—  Mathematics & Statistics

Algebra and Combinatorics Seminar, Spring 2015

Wednesdays 3:40-4:30 BVM hall 11th floor seminar room

Next semester
Previous semesters

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)
TBA
18

25
Tony Giaquinto (Loyola)
April
1 (near Easter)


8
Andrew Carroll (DePaul)

15


22
Stefan Catoiu (DePaul University)
TBA
29
Christophe Hohlweg (Universite du Quebec a Montreal)


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 on-campus 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

Feb 4: Alexander Stolin (Gothenburg), Classification of quantum groups and Lie bialgebras

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 quasi-Frobenius Lie subalgebras of sl(n,F). In the second and third cases, a Belavin-Drinfeld cohomology can be introduced which enables one to classify Lie bialgebras on sl(n,F), up to gauge equivalence. The Belavin-Drinfeld untwisted and twisted cohomology sets associated to an r-matrix are computed. For the Cremmer-Gervais r-matrix in sl(3), we also construct a natural map of sets between the total Belavin-Drinfeld 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 Kashiwara-Nakashima 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 structure-preserving 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.