Deformation coproducts and differential maps∗

Let T be the Itô Hopf algebra over an associativa algebra L into which the universal enveloping algebra U of the commutator Lie algebra L is embedded as the subalgebra of symmetric tensors. We show that there is a one-to-one correspondence between deformations ∆[h] of the coproduct in T and pairs ( → d [h], ← d [h]) of right and left differential maps which are deformations of the differential maps for T [6]. Corresponding to the multiplicativity and coassociativity of ∆[h], → d [h] and ← d [h] satisfy the Leibniz-Itô formula and a mutual commutativity condition. ∆[h] is recovered from → d [h] and ← d [h] by a generalised Taylor expansion. As an illustrative example we consider the differential maps corresponding to the quantisation of quasitriangular commutator Lie bialgebras of [7].