JOINT
ALGEBRA SEMINAR
Fall 2006
TIME: Tuesday
3:00-4:00 at
Carleton; Thursday
3:20-4:20 at the University
of Ottawa.
LOCATION: Either at Carleton
University (HP 4369) or at the University
of Ottawa (KED room B-005),
depending on the speaker
Saturday Sept. 30
Algebra Day at Carleton University:
http://www.math.carleton.ca/AlgDay.html
Tuesday, October 10, at Carleton
University
Speaker: Vlasta Dlab
(Carleton)
Title: Approximations of
algebras by standardly stratified algebras
Abstract: can be found here
Thursday, October 19, at the University
of Ottawa
Speaker: Ottmar Loos
(Universitaet Innsbruck)
Title: Symmetric
compositions over rings
Abstract: Symmetric
compositions are non-associative and in general non-unital algebras with a
quadratic form q satisfying the composition law
q(x y) = q(x) q(y) and the associativity property b(xy, z) = b(x, yz)
where b(x,y) = b(x+y)- b(x) - b(y) is the polar form of q.
Symmetric compositions are closely related to composition
algebras in the usual sense (quaternions and octonions) and to the phenomenon
of triality for the universal covering group of the orthogonal group in
dimension 8. Despite their innocuous-looking definition, they exhibit strange
phenomena in characteristic 3.
The talk will give an outline of the historic development and
then present some new results for symmetric compositions over commutative
rings.
Tuesday, October 24, at Carleton
University
Speaker: Ben Steinberg (Carleton
University)
Title: Integer
programming over right-angled Artin groups
Abstract: Integer
programming is really the question of determining membership in finitely
generated submonoids of free abelian groups. So the membership problem in
finitely generated submonoids of groups can be viewed as a non-commutative
analogue of integer programming.
In this talk we characterize which right-angled Artin groups have
decidable membership problem for finitely generated submonoids in terms of
their commutation graphs. In the process we provide the first example of
a finitely presented group with an undecidable membership problem for finitely
generated submonoids but a decidable membership problem for finitely generated
subgroups.
This is joint work with Markus Lohrey (Stuttgart).
Thursday, November 2, at the University
of Ottawa
Speaker: Maribel Tocon (University
of Ottawa)
Title:
Graded-simple Lie algebras of type B_2
Abstract: Generalizations
of affine Lie algebras naturally lead to root-graded Lie algebras which
also have a second complatible grading by an abelian group. One of the
important features of this type of Lie algebras is that other types of
nonassociative algebras are required for their description.
In this talk, we will describe graded-simple Lie algebras of type B_2,
under some natural assumptions on the abelian group, by means of the
classification of graded-simple Jordan
pairs covered by a triangle.
Tuesday, November 7 and 14, at Carleton
University
Speaker: Yuly Billig (Carleton
University)
Title: Solving
non-commutative differential equations in vertex algebras
Abstract: In this talk
we will show how the vertex algebra technique can be used to study
representations for the Lie algebra $W_2$ of vector fields ona 2-dimensional
torus.
In a famous 1978 paper Lepowsky and Wilson showed that a module for an
infinite-dimensional Heisenberg algebra admits the action of a much larger
affine Kac-Moody algebra. It turns out that for $W_2$ we have an analogous
situation: $W_2$ has a loop algebra $\tilde{sl}_2$ as a subalgebra, and for a
family of irreducible representations of $\tilde{sl}_2$ the action can be
extended to all of $W_2$. This can be done by solving certain differential equations
in a vertex algebra associated with $W_2$. Unlike the case of the Heisenberg
algebra, the positive/negative parts of the loop algebra are non-commutative,
which makes solving these equations rather challenging.
This is a joint work with Alex Molev and Ruibin Zhang (Univ.
of Sydney).
Thursday, November 23, at the University
of Ottawa
Speaker: Daniel Daigle (University
of Ottawa)
Title:On polynomials f(X,Y,Z) annihilated by two locally nilpotent derivations.
Abstract: I will explain the relation between the class of polynomials mentioned in the title and the open problem which asks for a description of the
automorphisms of the k-algebra k[X,Y,Z], where k is a field of characteristic zero. I will also give a characterization of the polynomials (X,Y,Z) of the title.
