Deformation coproducts and differential maps
Volume 188 / 2008
Abstract
Let $\mathcal{T}$ be the Itô Hopf algebra over an associative algebra $ \mathcal{L}$ into which the universal enveloping algebra $\mathcal{U}$ of the commutator Lie algebra $\mathcal{L}$ is embedded as the subalgebra of symmetric tensors. We show that there is a one-to-one correspondence between deformations $\Delta [h]$ of the coproduct in $\mathcal{T}$ and pairs $(\mathop{\mathrel{d}}\limits^\to[h],$ $\mathop{\mathrel{d}}\limits^\gets [h])$ of right and left differential maps which are deformations of the differential maps for $ \mathcal{T}$ [Hudson and Pulmannová, J. Math. Phys. 45 (2004)]. Corresponding to the multiplicativity and coassociativity of $\Delta [h],$ $\mathop{\mathrel{d}}\limits^\to[h]$ and $ \mathop{\mathrel{d}}\limits^\gets[h]$ satisfy the Leibniz–Itô formula and a mutual commutativity condition. $\Delta [h]$ is recovered from $\mathop{\mathrel{d}}\limits^\to[h]$ and $ \mathop{\mathrel{d}}\limits^\gets [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 [Hudson and Pulmannová, Lett. Math. Phys. 72 (2005)].
