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Quantum Groups & Braid Groups
and other topics in Lie Theory
Titles and abstracts.
All talks will be held in Carslaw 175
Main speakers
| François Digne |
Amiens |
Wednesday 11:30-12:30
Thursday 10:30-11:30 |
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Presentations of braid groups
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Recently new presentations for braid groups and their
generalizations associated to arbitrary finite Coxeter groups
(Artin-Tits groups of spherical type) were discovered.
These new "dual" presentations, as well as the
classical ones, lead to interesting submonoids of the
Artin-Tits groups which are lattices for the order given by
(left or right) divisibility (the "Garside property"). The dual
presentations are also useful for computing centralizers.
In the case of the general Artin groups associated to arbitrary
Coxeter groups the classical monoids do not have the Garside
property. It is hoped that the dual monoid behaves better
with respect to divisibility. Currently this is known only for
affine type Ãn.
In these two lectures we will describe the general setup
for such presentations and explain some of the results and
their consequences, reviewing work by Dehornoy, Bessis,
Michel, Paris, Picantin and the speaker.
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| Anthony Joseph |
Weizmann Institute |
Wednesday 10:00-11:00
Thursday 9:00-10:00
Friday 9:00-10:00 |
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Primitive ideals
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We start by reviewing the developments in primitive
ideal theory. This includes, notably, Duflo's theorem,
the theory of (left) cells and Goldie rank
polynomials. Then we describe some recent work with
Walter Borko describing the decomposition of the
primitive spectrum into sheets. Finally, a number of
open problems will be discussed.
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Primitive ideals and the Springer correspondence.
We describe some remarkable and unexpected
relationships between primitive ideal theory and the
geometry of the flag variety. An earlier result
relating Goldie rank polynomials of primitive
quotients to characteristic polynomials of orbital
varieties is reviewed. We then describe recent results
obtained with Vladimir Hinich relating left cells of
geometric cells. A notable point is that in both cases
the objects in questions are computed in the same
fashion but with different starting data. This
rationalizes in an elegant fashion the slight
difference between the representation theory and the
geometry
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Crystals and Demazure flags.
We describe a Demazure module and it origins in
geometry. We briefly outline the theory of crystals
and note that this leads to the most comprehensive
proof of the Demazure character formula. Finally we
describe recent work concerning the existence of
Demazure flags (excellent filtrations) for appropriate
tensor products in all characteristics and for all Kac
Moody algebras in the symmetric simply laced case.
This relies heavily on Kashiwara's globalization
technology as well as on a positivity result of
Lusztig.
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Invited speakers
| Peter Bouwknegt |
University of Adelaide |
Friday 3:30-4:30 |
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T-duality
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T-duality, in its simplest form, is the R to 1/R symmetry
of String Theory compactified on a circle of radius R. It
can be generalised to manifolds which admit circle actions
(e.g. circle bundles) or, more generally, torus actions.
In the case of nontrivial circle bundles, T-duality
relates manifolds of different topology and in particular
provides isomorphisms between the twisted cohomologies and
twisted K-theories of these manifolds. In this talk we
will discuss these developments as well as provide some
examples for torus bundles over flag manifolds.
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| Michael Cowling |
UNSW |
Friday 10:30-11:30 |
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The fundamental theorem of projective geometry is also a
theorem in analysis
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The fundamental theorem of projective geometry states that
maps on a spherical Tits building come from actions of the
associated semisimple algebraic group. The aim of this
talk is to convince the audience that in the real case,
there is a local version of the theorem, which is based on
analytic methods.
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| Alexei Davydov |
University of Macquarie |
Friday 2:00-3:00 |
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Braids and symplectic groups
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It is well known that braid groups act naturally on
(powers of) objects of a braided monoidal category. We
describe a braided monoidal category giving rise to braid
group representations by symplectic matrices studied in
(B. Wajnryb, A braidlike presentation of Sp(n,p)
Israel J. Math. 76 (1991), no. 3, 265--288). In contrast
to the "standard" examples of braided monoidal categories,
such as categories of representations of quantum groups,
the monoidal structure in our example is given by a sum of
vector spaces rather than a tensor product. The braiding is
given by a simple formula which allows easy
generalizations leading to new finite quotients of braid
groups.
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| Tony Dooley |
UNSW |
Wednesday 2:00-3:00 | |
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Analysis on harmonic extensions of H-type groups
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The study of analysis on the upper half plane and its
relationship with analysis on the line may be generalised
to the study of the nilpotent component N of a semisimple
Lie group as the boundary of the symmetric space G/K. In
a similar way, a group of Heisenberg type serves as the
boundary of a generalised Siegel domain.
