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Friedrich Wagemann

Publications and source records attributed to Friedrich Wagemann.

45 records · Page 3Linked to original sources
arXiv2007

Obstruction classes of crossed modules of Lie algebroids and Lie groupoids linked to existence of principal bundles

Let K be a Lie group and P be a K-principal bundle on a manifold M. Suppose given furthermore a central extension 1\to Z\to \hat{K}\to K\to 1 of K. It is a classical question whether there exists a \hat{K}-principal bundle \hat{P} on M such that \hat{P}/Z is isomorphic to P. Neeb defines in this context a crossed module of topological Lie algebras whose cohomology class [ω_{\rm top alg}] is an obstruction to the existence of \hat{P}. In the present paper, we show that [ω_{\rm top alg}] is up to torsion a full obstruction for this problem, and we clarify its relation to crossed modules of Lie algebroids and Lie groupoids, and finally to gerbes.

math.AT↗
arXiv2007

Lie group structures on groups of smooth and holomorphic maps on non-compact manifolds

We study Lie group structures on groups of the form C^\infty(M,K)}, where M is a non-compact smooth manifold and K is a, possibly infinite-dimensional, Lie group. First we prove that there is at most one Lie group structure with Lie algebra C^\infty(M,k) for which the evaluation map is smooth. We then prove the existence of such a structure if the universal cover of K is diffeomorphic to a locally convex space and if the image of the left logarithmic derivative in Ω^1(M,k) is a smooth submanifold, the latter being the case in particular if M is one-dimensional. We also obtain analogs of these results for the group O(M,K) of holomorphic maps on a complex manifold with values in a complex Lie group. We show that there exists a natural Lie group structure on O(M,K) if K is Banach and M is a non-compact complex curve with finitely generated fundamental group.

math.DG↗
arXiv2006

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.

math.KT↗
arXiv2000

A Two Dimensional Analogue of the Virasoro Algebra

The author regrets that after discussion of the results with B. L. Feigin a serious error has shown up. The proof of the results we presented in the paper is wrong. We thank B. L. Feigin for the discussion.

math-ph↗
arXiv2000

Differential Graded Cohomology and Lie algebras of Holomorphic Vector Fields

The Dolbeault resolution of the sheaf of holomorphic vector fields $Lie$ on a complex manifold $M$ relates $Lie$ to a sheaf of differential graded Lie algebras, known as the Frölicher-Nijenhuis algebra $g$. We establish - following B. L. Feigin - an isomorphism between the differential graded cohomology of the space of global sections of $g$ and the hypercohomology of the sheaf of continuous cochain complexes of $Lie$. We calculate this cohomology up to the singular cohomology of some mapping space. We use and generalize results of N. Kawazumi on complex Gelfand-Fuks cohomology. Applications are - again following B. L. Feigin - in conformal field theory, and in the theory of deformations of complex structures. In an erratum to this paper, we admit that the sheaf of continuous cochains of a sheaf of vector fields with values in the ground fields does not make much sense. The most important cochains (like evaluations in a point or integrations over the manifold) do not come from sheaf homomorphisms. The main result of the above article (theorem 7) remains true.

math-ph↗
arXiv2000

Erratum: Some Remarks on the Cohomology of Krichever-Novikov Algebras

We discovered that only a weakened version of the main lemma is true. We state the right version, and the remaining open problem: Is it possible to approximate holomorphic vector fields (or more generally, sections in a line bundle) on an open Riemann surface of finite type (i.e. a compact one without a finite number of points) by meromorphic vector fields (where the poles are supposed to be in the distinguished points) ? We know that this is true for functions.

math-ph↗
arXiv2000

On the Leibniz cohomology of vector fields

I. M. Gelfand and D. B. Fuks have studied the cohomology of the Lie algebra of vector fields on a manifold. In this article, we generalize their main tools to compute the Leibniz cohomology, by extending the two spectral sequences associated to the diagonal and the order filtration. In particular, we determine some new generators for the diagonal Leibniz cohomology of the Lie algebra of vector fields on the circle.

math.KT↗
arXiv2000

A Crossed Module giving the Godbillon-Vey Cocycle

We find crossed modules, i.e. certain 4 term exact sequences, associated to the Godbillon-Vey class for W_1, Vect(S^1), Vect_{1,0}(Σ) and Hol(Σ_r), i.e. for the Lie algebras of formal vector fields in 1 variable, vector fields on the circle, differentiable vector fields of holomorphic type on a Riemann surface and holomorphic vector fields on an open Riemann surface.

math-ph↗