LIE GROUPS, ALGEBRAIC GROUPS, QUANTUM GROUPS
AND THEIR REPRESENTATIONS
TITLES AND ABSTRACTS
Peter Bouwknegt
D-branes and (twisted) K-theory
Informal talk on my recent work with Mathai, for those
who haven't heard it yet.
Michel Broué
Lie groups over fields with x elements ?
Jie Du
Cellular systems and stratified algebras
Graham-Lehrer's idea of cellular bases is generalized to give
a unified treatment for quasihereditary algebras, cellular algebras and
stratified algebras.
Nick Dungey
Heat kernels on some solvable Lie groups
We consider the question of estimating the large-time
behaviour of the heat kernel of a suitable differential operator
H on a solvable Lie group G (where G is assumed to have
polynomial volume growth). Many results are known if H is
a second-order sublaplacian, or if G is nilpotent and H
has any (even) order. We describe some recent results dealing
with non-nilpotent G and arbitrary order, and sketch connections
with homogenization theory and with the representation theory of G.
Mike Eastwood
Some special geometry in dimension six
What happens if you modify the notion of contact manifold
by allowing the contact distribution to have codimension two? There
are special features in dimension six.
Omar Foda
Tableaux and rigged configurations
Young tableaux are central objects in combinatorics, representation
theory, the theory of symmetric functions, and related subjects. In
the mid 80's, Tolya Kirillov discovered a deep and by-now classical
bijection between tableaux and sets of labelled partitions called
'rigged configurations'.
I would like to explain Kirillov's bijection in detail, and, if time
permits, outline some of its recent applications, particularly in
representation theory of affine and Virasoro algebras.
Andrew Francis
Centralizers of parabolic subalgebras in Iwahori-Hecke algebras
The minimal basis theory for centres of an Iwahori-Hecke algebra of a
finite Coxeter group can be extended to find integral (minimal) bases for
centralizers of parabolic subalgebras. The approach for centres depends on
some properties of conjugacy classes in the finite Coxeter group, and the
extension to centralizers requires an extension of these to "twisted"
J-conjugacy classes: the classes generated by the twisted action of a
parabolic subgroup WJ.
There are several applications of the minimal basis I hope to mention, such
as a simple expression for the elementary symmetric functions of Murphy
operators in type A; some symmetry results on the images of class
elements under irreducible charaters (any type) and on coefficients in
norms (type A); and some nice properties elements of the minimal basis
have under projection to a parabolic subalgebra
Mark Gould
Generalised Lie Algebras
A new Definition of Generalised Lie Algebras will be presented
based on the theory of co-associative co-algebras. These structures
satisfy a generalised form of antisymmetry and the Jacobi identity for
the generalised Lie bracket and a suitable definition of universal
enveloping algebra is obtained which is automatically a Hopf algebra.
Rod Gover
Electromagnetism, Elliptic Systems and Tractor Calculus
In the classical theory of electromagnetism and in quantised versions
thereof the ``Lorentz gauge'' has often been used to restrict the
gauge freedom of the field potential. However this gauge equation is
not conformally invariant even though the field equations themselves
are. In 1984 M.G. Eastwood and M. Singer produced an alternative
gauge equation that is conformally invariant on solutions of the field
equations. Their system can be very elegantly described in terms of a
new tool in differential geometry called tractor calculus.
Investigating why this is the case has revealed a rather beautiful
picture involving some elementary Lie algebra representation theory.
Here via purely algebraic tools we can see not only how to predict the
Eastwood-Singer construction but how to manufacture similar gauge
systems for other field equations. Switching to the positive signature
setting we see the gauge operators naturally complete the field operators
to elliptic differential operators.
This is collaborative work with Tom Branson (University of Iowa).
A host algebra generalises
a group algebra in a way which allows one to give meaning to
the concept of a group algebra for general topological groups.
It is defined for a unital C*-algebra F
and a proper subset of its states S0
within which one wants to
keep the analysis (an interesting situation also for quantum
physics, due to selection criteria for physical states).
To be precise, a host algebra is a C*-algebra L
and two embeddings F\subsetE\supsetL
into a larger C*-algebra E such that the states on L
extend uniquely to F, and this extension defines a bijection
between S0 and the whole state space of L.
