Dynamical Weyl groups
by Etingof, Pavel
Consider the space of rational functions on the Cartan subalgebra of a
simple Lie algebra g with values in a finite dimensional representation
of the corresponding quantum group. It turns out that there exists a natural
representation of the braid group of
g on this space. This representation is a deformation of the
quantum Weyl group of Soibelman and Lusztig, which may be called a "dynamical"
Weyl group, since it is intimately related with solutions of the dynamical
Yang-Baxter equation of Gervais, Neveu, and Felder. I will talk about the
definition of this object, and its applications, developed in my joint
work with A. Varchenko.
Is a finite group determined by its tensor category of representations?
by Gelaki, Shlomo
It is well known that if two finite groups have the same symmetric tensor
categories of complex representations then they are isomorphic. In a joint
work with P. Etingof we studied the following question: when do two finite
groups G_1, G_2 have the same tensor categories of complex representations
(without regard for the commutativity constraint)? We called two groups
with such property isocategorical. In my talk I will present our example
of two groups which are isocategorical but not isomorphic: the affine
symplectic group of a vector space over the field of two elements, and an
appropriate "affine pseudosymplectic group" introduced by R. Griess. On
the other hand, I will explain our classification of groups isocategorical
to a given group. This classification implies in particular that if G has
no nontrivial normal abelian subgroups of order 2^{2m} then any group
isocategorical to G must actually be isomorphic to G.
From crossed sets to pointed Hopf algebras
by Graña, Matias
This is joint work with N. Andruskiewitsch.
Crossed sets are particular cases of quandles, and hence of
set-theoretical solutions to the braid equation.
They have a cohomology theory attached, which has been used to
provide invariants of knots. We use the second cohomology group to
produce Yetter-Drinfeld modules over groups which, in turn, give
rise to Nichols algebras and a fortiori pointed Hopf algebras.
In the talk we will define the second non-abelian cohomology
group and show how it can be used to classify extensions.
This naturally leads to the definition of simple crossed sets,
a classification of which is harder than the classification of
simple groups. We will present different methods for constructing
crossed sets, computations of their first abelian cohomology groups
and of Nichols algebras of some of these.
Bitableaux bases and presentations of the quantum coordinate algebra of semisimple groups
by Iglesias, Rodrigo
The quantum coordinate algebra Fq (G) is a deformation of the algebra
F(G) of ``polynomial functions'' on a complex semisimple group G.
More precisely, the dual of the quantum enveloping algebra Uq(g) (the
``simply connected'' version) corresponding to the complex semisimple
Lie algebra g, has a natural algebra structure. The algebra Fq(G)
is defined as the subalgebra of the dual of Uq(g) generated by the
matrix coefficients corresponding to finite dimensional representations
of Uq(g). When q -->1 we recover the classical coordinate algebra
F(G).
In some cases (for example for g = sl(n)) it is well known how to
give a presentation of Fq(G) but the procedure relies on the particular
geometric description of Fq(SL(n)). Our aim is to describe a different
method to obtain a presentation of Fq(G) which works for an arbitrary
semisimple Lie algebra g.
The method is based on a Peter-Weyl-type decomposition of Fq(G) (the
left-right action endow Fq(G) with a structure of (Uq(g) x Uq(g))-module
algebra) and on an aplication of Kashiwara's theory of crystal bases
which enables one to observe the behavior of Fq(G) when q -->0. In
the limit q -->0 the left-right action and the multiplication in Fq(G)
turn to be drastically simpler: we can find a monomial linear base
of Fq(G) such that the structure constants have simple combinatorial
description. This monomial base is parametrized by pairs of generalized
standard tableaux (as defined by P. Littelmann, Adv. Math. 124, 1996).
This cristalized version of Fq(G) can be exploited to obtain a monomial
base and a presentation of Fq(G) for a generic q.
Fusion product tensor products and the q-Verlinde numbers
by Kedem, Rinat
The fusion tensor product is a filtered product of finite
dimensional representations of some affine algebra (we consider affine
sl(2)) defined by Feigin. We explain how this can be used to define a
graded version of the Verlinde algebra, using certain coinvariants of
integrable modules.
Symmetric quantum Lie operations and differential calculi
by Kharchenko, Vladislav
The notion of quantum Lie operation naturally appears in line with
the Friedrichs criteria for Lie polynomials (Journal of Algebra, 217,
188-228, 1999). We prove that the (n-2)!-dimensional space of
generic quantum Lie operations has a basis of the symmetric ones.
In the general case almost always a basis of symmetric operations
exists. All exceptional cases are found. We investigate in details
a left covariant first order differential calculus that naturally
arises on every skew primitively generated Hopf algebra with a week
homogeneity condition. By means of the P.M. Cohn theory we show that
the subalgebra of constants for the cover free differential algebra
is a free algebra and an ad-invariant left coideal. We prove density
and structural theorems for the operator algebra generated by
partial derivatives. If the given algebra is finitely generated
then every differential left ideal is generated by constants,
a non-commutative Taylor series decomposition formula is valid,
and the category of locally nilpotent modules over the operator
algebra is semisimple with the only simple object that is isomorphic
to the optimal algebra as a module. We find a necessary and
sufficient condition for a 1-form to be a complete differential.
On braided cocommutative Hopf algebras
by Natale, Sonia
Let H be a semisimple quasitriangular Hopf algebra over an
algebraically closed field k of characteristic zero. Let A be Hopf
algebra in the braided category of left H-modules such that A is
cocommutative in the braided sense. We show that there exist a
super-cocommutative Hopf superalgebra A' and a finite group G
acting on A' by automorphism such that A can be obtained from
A' by twisting the multiplication and the comultiplication by means of a
2-cocycle on G. Since the structure of super-cocommutative Hopf
superalgebras is known, this allows us to give a description of
the structure of A.
Hopf algebras related to the permutahedra and the associahedra
by Ronco, María
The graded vector space $K[P_{\infty }]$, spanned by the faces of
the permutahedra, has a natural structure of associative algebra,
given by the shuffles product. It is possible to define different
coproducts on this vector space, in such a way that $K[P_{\infty }]$
becomes a noncommutative and non cocommutative graded Hopf algebra.
These estructures induce different structures of Hopf algebra on the
vector space spanned by all planar rooted trees ${\cal P}^K$, which
are related with the Koszul\rq s operads $Pasc $, defined by F. Chapoton,
and $Tridend $, defined by J.-L. Loday and M. Ronco.
On the Drinfeld problem of quantization of Lie bialgebras
by Stolin, Alexander
We find certain functional identities for the Gauss q-power
function of a sum of q-commuting variables. Then we use
these identities to get explicit quantizations of all
rational and trigonometric solutions of of the classical
Yang-Baxter equation in sl(2). Then we compute the
corresponding deformed trigonometric and rational quantum
R-matrices.
Multisegments and Multipartitions
by Vazirani, Monica
The irreducible representations of the symmetric group $S_n$
are parameterized by partitions of $n$.
Thus we can use tuples of partitions to parameterize
the irreducible representations of the wreath product $S_n \wr \Z$.
The affine Hecke algebra is a $q$-deformation of $S_n \wr \Z$,
and Bernstein-Zelevinsky showed that its irreducible representations,
and hence the irreducible representations of $S_n \wr \Z$,
are parameterized by objects called multisegments.
We define a map from multisegments to multipartitions
that matches up the two parameterizations.
The multipartitions in the image of this map have a particularly simple
description in the language of Kashiwara's crystal graphs.
Sesiones 1ra semana