From Bialgebras to Operads. Quantum Line and Cooperad of Correlation Functions

Abstract

A q-line is a simple example of a braided Hopf algebra. This is just an algebra of polynomials kq[z] with primitive generator and q-deformed statistics.The (co)action of a q-line on an algebra is a q-derivation. We construct an operad and a cooperad from a bialgebra. In the case of a q-line, this construction is related to the cooperad of correlation functions of I. Kriz et al., which describes vertex algebras.Modules over the factor-algebra kq[z]/(z N ) are N -complexes. We consider a homotopical category of N -complexes as an example of the q-analog of Maltsiniotis' strongly triangulated category.The general constructions are considered in the context of iterated monoidal categories with unbiased lax tensor products described in the terms of the Gray tensor products of 2-fold categorical operads of sequential trees Tree . K e y w o r d s: bialgebra, operad, unbiased tensor products, multitensor category, vertex algebra.