Damek and Ricci showed that these spaces are harmonic,
although they are not necessarily symmetric spaces, thus
resolving an old conjecture of Lichnerowicz. Zhang and I
recently gave a characterisation of the positive definite
functions on these spaces.
I will give an introduction to the main results and
discuss some open problems.
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| James East |
University of Sydney |
Friday 1:00-2:00 |
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The factorizable braid monoid
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In this talk we discuss a new braid monoid which we call the
factorizable braid monoid, and denote by
FBn. This monoid is a
preimage of Fn (the monoid of uniform block bijections of
an n-set) in the same natural way that the braid group is a preimage
of the symmetric group. We will define
FBn geometrically, and
also describe how it may be constructed from the braid group and
Eqn (the join semilattice of equivalence relations on an
n-set). This allows us to find a presentation of
FBn, and discover
connections with the singular braid monoid. We conclude by showing how
to define FBW for an
arbitrary Coxeter group W. This involves a new interpretation of
Eqn in terms of the Coxeter complex of the symmetric
group.
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| Vyachesla Futorny |
University of Sydney |
Thursday 1:00-2:00 |
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Representations of affine Lie superalgebras
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We discuss the classification problem for the irreducible
modules which have non-zero level and finite-dimensional weight
spaces of the affine Lie superalgebras and also the
classification of all irreducible weakly integrable
modules (in the sense of Kac and Wakimoto). This talk is
based on joint work with Senapati Eswara Rao.
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| Anthony Henderson |
University of Sydney |
Wednesday 4:30-5:30 |
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Representations of wreath products on cohomology of
De Concini-Procesi compactifications
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Any finite complex reflection group W acts on the
projective hyperplane complement of its reflecting
hyperplanes, and thus the cohomology groups of this
hyperplane complement are representations of W;
their characters have been computed in many cases by
Lehrer and others. It is an interesting problem to do the
same for the "wonderful" compactification of the
hyperplane complement defined by De Concini and Procesi.
If W=Sn, this compactification is the
moduli space of stable genus 0 curves with
n+1 marked points, and the problem was solved by
Ginzburg and Kapranov. I will explain the solution when
W is the wreath product of Sn
with the group of rth roots of unity.
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| Ngau Lam |
National Cheng Kung University |
Thursday 3:30-4:30 |
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Unitarizable representations of Lie superalgebras and
their formal characters
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The oscillator representations of the classical Lie
superalgebras gl(m|n) ops(m|2n) and their direct
limits are unitarizable. We decompose the tensor powers of
the oscillator representations into unitarizable
irreducible highest weight representations. The
multiplicities of the irreducible representations are
given explicitly, and their formal characters are also
obtained. Our study makes essential use of generalized
Howe dualities between Lie superalgebras and classical Lie
groups.
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| Yucai Su |
Shanghai Jiaotong University |
Thursday 2:00-3:00 |
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Quasifinite representations of Lie algebras related of the
Virasoro algebra
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Let Γ be a free Abelian group, let L = ∑α
in Γ Lα be a
Γ-graded Lie algebra over the field of complex numbers
(each homogeneous space may be infinite-dimensional) such that
L0 is commutative. An L-module V is quasifinite if
V = ∑α in ΓVα is
Γ-graded such that Vα is finite
dimensional and Lα Vβ is
contained in Vα+β , for all
α, β in Γ.
The purpose of this talk is to give
a classification of irreducible quasifinite modules, and to
determine the unitary ones, over the higher rank Virasoro
algebras and the Lie algebras of Weyl type and Block type,
which are related to the Virasoro algebra. In particular, it
is obtained that an irreducible quasifinite module over an
above-mentioned Lie algebra is a highest/lowest weight module
or a module of the intermediate series except it is a higher
rank Virasoro algebra.
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| George Willis |
University of Newcastle |
Wednesday 3:00-4:00 |
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The direction of an automorphism of a totally disconnected group
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Many totally disconnected locally compact groups have an
associated `structure at infinity'. In the case of the
automorphism group of a homogeneous tree it is the space
of ends of the tree and in the case of a simple p-adic
Lie group it is the Bruhat-Tits building at infinity. The
talk will describe a extension of these cases to general
totally disconnected groups based on the idea of the
direction of an automorphism.
This is joint work with Udo Baumgartner.
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