420 research outputs found
Clifford algebras and the Duflo isomorphism
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
Clifford algebras. In this context, Duflo's theorem generalizes to a statement
in equivariant cohomology
The basic gerbe over a compact simple Lie group
Let be a compact, simply connected simple Lie group. We give a
construction of an equivariant gerbe with connection on , with equivariant
3-curvature representing a generator of . Technical tools
developed in this context include a gluing construction for gerbes and a theory
of equivariant bundle gerbes.Comment: 19 pages. To appear in L'Enseignement Mathematiqu
Witten's formulas for intersection pairings on moduli spaces of flat -bundles
In a 1992 paper, Witten gave a formula for the intersection pairings of the
moduli space of flat -bundles over an oriented surface, possibly with
markings. In this paper, we give a general proof of Witten's formula, for
arbitrary compact, simple groups, and any markings for which the moduli space
has at most orbifold singularities.Comment: 46 page
Dirac actions and Lu's Lie algebroid
Poisson actions of Poisson Lie groups have an interesting and rich geometric
structure. We will generalize some of this structure to Dirac actions of Dirac
Lie groups. Among other things, we extend a result of Jiang-Hua-Lu, which
states that the cotangent Lie algebroid and the action algebroid for a Poisson
action form a matched pair. We also give a full classification of Dirac actions
for which the base manifold is a homogeneous space , obtaining a
generalization of Drinfeld's classification for the Poisson Lie group case.Comment: 41 pages. To appear in "Transformation Groups
Lectures on pure spinors and moment maps
The goal of this article is to give an elementary introduction to Dirac
geometry and group-valued moment maps, via pure spinors. The material is based
on my lectures at the summer school on 'Poisson geometry in Mathematics and
Physics' at Keio University, June 2006
On Riemann-Roch Formulas for Multiplicities
A Theorem due to Guillemin and Sternberg about geometric quantization of
Hamiltonian actions of compact Lie groups on compact Kaehler manifolds says
that the dimension of the -invariant subspace is equal to the Riemann-Roch
number of the symplectically reduced space. Combined with the shifting-trick,
this gives explicit formulas for the multiplicities of the various irreducible
components. One of the assumptions of the Theorem is that the reduction is
regular, so that the reduced space is a smooth symplectic manifold. In this
paper, we prove a generalization of this result to the case where the reduced
space may have orbifold singularities. Our proof uses localization techniques
from equivariant cohomology, and relies in particular on recent work of
Jeffrey-Kirwan and Guillemin. Since there are no complex geometry arguments
involved, the result also extends to non Kaehlerian settings.Comment: 21 pages, AMS-LaTe
Maslov indices for periodic orbits
It is shown that there is a generalization of the Conley-Zehnder index for
periodic trajectories of a classical Hamiltonian system from
the case to arbitrary symplectic manifolds. As it turns out, it is
precisely this index which appears as a Maslov phase in the trace formulas by
Gutzwiller and Duistermaat-Guillemin. Contribution presented at the XIX ICGTMP
Salamanca June 92.Comment: 5 pages, THEP 92/2
Tilings defined by affine Weyl groups
Let W be a Weyl group, presented as a crystallographic reflection group on a
Euclidean vector space V, and C an open Weyl chamber. In a recent paper,
Waldspurger proved that the images (id-w)(C), for Weyl group elements w, are
all disjoint, and their union is the closed cone spanned by the positive roots.
We show that similarly, if A is the Weyl alcove, the images (id-w)(A), for
affine Weyl group elements w, are all disjoint, and their union is V.Comment: 9 pages, 3 figure
On the quantization of conjugacy classes
Let G be a compact, simple, simply connected Lie group. A theorem of
Freed-Hopkins-Teleman identifies the level k fusion ring R_k(G) of G with the
twisted equivariant K-homology at level k+h, where h is the dual Coxeter
number. In this paper, we review this result using the language of
Dixmier-Douady bundles. We show that the additive generators of the group
R_k(G) are obtained as K-homology push-forwards of the fundamental classes of
conjugacy classes in G.Comment: 32 page
Twisted K-homology and group-valued moment maps
Let G be a compact, simply connected Lie group. We develop a `quantization
functor' from pre-quantized quasi-Hamiltonian G-spaces at level k to the fusion
ring (Verlinde algebra) R_k(G). The quantization Q(M) is defined as a
push-forward in twisted equivariant K-homology. It may be computed by a fixed
point formula, similar to the equivariant index theorem for Spin_c-Dirac
operators. Using the formula, we calculate Q(M) in several examples.Comment: 38 page
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