Double Lie Algebroids and Second-Order Geometry, II

✍ Scribed by K.C.H. Mackenzie

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 277 KB
• Volume: 154
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1999.1892

✦ Synopsis

In a previous paper [11] we proposed a Lie theory for double Lie groupoids based on the known Lie theory of ordinary Lie groupoids, on various constructions in Poisson geometry, and on the special features of the differential geometry of the tangent bundle. That paper [11] gave the first of the two steps involved in constucting the double Lie algebroid of a double Lie groupoid and one purpose of the present paper is to give the more difficult second step.

Ordinary Lie algebroids may be viewed both as generalizations of Lie algebras and therefore as vehicles for a generalized Lie theory [12] and as abstractions of the tangent bundle of an ordinary manifold see [25,19]. The differentiation process given here includes as special cases: (i) the process of passing from a double Lie group [9] or matched pair of Lie groups [20] to the corresponding double Lie algebra or matched pair of Lie algebras, (ii) the (related) process of obtaining a Lie bialgebra from a Poisson Lie group or, more generally, a Lie bialgebroid from a Poisson groupoid [18], and (iii) the process of obtaining a pair of compatible partial connections from an affinoid [24]. However, from our point of view the fundamental process is that of obtaining the double (iterated) tangent bundle T 2 M=T(TM ) from the double groupoid structure on M 4 , where elements of M 4 are regarded as the corners of an empty square. All these processes yield double Lie algebroids and we show that the calculus possible for the double tangent bundle applies to them all.

In [11] it was shown how a single application of the Lie functor to a double Lie groupoid S produces an LA-groupoid, that is, a Lie groupoid object in the category of Lie algebroids. If one applies the Lie functor to,