Document Zbl 0961.58009

Integration of Lie bialgebroids. (English) Zbl 0961.58009

Topology 39, No. 3, 445-467 (2000).

In this paper the authors show that, if \(G\) is an \(\alpha\)-simply connected , and if \((AG, A^*G)\) form a (where \(AG\) is the Lie algebroid of \(G\)), then there exists a unique Poisson structure on \(G\) such that \(G\) becomes a Poisson groupoid with bialgebroid \((AG, A^*G)\).
The paper initially develops some properties of affine on Lie groupoids. The results here are that the of affine multivector fields is affine, and that an affine multivector field on an \(\alpha\)-connected groupoid vanishes everywhere if it vanishes on the units and its derivative is zero. (Here, the derivative of an affine \(k\)-vector field \(D\) is the map \(dD\) from sections of the algebroid \(AG\) to sections of \(\bigwedge^k AG\) where, if the vector field \(\overline{X}\) is the right translation of the section \(X\), then \(dD(X)\) is the section whose right translation is the vector field \(L_{\overline{X}} D\).) These results are applied in the main theorem to show that the proposed Poisson bivector \(\pi\) satisfies \([\pi, \pi] = 0\) (although it is not immediately clear that the \(\alpha\)-connectedness requirement on \(G\) is always satisfied).
Results from an earlier paper by the same authors [K. C. H. Mackenzie and P. Xu, Duke Math. J. 73, No. 2, 415-452 (1994; Zbl 0844.22005) ] are recalled, and a theorem on the integration of Lie algebroid morphisms to Lie groupoid morphisms is proved in an appendix. The proposed Poisson bivector comes from integrating the Lie algebroid morphism \(j_G \circ \pi^{\sharp}_{AB} \circ j'_G : AT^*G \rightarrow ATG\), where \(\pi_{AG}\) is the Poisson tensor on \(T^*AG\), \(j_G : TAG \rightarrow ATG\) is the restriction of the canonical involution on \(TTG\), and \(j'_G\) is (essentially) its dual.
The final section of the paper uses the main result to show that, if \(P\) is a where the Lie algebroid \(T^*P\) integrates to an \(\alpha\)-simply connected groupoid \(G\), then \(G\) becomes a symplectic groupoid in a natural way.