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Integration problem for LA-groupoids

Establish necessary and sufficient conditions under which an LA-groupoid integrates to a double Lie groupoid, and develop a general integration procedure; in particular, determine whether every LA-groupoid admits an integration to a double Lie groupoid.

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Background

The paper develops a framework relating double quasi Poisson groupoids and quasi Poisson LA-groupoids via differentiation. While Theorem A.1 (here referenced as Theorem \ref{thm:qpla}) shows how to pass from a double quasi Poisson groupoid to a quasi Poisson LA-groupoid by differentiation, the reverse direction requires integrating an LA-groupoid to a double Lie groupoid.

The authors highlight that this integration problem for LA-groupoids has not been solved in general. They provide partial positive results in special cases (e.g., over unit groupoids with integrable Lie algebroids), underscoring that a complete, general solution to the integration of LA-groupoids remains unknown.

References

The inverse operation involves the integration problem for LA-groupoids which is still an open question, see .

— Homological vector fields over differentiable stacks (2403.14871 - Álvarez et al., 21 Mar 2024) in Appendix, Section “Double quasi Poisson groupoids,” Subsection “Integration of quasi Poisson LA-groupoids over a unit groupoid”