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Algorithmic computations of Lie algebras cohomologies

Šilhan, Josef (2003)

Proceedings of the 22nd Winter School "Geometry and Physics"

From the text: The aim of this work is to advertise an algorithmic treatment of the computation of the cohomologies of semisimple Lie algebras. The base is Kostant’s result which describes the representation of the proper reductive subalgebra on the cohomologies space. We show how to (algorithmically) compute the highest weights of irreducible components of this representation using the Dynkin diagrams. The software package L i E offers the data structures and corresponding procedures for computing...

An application of principal bundles to coloring of graphs and hypergraphs

Milgram, James R., Zvengrowski, Peter (1994)

Proceedings of the Winter School "Geometry and Physics"

An interesting connection between the chromatic number of a graph G and the connectivity of an associated simplicial complex N ( G ) , its “neighborhood complex”, was found by Lovász in 1978 (cf. L. Lovász [J. Comb. Theory, Ser. A 25, 319-324 (1978; Zbl 0418.05028)]). In 1986 a generalization to the chromatic number of a k -uniform hypergraph H , for k an odd prime, using an associated simplicial complex C ( H ) , was found ([N. Alon, P. Frankl and L. Lovász, Trans. Am. Math. Soc. 298, 359-370 (1986; Zbl 0605.05033)],...

An infinite dimensional version of the third Lie theorem

Rybicki, Tomasz (2002)

Proceedings of the 21st Winter School "Geometry and Physics"

The concept of evolution operator is used to introduce a weak Lie subgroup of a regular Lie group, and to give a new version of the third Lie theorem. This enables the author to formulate and to study the problem of integrability of infinite-dimensional Lie algebras. Several interesting examples are presented.

An introduction to algebraic K-theory

Ausoni, Christian (2001)

Proceedings of the 20th Winter School "Geometry and Physics"

This paper gives an exposition of algebraic K-theory, which studies functors K n : Rings → Abelian Groups , n an integer. Classically n = 0 , 1 introduced by Bass in the mid 60’s (based on ideas of Grothendieck and others) and n = 2 introduced by Milnor [Introduction to algebraic K-theory, Annals of Math. Studies, 72, Princeton University Press, 1971: Zbl 0237.18005]. These functors are defined and applications to topological K-theory (Swan), number theory, topology and geometry (the Wall finiteness obstruction to a CW-complex being finite,...

An introduction to Cartan Geometries

Sharpe, Richard (2002)

Proceedings of the 21st Winter School "Geometry and Physics"

A principal bundle with a Lie group H consists of a manifold P and a free proper smooth H -action P × H → P . There is a unique smooth manifold structure on the quotient space M = P / H such that the canonical map π : P → M is smooth. M is called a base manifold and H → P → M stands for the bundle. The most fundamental examples of principal bundles are the homogeneous spaces H ⊂ G → G / H , where H is a closed subgroup of G . The pair ( 𝔤 , 𝔥 ) is a Klein pair. A model geometry consists of a Klein pair ( 𝔤 , 𝔥 ) and a Lie group H with Lie algebra 𝔥 . In this...

Aspects of parabolic invariant theory

Gover, Rod A. (1999)

Proceedings of the 18th Winter School "Geometry and Physics"

A certain family of homogeneous spaces is investigated. Basic invariant operators for each of these structures are presented and some analogies to Levi-Civita connections of Riemannian geometry are pointed out.

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