Irreducible holonomy representations. (English) Zbl 1037.53034

Schwachhöfer, Lorenz J.

Slovák, Jan (ed.) et al., The proceedings of the 20th winter school “Geometry and physics”, Srní, Czech Republic, January 15–22, 2000. Palermo: Circolo Matematico di Palermo. Suppl. Rend. Circ. Mat. Palermo, II. Ser. 66, 59-81 (2001).

This is a nice survey of the problem of classification of the irreducible holonomy groups of torsion free affine connections. This classification was finally obtained by S. Merkulov and the author [Ann. Math. (2) 150, 77–149 (1999; Zbl 0992.53038); Oxford University Press. 395–402 (1998; Zbl 0907.53018)]. In this paper the author in particular stresses his attention on the Merkulov twistor space associated to each holomorphic torsion free connection on a complex manifold and construction of special symplectic holonomies introduced by Chi, Merkulov, and the author.
For the entire collection see [Zbl 0961.00020].

Reviewer: Iskander A. Taimanov (Novosibirsk)

MSC:

53C29	Issues of holonomy in differential geometry
53C05	Connections, general theory
53C28	Twistor methods in differential geometry
53C17	Sub-Riemannian geometry
17B10	Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)

Keywords:

holonomy groups; representation theory; twistors; Poisson manifolds