Related papers: Lie theory and the Chern-Weil homomorphism

Let G be a connected Lie group with Lie algebra g. The Duflo map is a vector space isomorphism between the symmetric algebra S(g) and the universal enveloping algebra U(g) which, as proved by Duflo, restricts to a ring isomorphism from…

Let $\mathbb{X}=[X_1\rightrightarrows X_0]$ be a Lie groupoid equipped with a connection, given by a smooth distribution $\mathcal{H} \subset T X_1$ transversal to the fibers of the source map. Under the assumption that the distribution…

We propose the notion of a supercategory as an alternative approach to supermathematics. We show that this setting is rich to carry out many of the basic constructions of supermathematics. We also prove generalizations of a number of…

The theory of principal $G$-bundles over a Lie groupoid is an important one, unifying the various types of principal $G$-bundles, including those over manifolds, those over orbifolds, as well as equivariant principal $G$-bundles. In this…

We prove that Chern-Weil forms are the only natural differential forms associated to a connection on a principal G-bundle. We use the homotopy theory of simplicial sheaves on smooth manifolds to formulate the theorem and set up the proof.…

This article summarizes joint work with A. Alekseev (Geneva) on the Duflo isomorphism for quadratic Lie algebras. We describe a certain quantization map for Weil algebras, generalizing both the Duflo map and the quantization map for…

In this short review article we sketch some developments which should ultimately lead to the analogy of the Chern-Weil homomorphism for principal bundles in the realm of non-commutative differential geometry. Principal bundles there should…

We describe the Cartan and Weil models of twisted equivariant cohomology together with the Cartan homomorphism among the two, and we extend the Chern-Weil homomorphism to the twisted equivariant cohomology. We clarify that in order to have…

Chevalley's theorem states that for any simple finite dimensional Lie algebra G (1) the restriction homomorphism of the algebra of polynomials on G onto the Cartan subalgebra H induces an isomorphism between the algebra of G-invariant…

A cohomology theory, associated to a $n$-Lie algebra and a representation space of it, is introduced. It is observed that this cohomology theory is qualified to encode the generalized derivation extensions, and that it coincides, for $n=3$,…

In this paper we extend the Chern-Weil-Lecomte characteristic map to the setting of $L_{\infty}$-algebras. In this general framework, characteristic classes of $L_{\infty}$-algebra extensions are defined by means of the Chern-Weil-Lecomte…

We view the space of cotraces in the structural coalgebra of a principal coaction as a noncommutative counterpart of the classical Cartan model. Then we define the cyclic-homology Chern-Weil homomorphism by extending the Chern-Galois…

We introduce a notion of secondary characteristic classes of Lie algebra extensions. As a spin-off of our construction we obtain a new proof of Lecomte's generalization of the Chern-Weil homomorphism.

Classical Chern-Weil theory proves existence of a ring homomorphism - the Chern character - from the Grothendieck ring of locally free finite rank A-modules to the algebraic de Rham cohomology of A. In this paper we prove existence of a…

Chern-Weil theory provides for each invariant polynomial on a Lie algebra g a map from g-connections to differential cocycles whose volume holonomy is the corresponding Chern-Simons theory action functional. Kotov and Strobl have observed…

For a compact Lie group acting on a smooth manifold, we define the differential cohomology of a certain quotient stack involving principal bundles with connection. This produces differential equivariant cohomology groups that map to the…

In this article, we study the Chern-Weil theory for Hopf-Galois extensions originally introduced by Hajac and Maszczyk in the context of coalgebra extensions. We show that the cyclic homology Chern-Weil homomorphism defines natural…

A free differential algebra is generalization of a Lie algebra in which the mathematical structure is extended by including of new Maurer-Cartan equations for higher-degree differential forms. In this article, we propose a generalization of…

In this note, making use of noncommutative $l$-adic cohomology, we extend the generalized Riemann hypothesis from the realm of algebraic geometry to the broad setting of geometric noncommutative schemes in the sense of Orlov. As a first…

Chern-Weil and Chern-Simons theory extend to certain infinite-rank bundles that appear in mathematical physics. We discuss what is known of the invariant theory of the corresponding infinite-dimensional Lie groups. We use these techniques…