Deformation coproducts and differential maps

R. L. Hudson¹ ; S. Pulmannová²

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

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