Deformation coproducts and differential maps

R. L. Hudson, and S. Pulmannová. "Deformation coproducts and differential maps." Studia Mathematica 188.1 (2008): 1-16. <http://eudml.org/doc/286560>.

@article{R2008,
abstract = {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)].},
author = {R. L. Hudson, S. Pulmannová},
journal = {Studia Mathematica},
keywords = {deformed coproduct; Hopf algebra; Ito algebra; differential map; quantisation},
language = {eng},
number = {1},
pages = {1-16},
title = {Deformation coproducts and differential maps},
url = {http://eudml.org/doc/286560},
volume = {188},
year = {2008},
}

TY - JOUR
AU - R. L. Hudson
AU - S. Pulmannová
TI - Deformation coproducts and differential maps
JO - Studia Mathematica
PY - 2008
VL - 188
IS - 1
SP - 1
EP - 16
AB - 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)].
LA - eng
KW - deformed coproduct; Hopf algebra; Ito algebra; differential map; quantisation
UR - http://eudml.org/doc/286560
ER -