This paper gives a unified approach to these structures in the context of operads. Generalizing polarizations for polynomials in invariant theory to operads leads to linearly compatible operads. Partitioning polarizations into foliations gives matching operads which further yields total compatible operads under an invariance condition. For unary/binary quadratic operads, linear compatibility and total compatibility are in Koszul dual, and the matching compatibilities are Koszul self-dual among themselves. For binary quadratic operads, these three compatible operads can be achieved by taking Manin products. For some finitely generated binary quadratic operad, Koszulity is preserved under taking the compatibilities.