From algebraic to analytic double product integrals

The algebraic theory of double product integrals and particularly its role in the quantisation of Lie bialgebras is described. When the underlying associative algebra is that of the Itˆo differentials of quantum stochastic calculus such product integrals are formally represented as operators which are infinite sums of iterated integrals in Fock space. In this paper we describe some of the analytic problems encountered in making such sums rigourously meaningful, as well as the expected properties of such analytic double product integrals.