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Poisson-Lie groupoids and the contraction procedure. (English) Zbl 1331.53114
Ciccoli, Nicola (ed.) et al., From Poisson brackets to universal quantum symmetries. Selected papers of the workshop, IMPAN, Warsaw, August 18–22, 2014. Warsaw: Polish Academy of Sciences, Institute of Mathematics (ISBN 978-83-86806-29-4/pbk). Banach Center Publications 106, 35-46 (2015).
Summary: On the level of Lie algebras, the contraction procedure is a method to create a new Lie algebra from a given Lie algebra by rescaling generators and letting the scaling parameter tend to zero. One of the most well-known examples is the contraction from \(\mathfrak{su}(2)\) to \(\mathfrak e(2)\), the Lie algebra of upper-triangular matrices with zero trace and purely imaginary diagonal. In this paper, we will consider an extension of this contraction by taking also into consideration the natural bialgebra structures on these Lie algebras. This will give a bundle of central extensions of the above Lie algebras with a Lie bialgebroid structure having transversal component. We consider as well the dual Lie bialgebroid, which is in a sense easier to understand, and whose integration can be explicitly presented.
For the entire collection see [Zbl 1332.81016].
MSC:
53D17 Poisson manifolds; Poisson groupoids and algebroids
17B62 Lie bialgebras; Lie coalgebras
16S80 Deformations of associative rings
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