nLab Lie theory

Classical Lie theory and beyond

Lie theory (wikipedia: Lie theory) is the study of Lie groups, Lie algebras, their actions as groups of transformations, their representations, their cousins (Chevalley groups, quantum groups, Banach Lie groups, Lie bialgebras, universal enveloping algebras) and categorifications (higher Lie groups, $L_\infty$-algebras etc.), including horizontal: Lie groupoid, Lie algebroid, $L_\infty$-algebroid.

Lie group is a group internal to Diff. The Lie algebras over real or complex numbers appear as the linearization of real or complex Lie groups; the infinitesimal version of Lie groups are local Lie groups. The relation between real Lie algebras and real Lie groups was established by Lie's three theorems and the study of this correspondence is the classical Lie theory in narrow sense.

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 $\infty$-Lie groups and $\infty$-Lie groupoids. This is described below.

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

Lie groupoids versus 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.

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

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

Related entries

Smooth spaces and smooth groups: homogeneous space, Lie group, Lie groupoid, Morita equivalence of Lie groupoids, NQ-supermanifold, orbifold, differentiable stack, generalized smooth algebra, generalized smooth space, Lie infinity-groupoid, local Lie group, Poisson-Lie group, symmetric space

Algebraic cousins and tools: algebraic group Chevalley-Eilenberg algebra, CoDGCA, coalgebra, L-infinity-algebra, Lie algebra, Lie algebroid, Lie infinity-algebroid, universal enveloping algebra, PBW theorem, Ado's theorem, Hausdorff series, Duflo isomorphism, coexponential map, free graded co-commutative coalgebra, quantum group, Chevalley group, Lie bialgebra, Leibniz algebra

Geometry on Lie groups and related spaces: integration over supermanifolds, space and quantity, superdifferential form, volume of a Lie groupoid, Maurer-Cartan form, invariant vector field

Lie theory in narrow sense (Lie group(oid)s vs. algebr(oid)s): tangent Lie algebra, tangent Lie algebroid, Lie integration, Lie theory for stacky Lie groupoids, Lie's three theorems, exponential map

Applications where Lie objects appear: matrix theory, connection on a bundle, BRST complex, BV theory, examples for Lagrangian BV, fundamental groupoid, geometric quantization, invariant theory, representation theory, conformal field theory

There are also many special kinds of Lie groups (semisimple, solvable, nilpotent, general linear, special linear, metaplectic group, unitary group etc.), kinds of Lie algebras (Kac-Moody, semisimple, affine Lie algebras, Virasoro Lie algebra…), structure notions on Lie groups and algebras (maximal torus, Killing form, Dynkin diagram, root system, Cartan subalgebra, Gauss decomposition, Bruhat decomposition…), and versions and tools of representation theory (Young diagram, symmetric function, Schur function, combinatorial representation theory, geometric representation theory, weight lattice, induced representation, Verma module, Langlands program), see those entries.

References

General

• A. L. Onishchik (ed.) Lie Groups and Lie Algebras

  • I. A. L. Onishchik, E. B. Vinberg, Foundations of Lie Theory,

  • II. V. V. Gorbatsevich, A. L. Onishchik, Lie Transformation Groups

  Encyclopaedia of Mathematical Sciences, Volume 20, Springer 1993

• Joachim Hilgert, Karl-Hermann Neeb, Structure and Geometry of Lie Groups, Springer Monographs in Mathematics, Springer-Verlag New York, 2012 (doi:10.1007/978-0-387-84794-8)

• Andrei Okounkov, New worlds for Lie theory, talk at ICM 2018 (pdf)

Lie 1-groupoids vs. Lie algebroids

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 categorical

See at Lie integration.

category: Lie theory