The main examples are of course group algebras
but these are not the only examples.
We study the general existence question for a host algebra of a given
pair (F,S0) we show that given a host algebra
one can do integral decompositions of states in S0
in terms of other states in S0 and we show that if
one does induction of representations via host algebras, one stays
within the class of representations with the right continuity
properties w.r.t. S0. Moreover, if
S0 is a folium, then up to a central algebra, one
can always construct a host algebra, but this central algebra can be
an obstruction to the existence of a host algebra.
Joost van Hamel
Equivariant Borel-Moore homology
The Localisation Theorem in equivariant cohomology is an important tool
to establish relations between the cohomology of a space with a torus
action and its set of fixed points. For manifolds the localisation map
is not quite compatible with Gysin maps ('integration'). I will define
equivariant Borel-Moore homology, and show that in this theory
localisation is compatible with push-forward. For manifolds this gives a
nice way of looking at the localisation formula for Gysin maps.
Karl Hofmann
Transitive actions of compact groups and topological dimension
I report here on work with Sidney A. Morris.
We define on a new topological topological dimension function
taking also transfinite values.
It provides a useful parameter for homogeneous
spaces of compact groups.
We prove that such a space contains a cube
of the right dimenson and no bigger ones.
The tools include the Lie algebra and the
exponential function for arbitrary
compact groups. The results hold also for
homogeneous spaces of locally compact groups.
In the process
we generalize a Theorem of Iwasawa on the
structure of locally compact groups which
is of independent interest as well.
Peter Jarvis
Local invariants of composite quantum systems
The characterisation of states of
composite quantum systems up to unitary transformations on each subsystem
is considered, with emphasis on group theory. In particular, for the
density matrix of two q-bit systems, a
generating function for local invariants is computed.
The corresponding Cohen-Macauley structure of the
invariant ring is investigated in detail.
[Joint work with RC King (University of Southampton).]
We review self-duality of nonlinear electrodynamics
and its extension to several Abelian gauge fields
(such as dynamical systems with gauge (p-1)-forms in d = 2p
space-time dimensions or systems with (p-1)-forms
and (d-p-1)-forms for any d) coupled to scalar fields.
The self-duality equations, which have to be satisfied
by the Lagrangian of any self-dual system, and their
solutions are discussed. Important examples of self-dual
systems are: (i) the Born-Infeld action (D3-brane action),
(ii) models for partial supersymmetry breaking in four
space-time dimensions.
Gus Lehrer
Cohomological group actions via counting rational points
Max Lohe
q-deformed Heisenberg algebras
We define position and momentum operators, in 3 dimensions, as vector
operators with respect to the quantum group SUq(2), and impose
algebraic relations which generalize the canonical commutation
relations, and so form a q-deformed Heisenberg algebra. From these
operators we construct, by vector coupling, the generators of the
quantum group which we interpret as orbital angular momentum
generators. We discuss the invariance properties of the constructed
algebra and possible physical interpretations.
Ben Martin
Spaces of representations of finite groups
Let F={f1,...,fr}
be a finite group and let G be a reductive
algebraic group. The set of representations (i.e. homomorphisms)
\rho: F -> G may be regarded as a Zariski-closed subvariety
of Gr. Recently I proved that there are only finitely many
Zariski-closed conjugacy classes of representations from F to
G. In the
special case G=GLn(k), this becomes the well-known
result that
there are only finitely many isomorphism classes of n-dimensional
semisimple F-modules.
The proof uses work of Vinberg, who considered only fields of characteristic
zero. I will discuss Vinberg's ideas and explain how to generalise them to
nonzero characteristic.
Andrew Mathas
Equating decomposition numbers for different primes
The main outstanding problem in the modular representation
theory of the symmetric groups is the determination of
their p-modular decomposition matrices.
As an experiment, Gordon James and I started to compute the
decomposition matrices of the symmetric groups in characteristic
5; the first problem that we were unable to resolve was the
multiplicity of the simple module D(12,9) in the Specht module
S(8,8,4,1); all that we could determine was that this multiplicity
was either 1 or 2. (Thanks to a computer calculation of Lübeck
and Müller, we now know the answer is 1.)
In the process of this investigation we noticed the striking
similarity between the following submatrices of the 3-modular
decomposition matrix of Sym(11) and the 5-modular decomposition
matrix of Sym(21).
