143 research outputs found
Clifford algebras and the classical dynamical Yang-Baxter equation
We describe a relationship of the classical dynamical Yang-Baxter equation
with the following elementary problem for Clifford algebras: Given a vector
space with quadratic form , how is the exponential of an element in
under exterior algebra multiplication related to its exponential
under Clifford multiplication
On the Kashiwara-Vergne conjecture
Let be a connected Lie group, with Lie algebra . In 1977, Duflo
constructed a homomorphism of -modules , which restricts
to an algebra isomorphism on invariants. Kashiwara and Vergne (1978) proposed a
conjecture on the Campbell-Hausdorff series, which (among other things) extends
the Duflo theorem to germs of bi-invariant distributions on the Lie group .
The main results of the present paper are as follows. (1) Using a recent
result of Torossian (2002), we establish the Kashiwara-Vergne conjecture for
any Lie group . (2) We give a reformulation of the Kashiwara-Vergne property
in terms of Lie algebra cohomology. As a direct corollary, one obtains the
algebra isomorphism , as well as a more general
statement for distributions.Comment: 18 pages, final version, to be published in Inventiones Mat
Dirac structures and Dixmier-Douady bundles
A Dirac structure on a vector bundle V is a maximal isotropic subbundle E of
the direct sum of V with its dual. We show how to associate to any Dirac
structure a Dixmier-Douady bundle A, that is, a Z/2Z-graded bundle of
C*-algebras with typical fiber the compact operators on a Hilbert space. The
construction has good functorial properties, relative to Morita morphisms of
Dixmier-Douady bundles. As applications, we show that the `spin' Dixmier-Douady
bundle over a compact, connected Lie group (as constructed by Atiyah-Segal) is
multiplicative, and we obtain a canonical `twisted Spin-c-structure' on spaces
with group valued moment maps.Comment: 41 page
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