CHAPTER 1
Introduction
The general idea of deforming a given theory by use of its symmetries goes back
to Drinfel’d. One paradigm being that the data of a Drinfel’d twist based on a bi-
algebra acting on an associative algebra A, produces an associative deformation
of A. In the context of Lie theory, one considers for instance the category of
module-algebras over the universal enveloping algebra U(g) of the Lie algebra g of
a given Lie group G. In that situation, the notion of Drinfel’d twist is in a one
to one correspondence with the one of left-invariant formal star-product
ν
on the
space of formal power series
C∞(G)[[ν]],
see [14]. Disposing of such a twist, every
U(g)-module-algebra A may then be formally deformed into an associative algebra
A[[ν]]. It is important to observe that, within this situation, the symplectic leaf B
through the unit element e of G in the characteristic foliation of the (left-invariant)
Poisson structure directing the star-product ν, always consists of an immersed
Lie subgroup of G. The Lie group B therefore carries a left-invariant symplectic
structure. This stresses the importance of symplectic Lie groups (i.e. connected Lie
groups endowed with invariant symplectic forms) as semi-classical approximations
of Drinfel’d twists attached to Lie algebras.
In the present memoir, we address the question of designing non-formal Drin-
fel’d twists for actions of symplectic Lie groups B that underly negatively curved
ahlerian Lie groups, i.e. Lie groups that admit a left-invariant ahlerian structure
of negative curvature. These groups exactly correspond to the normal j-algebras
defined by Pyatetskii-Shapiro in his work on automorphic forms [22]. In particular,
this class of groups contains all Iwasawa factors AN of Hermitian type simple Lie
groups G = KAN.
Roughly speaking, one looks for a smooth one-parameter family of complex
valued smooth two-point functions on the group, {Kθ}θ∈R C∞(B × B, C), with
the property that, for every strongly continuous and isometric action α of B on a
C∗-algebra
A, the following formula
(1) a
α
θ
b :=
B×B
Kθ(x, y) αx(a) αy(b) dx dy ,
defines a one-parameter deformation of the
C∗-algebra
A.
The above program was realized by Marc Rieffel in the particular case of the
Abelian Lie group B =
R2n
in [26]. More precisely, Rieffel proved that for any
strongly continuous and isometric action of
R2n
on any Fr´ echet algebra A, the
associated Fr´ echet subalgebra
A∞
of smooth vectors for this action, is deformed
by the rule (1), where the two-point kernel there, consists of the Weyl symbol
composition kernel:
Kθ(x, y) :=
θ−2n
exp
i
θ
ω0(x,
y) ,
1
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