18,3 | 1 10,1 | 1
17,4 | 1 1 9,2 | 1 1
13,8 | . 1 1 7,4 | . 1 1
13,4^2 | 1 1 1 1 7,2^2 | 1 1 1 1
12,9 | . . 1 . 1 6,5 | . . 1 . 1
12,4^2,1 | 1 1 1 1 1 1 6,2^2,1 | 1 1 1 1 1 1
8^2,5 | . . 1 1 1 . 1 4^2,3 | . . 1 1 1 . 1
8^2,4,1 | . 1 1 1 1 1 1 1 4^2,2,1 | . 1 1 1 2 1 1 1
n=21 and p=5 n=11 and p=3
In this talk we will give some explanation as to why these
decomposition matrices are are almost identical. The answer is
given by a general result about the decomposition matrices of the
Iwahori-Hecke algebras of the symmetric group in characteristic
zero.
David McAnally
Some New Evaluation Formulae for Heckman-Opdam Polynomials
Heckman-Opdam polynomials are eigenfunctions of a set of commuting
Cherednik (or Dunkl-Cherednik) operators, which are based on root
systems of simple Lie algebras. These polynomials form a natural
basis on which to give the action of a degenerate double affine
Hecke algebra, and their symmetrizations generalize the group
characters for the corresponding compact groups. New formulae are
obtained for evaluation at the identity after symmetrization and
antisymmetrization with respect to certain subgroups of the Weyl
group.
Ian McArthur
Kappa symmetry in coset superspaces
It has been realized in recent years that superstring theories
have a much richer structure than previously believed and contain, in
addition to fundamental strings, higher dimensional spatially extended
structures that go by the generic name of "branes." The covariant low
energy effective actions which describe the long wavelength fluctuations of
branes are generalizations of the Green-Schwarz action for superstrings, in
which spacetime supersymmetry is manifest. Just as in the Green-Schwarz
action, these low energy effective actions have a mysterious local
fermionic symmetry (called kappa-symmetry) whose presence ensures equality
of the number of physical bosonic and fermionic degrees of freedom. For
the case in which the background superspace in which the brane is embedded
is a coset superspace and the bosonic degrees of freedom on the brane
worldvolume are scalar fields, I will discuss a geometric interpretation of
the kappa symmetry in terms of the action of a supergroup on the coset
superspace.
Alex Molev
Representations of quantum affine algebras and Yangians
Finite-dimensional irreducible representations of both the quantum
affine algebras and Yangians are completely described by their highest
weights (Drinfeld and Chari-Pressley). However, the structure of the
general representations still remains unknown. Recently, an
irreducibility criterion for tensor products of the fundamental
representations was obtained by different methods (Akasaka-Kashiwara,
Varagnolo-Vasserot, Frenkel-Mukhin). In the case of A type it is
possible to obtain a more general criterion of irreducibility of the
evaluation modules and the so-called skew modules which will be
discussed in the talk. These criteria allow one to explicitly
construct a large class of representations.
Sidney Morris
A Structure Theorem on Compact Groups
I report here on work with Karl Heinrich Hofmann.
We prove a new structure theorem for compact groups which we call
the Countable Layer Theorem. It expresses any compact group G
in terms of a compact connected abelian group and simple
compact groups. One corollary is a topological
decomposition theorem which yields a very explicit
calculation for the topological weight of a compact group in terms of
its connected component G0
and algebraic invariants of the factor groups
G/G0.
Michael Murray
Geometric objects representing cohomology
Yes, it's more gerbes. This time
I want to talk about the structure of the hierarchy
of all gerbes and they way they fit together. This
is joint work with Danny Stevenson and Stuart Johnson.
Vladimir Pestov
Amenability of induced representations: an application of the
concentration technique
We will describe a link (in essence discovered by Gromov and
Milman in 1983) between amenability of groups and unitary
representations, on the one hand, and the phenomenon of concentration of
measure on high-dimensional structures, on the other, and show how
concentration can be applied to the 1990 problem by Bekka about
amenability of induced representations of locally compact groups.
