Let 𝒯 be the Itô Hopf algebra over an associative 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 Δ[h] of the coproduct in 𝒯 and pairs (d⃗[h],d⃖[h]) of right and left differential maps which are deformations of the differential maps for 𝒯 [Hudson and Pulmannová, J. Math. Phys. 45 (2004)]. Corresponding to the multiplicativity and coassociativity of Δ[h], d⃗[h] and d⃖[h] satisfy the Leibniz-Itô formula and a mutual commutativity condition. Δ[h] is recovered from d⃗[h] and d⃖[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)].
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm188-1-1,
author = {R. L. Hudson and S. Pulmannov\'a},
title = {Deformation coproducts and differential maps},
journal = {Studia Mathematica},
volume = {187},
year = {2008},
pages = {1-16},
zbl = {1151.53069},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm188-1-1}
}
R. L. Hudson; S. Pulmannová. Deformation coproducts and differential maps. Studia Mathematica, Tome 187 (2008) pp. 1-16. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm188-1-1/