Contents

Classical Lie theory and beyond

Classical Lie theory is the theory of groups internal to Diff – Lie groups – and their relation to their linear approximation – Lie algebras. The relation between Lie algebras and Lie groups was established by Lie's three theorems.

In a categorical context there are the usual two generalization of Lie groups: the horizontal categorification – or oidification – which adds more objects and leads to Lie groupoids; and the vertical categorification which adds higher morphisms and leads to $\mathrm{\infty }$-Lie groups and $\mathrm{\infty }$-Lie groupoids. This is described below.

In the course of these categorifications one usually finds that working internal to $\mathrm{Diff}$ is too restrictive for many purposes. Therefore higher Lie theory is often considered internal to generalized smooth spaces.

Oidified Lie theory: Lie groupoids and Lie algebroids

Oidification leads from Lie groups to Lie groupoids and from Lie algebras to Lie algebroids.

When it was found that Lie algebroids integrate to Lie groupoids by mapping paths into them, not only a “well known” but apparently also well forgotten way to integrate Lie algebras by means of paths was rediscovered, but also the idea to integrate higher Lie algebroids by mapping higher dimenmsional paths into them was clearly suggested.

Literature

The study of Lie theory of Lie groupoids and Lie algebroids was notably motivated by Alan Weinstein’s studies of geometric quantization of Poisson manifolds, as well as by Kirill Mackenzie’s work.

The canonical review for the integration theory of Lie algebroids (and hence for Lie algebras by the path integration method) is:

• Marius Crainic, Rui Loja Fernandes, Lectures on Integrability of Lie Brackets, (arXiv,blog).

This focuses on integration to Lie groupoids proper, i.e. to integration internal to manifolds. In contrast to Lie algebras, not every Lie algebroid integrates to such a proper Lie groupoid, though: the space of morphisms of the Lie groupoid is a quotient of paths in the Lie algebroid by Lie algebroid homotopies, and this quotient may not exist as a manifold. (Crainic and Fernandes discuss the obstruction in detail.) But the quotient of course always exists as a weak quotient or homotopy quotient or stacky quotient? itself. The result of realizing the space of morphisms of the integrated Lie algebroid as such a stacky quotient has been studied by Chenchang Zhu: Lie theory for stacky Lie groupoids

Higher Lie theory: $\mathrm{\infty }$-Lie groupoids and $L_{\mathrm{\infty }}$-algebroids

bla bla bla infinity-groupoid bla bla bla Lie infinity-groupoid bla bla bla Lie infinity-algebroid bla bla bla

Literature

• Sullivan,…

• Ševera,…

• Getzler,…

• Henriques,…

• Ševera,…

Entries related to Lie theory

• BRST complex
• BV theory
• CartSp
• Chevalley-Eilenberg algebra
• CoDGCA
• L-infinity-algebra
• Lie algebra
• Lie algebroid
• Lie groupoid
• Lie infinity-algebroid
• Lie integration
• Lie theory for stacky Lie groupoids
• Lie's three theorems
• Morita equivalence
• NQ-supermanifold
• coalgebra
• differentiable stack
• examples for Lagrangian BV
• free graded co-commutative coalgebra
• fundamental groupoid
• generalized smooth algebra
• generalized smooth space
• Lie infinity-groupoid
• integration over supermanifolds
• space and quantity
• superdifferential form
• volume of a Lie groupoid