This article is to contribute to the classification of finite-dimensional complex pointed Hopf algebras with Weyl groups of E6, E7, F4, G2. Many papers are about the classification of finite dimensional pointed Hopf algebras, for example, [AS98, AS02, AS00, AS05, He06, AHS08, AG03, AFZ, AZ07, Gr00, Fa07, AF06, AF07, ZZC, ZC]. In these research ones need the centralizers and character tables of ...

There are also four appendices. Let K be a field of characteristic 0, and let C be a commutative K-algebra. which makes C into a Lie algebra, and is a biderivation (i.e. a derivation in each argument). The pair C, {−, −} is called a Poisson algebra. Poisson brackets arise in several ways. Example 1.1. Classical Hamiltonian mechanics. Here K = R, X is an even dimensional differentiable manifold ...

Using the concept of a twisted trace density on a cyclic groupoid, a trace is constructed on a formal deformation quantization of a symplectic orbifold. An algebraic index theorem for orbifolds follows as a consequence of a local Riemann–Roch theorem for such densities. In the case of a reduced orbifold, this proves a conjecture by Fedosov, Schulze, and Tarkhanov. Finally, it is shown how the K...

The aim of this paper is to show that any continuous ∗-homomorphism of L(C)(with twisted convolution as multiplication) into B(L(R)) is essentially a Weyl transform. From this we deduce a similar characterisation for the group Fourier transform on the Heisenberg group, in terms of convolution.

We extend properties of the weak order on finite Coxeter groups to Weyl groupoids admitting a finite root system. In particular, we determine the topological structure of intervals with respect to weak order, and show that the set of morphisms with fixed target object forms an ortho-complemented meet semilattice. We define the Coxeter complex of a Weyl groupoid with finite root system and show ...

we consider the semigroup $s$ of highest weights appearing in tensor powers $v^{otimes k}$ of a finite dimensional representation $v$ of a connected reductive group. we describe the cone generated by $s$ as the cone over the weight polytope of $v$ intersected with the positive weyl chamber. from this we get a description for the asymptotic of the number of highest weights appearing in $v^{otime...

We present a general method of deriving the effective action for conformal anomalies in any even dimension, which satisfies the Wess-Zumino consistency condition by construction. The method relies on defining the coboundary operator of the local Weyl group, gab → exp(2σ)gab, and giving a cohomological interpretation to counterterms in the effective action in dimensional regularization with resp...

All −1-type pointed Hopf algebras with Weyl groups of exceptional type are found. It is proved that every non −1-type pointed Hopf algebra is infinite dimensional. 2000 Mathematics Subject Classification: 16W30, 16G10 keywords: Quiver, Hopf algebra, Weyl group.

We present algorithms to factorize homogeneous polynomials in the first q-Weyl algebra and the first Weyl algebra. By homogeneous we mean homogeneous with respect to a Z-grading on the first (q-)Weyl algebra which will be introduced in this article. It turns out that the factorization can be almost completely reduced to commutative univariate factorization with some additional easy combinatoria...