On Lie algebra crossed modules

This article constructs a crossed module corresponding to the generator of the third cohomology group with trivial coefficients of a complex simple Lie algebra. This generator reads as , constructed from the Lie bracket [,] and the Killing form . The construction is inspired by the corresponding construction for the Lie algebra of formal vector fields in one formal variable on R, and its subalgebra sl_2(R), where the generator is usually called Godbillon-Vey class.Comment: 24 page

Deformations of Lie algebras of vector fields arising from families of schemes

Fialowski and Schlichenmaier constructed examples of global deformations of Lie algebras of vector fields from deforming the underlying variety. We formulate their approach in a conceptual way. Namely, we construct a stack of deformations and a morphism form the moduli stack of stable marked curves. The morphism associates to a family of marked curves the family of Lie algebras obtained by taking the Lie algebra of vertical vector fields on the family where one has extracted the marked points. We show that this morphism is almost a monomorphism by Pursell-Shanks theory.Comment: 20 page

The Federal Cartel Office Perspective

This article focuses on the German television (“TV”) market from an antitrust perspective, limited to some competition aspects of the technical and program side of the German TV market. On the technical side, we are in a situation of an emerging market for digital TV where a TV household needs a decoder in order to transfer digital TV signals into analog TV signals, because most households still have analog TV sets and also to descramble encrypted pay-TV signals for subscribers. The other issue, the program side, is more what competition authorities are dealing with, in particular the Bundeskartellamt in its most recent prohibition decision

On Hopf 2-algebras

Our main goal in this paper is to translate the diagram relating groups, Lie algebras and Hopf algebras to the corresponding 2-objects, i.e. to categorify it. This is done interpreting 2-objects as crossed modules and showing the compatibility of the standard functors linking groups, Lie algebras and Hopf algebras with the concept of a crossed module. One outcome is the construction of an enveloping algebra of the string Lie algebra of Baez-Crans, another is the clarification of the passage from crossed modules of Hopf algebras to Hopf 2-algebras.Comment: 26 pages, clarification of several statement