Organizers: Aaron Lauve, Peter Tingley
January  
14 
No talk 

21 
No talk 

28 
No talk 

February  
4 
Alexander Stolin (Gothenburg) 
Classification of quantum groups and Lie bialgebras 
11 ***IES 123*** 
Peter Tingley 
Crystals, PBW bases, and related combinatorics. 
18 
No talk 

25 ***IES 123*** 
Josh Hallam (Michigan State) 
Increasing Forests 
March  
4  No talk (spring break)  
11 
Shamil Shakirov (Berkeley) 
On Macdonald deformation of ReshetikhinTuraev TQFT 
18 


25 
Tony Giaquinto (Loyola) 
CANCELED 
April  
1 
Peter Tingley 
GelfandTsetlin bases and PBW bases 
8 
Andrew Carroll (DePaul) 
Moduli spaces for Schurtame algebras 
15 
teaching seminar conflict 

22 
Stefan Catoiu (DePaul University) 
Generalized Trigonometric Hopf Algebras
and Fermat's Equations 
29 
Christophe Hohlweg (Universite du Quebec a Montreal) 
ArtinTits Braid groups, low elements and weak order in Coxeter groups 
Directions 

IES (the institute for environmental sustainability) is located at the corner of W. Sheridan and N. Kenmore Avenues, Chicago, IL (map)
Parking is available oncampus for $7 in the Parking Garage (building P1 on the Lake shore campus map). To get to the Parking Garage, enter campus at the corner of West Sheridan Road and North Kenmore Avenue.
Abstracts 

Given an arbitrary field F of characteristic 0, we study Lie bialgebra structures on sl(n,F), based on the description of the corresponding classical double. For any Lie bialgebra structure, the classical double is isomorphic to the tensor product of sl(n,F) with either F[e]/e^2, F+F, or a quadratic field extension of F. In the first case, the classification leads to quasiFrobenius Lie subalgebras of sl(n,F). In the second and third cases, a BelavinDrinfeld cohomology can be introduced which enables one to classify Lie bialgebras on sl(n,F), up to gauge equivalence. The BelavinDrinfeld untwisted and twisted cohomology sets associated to an rmatrix are computed. For the CremmerGervais rmatrix in sl(3), we also construct a natural map of sets between the total BelavinDrinfeld twisted cohomology set and the Brauer group of the field F.
Feb 11: Peter Tingley (Loyola), Crystals, PBW bases, and related combinatorics.
Kashiwara's crystals are combinatorial objects used to study complex simple Lie groups and their highest weight representations. A crystal consists of an underlying set, which roughly indexes a basis for a representation, along with combinatorial operations related to the Chevalley generators. Here we are interested in types A and D, where the crystals are realized using Young tableaux and KashiwaraNakashima tableaux respectively. However the combinatorics is not always enough, and one would like to understand an actual basis corresponding to the crystal. This can be done using Lusztig's theory of PBW bases, and so there is a unique structurepreserving bijection between the PBW basis and the combinatorially constructed crystal. We describe that bijection explicitly. In type A the answer was to some extent known to experts, but in type D we believe it is new. I will explain as much of this as I can without assuming any prior knowledge of crystals or PBW bases, mainly through examples. This is joint work with two Loyola students (John Claxton and Adam Shultze), and also with Ben Salisbury.
Feb 25: Joshua Hallam (Michigan State), Increasing Forests
Let G be a graph with vertex set {1,2,...,n}.
We say a subtree of G is increasing if the labels along any path
starting at the minimum vertex increase. Moreover, we call a
subforest of G increasing if each of its components is an increasing tree.
We will show that the generating function for increasing forests of graph
always factors with nonnegative integer roots. We also give a
characterization of when the generating function is equal to the
graph's chromatic polynomial. We finish by generalizing this work to
simplicial complexes.
This joint work with Jeremy Martin and Bruce Sagan.
March 11: Shamil Shakirov (UC Berkeley),
On Macdonald deformation of ReshetikhinTuraev TQFT
The central quantity in the ReshetikhinTuraev approach to TQFT is the 6jsymbol of the quantum group U_q( sl_N ). This quantity satisfies several important algebraic identities, such as the Pentagon identity or the YangBaxter equation, which stand behind the topological invariance of ReshetikhinTuraev state sums. Most of these identities do not admit local deformations, except for one, which is sometimes called the Tetrahedral Duality. We will discuss this deformation, its relation to Macdonald theory, and conjectural relation to categorification of the ReshetikhinTuraev TQFT.
April 1: Peter Tingley (Loyola), GelfandTsetlin bases and PBW bases.
GelfandTsetlin bases are part of the history leading to the modern theory of canonical bases.
The idea is that, if you have a
family of nested semisimple algebras A_1,...A_n
such that, for any irreducible representation of any A_k,
its restriction to A_{k1} breaks up as a multiplicityfree sum of irreducible representations
of A_{n1}, then there is a unique (up to rescaling)
basis for any irreducible representation of A_n with
the property that it respects the decomposition into representations of any A_k.
Probably the most important example of this phenomenon is the algebras sl(n) with the usual embeddings.
I'll discuss that case in detail, explaining how it related to various combinatorial and algebraic things
we've seen in the seminar, specifically Young tableaux and PBW bases.
I'll then talk a bit about the other classical Lie algebras. The restrictions of so(n) to so(n1) are multiplicity free, so that case basically works, but there are complications. For sp(n) the restrictions are not multiplicity free, but some things still work.
April 8: Andrew Carroll (dePaul), Moduli spaces for Schurtame algebras
Over the last 40 years there have been a number of attempts to characterize the complexity of the category of modules over a (finitedimensional associative) algebra by means of invariant theory. More precisely, we seek to determine whether an algebra is tame or wild by considering rings of invariant functions, and the geometry of their moduli spaces. Such characterizations have been demonstrated in the context of path algebras of quivers, but attempts to generalize to bound quiver algebras have met resistance. I will describe these efforts, their success, their failures, and attempts to address these failures by defining the notion of Schur representation type. This is ongoing work with Calin Chindris.
April 22: Stefan Catoiu (DePaul),
Generalized Trigonometric Hopf Algebras and Fermat's Equation
It is wellknown that the Pythagorean equation x^2+y^2=z^2 has
infinitely many integer solutions. Fermat's Last Theorem says that the
equation x^n+y^n=z^n has no nontrivial solutions when n>2. Therefore,
despite its similar look, this is not an appropriate generalization of the
Pythagorean equation. Using Hopf algebras and generalized trigonometry we
produce the appropriate degree n generalization that has infinitely many
solutions. These are parametrized in the same way as the ones for the
Pythagorean equation.
April 29: Christophe Hohlweg (U. du Quebec, Montreal),
ArtinTits Braid groups, low elements and weak order in Coxeter groups
In this talk we will explain that the question of solving the conjugacy problem in the context of a general
ArtinTits Braid group reveals strong connections between the weak order of a Coxeter system (W,S), inversion sets of elements of W and small roots.
Small roots are the main ingredient introduced by Brink and Howlett in order to build a
'canonical automaton' that recognizes the language of reduced words of elements of W over S.
From small roots and inversion sets, we define a new finite class of elements in W called
'low elements.' These low elements are the key to prove that the smallest subset of W containing S, closed under join (for the right weak order) and suffix is finite, and by ricochet that finitely generated ArtinTits groups have a finite Garside family. Low elements seem rich in further applications in the study of infinite Coxeter groups, which will discuss if time allows.
This is based on joint works with Patrick Dehornoy and Matthew Dyer.