John Rice
Ginzburg's Construction of the Langlands Dual Group
The Langlands dual of a reductive group and automorphic
representations of the group are the two concepts necessary for the
formulation of the Langlands conjectures. Progress on the conjectures has
been hampered by the lack of a natural and transparent definition of the
dual group. In the late seventies such a definition was sought, without
success, in terms of Grothendieck's concept of Tannakian categories. Quite
separately, other ideas emerging from the Grothendieck school, derived
categories and perverse complexes, when applied to the flag manifold of a
reductive group, allowed the solution of the Kazhdan-Lusztig conjectures.
By applying these same ideas in the context of infinite dimensional
grassmannians Ginzburg achieved the long sought after 'Tannakian'
construction of the Langlands dual when the base field is the complex
numbers. Improvements by Mirkovic and Vilonen have extended this to general
fields. The seminar will provide an introduction and overview of the
construction and its underlying concepts.
Finlay Thompson
Permutation actions on tensor powers of the quaternions
The quaternions arise as the generator of the Brauer group of
the reals. Using this basic fact it is possible to define an interesting
finite group action on tensor powers of the algebra of quaternions by
permuting the module structures. These finite group actions have useful
applications to tensor calculus on four manifolds. In particular we give
some applications to study of non-flat local Lorentzian metrics.
Valery Tolstoy
Connection between Yangians and Quantum Affine Algebras
A simple explicit connection between Yangians and quantum affine
algebras is discussed. Starting from a quantum nontwisted affine
algebra Uq(g[u]), where g
is any finite-dimensional simple Lie
algebra, we construct a two-parameter deformation by a singular
transformation (at q=1). This deformation D_{q\eta}(g)
called Drinfeldian is a Hopf algebra, and it can be considered
as quatization of the universal enveloping algebra U(g[u]) in
the direction of a classical r-matrix which is a sum of
the simplest rational and trigonometrical r-matrices.
When the parameter $\eta$ is equal to zero the quantum algebra
D_{q\eta=0}(g) coincides with Uq(g[u]),
if the parameter q
goes to 1 the algebra D_{q=\!1\eta}(g) is isomorphic to the
Yangian Y_{\eta}(g). The Drinfeldian and the Yangian are defined
in terms of the Chevalley bases. These results can be easy generalized
to a supercase, i.e. when g is any classical simple Lie superalgebra.
Ilknur Tulunay
Cuspidal modules of finite general linear groups
In my talk, I will give the construction of irreducible cuspidal modules for a
Finite General Linear Group GLn(q) for all possible n and q.
Ole Warnaar
The Bailey lemma and admissible representations
of A1(1)
Bailey's lemma is a well-known tool in the theory
of q-series for proving Rogers-Ramanujan-type
identities. After giving an elementary introduction to the
Bailey lemma, I will show how the string functions
associated to admissible representations of A1(1) give
rise to a far-reaching extension of the lemma.
Brauer algebras were introduced by Brauer
in 1937 in order to extend classical Schur--Weyl duality to semisimple
algebraic groups of types B and C. More precisely, Schur--Weyl
duality means the following double centraliser property: Fix a field
k and two natural numbers n and r.
Then the general linear group
GLn(k) acts (diagonally) from the left on the vector space
(kn)x r whereas the symmetric group
Σr acts from
the right by permuting places of tensors. These two actions centralize
each other. Replacing in this setup the algebraic group GLn(k)
by an orthogonal group or a symplectic group, one has to replace the
symmetric group by the Brauer algebra if one wants to keep the double
centraliser property.
In this talk we consider the Brauer algebras as cellular algebras and
provide conditions for Brauer algebras to be quasi-Frobenius. This is
a characteristic-free approach to Brauer algebras.
Nanhua Xi
A partition of the Springer fibers BN for type
An-1, B2, G2, and
some applications
For type An-1, B2, G2
we will give a partition of
the Springer fibers BN and use the partition to compute
some equivariant K-groups.
Ruibin Zhang
Representations of Lie conformal algebras
Lie conformal algebras were introduced by Kac to provide an
axiomatic description of the quantum field theoretical notion
of operator product expansions. They are also closely related
to infinite dimensional Lie (super)algebras of Cartan type.
In this talk, we shall first briefly discuss structural features
of the simple Lie conformal algebras of finite type, then present
results on the classification and construction of the finite
irreducible representations of the W- and S-series of Lie
conformal algebras.