Deformation coproducts and differential maps

R. L. Hudson ¹   ; S. Pulmannová ²

¹ Department of Mathematical Sciences Loughborough University Loughborough, Leicestershire LE11 3TU, Great Britain
² Mathematical Institute Slovak Academy of Sciences Stefankova 49 81473 Bratislava, Slovakia

Studia Mathematica, Tome 188 (2008) no. 1, pp. 1-16

Résumé

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)].

Détail

DOI : 10.4064/sm188-1-1

Keywords: mathcal hopf algebra associative algebra mathcal which universal enveloping algebra mathcal commutator lie algebra mathcal embedded subalgebra symmetric tensors there one to one correspondence between deformations delta coproduct mathcal pairs mathop mathrel limits mathop mathrel limits gets right differential maps which deformations differential maps mathcal hudson pulmannov nbsp math phys corresponding multiplicativity coassociativity delta mathop mathrel limits mathop mathrel limits gets satisfy leibniz formula mutual commutativity condition delta nbsp recovered mathop mathrel limits mathop mathrel limits gets generalised taylor expansion illustrative example consider differential maps corresponding quantisation quasitriangular commutator lie bialgebras hudson pulmannov lett math phys