Winter 2007
TIME: Friday
10:30-11:30 at Carleton, 10:10-11:10 at the University
of Ottawa.
LOCATION: Either at Carleton
University (HP 4351) or at the University
of Ottawa (LMX 106), depending on the
speaker
Friday,
January 12, at the University of Ottawa
Speaker: Walter Burgess (University
of Ottawa)
Title: On the Cartan Determinant Conjecture and its converse.
Abstract: The talk will give a brief history of the Conjecture and some of the lines of attack on it. The Conjecture says that for, in particular, a finite dimensional algebra A, if the global dimension is finite then the determinant of its Cartan matrix of A is 1. Some new advances by Shiping Liu and Charles Paquette (from Sherbrooke) on the Conjecture and its converse will be described.
These inspired some joint work with Ahmad Mojiri comparing Liu and Paquette's algebras, called quasi-stratified algebras, with algebras such as left serial, standardly stratified or cellular. Constructions of quasi-stratified algebras are given showing that there are "lots" of them. One should keep in mind that the Cartan matrix has very little information about an algebra and its determinant even less; that this integer should contain any useful information is surprising.
Friday,
January 26, at the University of Ottawa
Speaker: Nicolas Guay
Title: Quantum and symplectic reflection
algebras
Abstract: TBA
Friday,
February 2, at the University of Ottawa
Speaker: Christophe Hohlweg (Fields)
Title: Noncommutative symmetric functions
for wreath products
Abstract: Permutations can be embedded with
a structure of graded Hopf algebra such that the descent algebra is a sub-Hopf
algebra. Here the product is the noncommutative analog of the induction of
characters, while the coproduct is the noncommutative analog of the restriction
of characters. This is the core of the theory of noncommutative symmetric
functions. It is well-known that symmetric functions have several
interpretations in representation theory. It turns out that most of these
interpretations have an analog in the noncommutative case. In this lecture, I
will explain how the theory of noncommutative symmetric functions is extended
to wreath products. (joint work with Pierre Baumann)
Tuesday, February 6, 9am, Room: TBA at the University
of Ottawa
Speaker: Nicholas Proudfoot (Columbia)
Title: Quivers
as presentations of projective varieties
Abstract: Much of the data of an algebraic
variety is encoded by its (derived) category of vector bundles, and it is
sometimes possible to 'present' this category as the (derived) category of
representations of an algebra. In such a situation, it is natural to consider
the moduli space of stable representations of the algebra, which depends
sensitively on your notion of stability. I will discuss the relationship
between this moduli space and the original variety.
Friday,
February 16, at Carleton University
Speaker: John Dixon (Carleton
University)
Title: How to find a random element of a group
Abstract: Many algorithms in computational
group theory require as input a sequence of random elements from the group in
question which are independent and (approximately) uniformly distributed. I
shall discuss some of the ways in which such elements can be generated, and
describe a new method of constructing an efficient random element generator for
a group specified by a set of generating elements (permutations or matrices,
for example). The theory underlying the method is based on properties of the
group algebra.
Friday,
March 2, at Carleton University
Speaker: Mahmood Sohrabi (Carleton
University)
Title: Groups Elementary Equivalent to a Free 2-nilpotent Group of Finite Rank
Abstract can be found here.
Friday,
March 9, at Carleton University
Speaker: Pavel Zalesski (UnB)
Title: Cohomology of profinite completions.
Abstract: We shall discuss a relation between cohomology of groups and cohomology of their profinite completions.
The central notion here is Serre's definition of a good group. A group $G$ is called good if the natural homomorphism
$H^n(\widehat G,M)\longrightarrow H^n(G,M)$ is an isomorphism for every finite module $G$-module $M$.
We shall give examples of good groups and not good groups and present recent results on the subject.
Friday,
March 16, at Carleton University
Speaker: Claus Koestler (Carleton
University)
Title: On noncommutative factorization properties of braid group representations
Abstract: Subfactor theory and free probability theory, respectively initiated by Jones and Voiculescu both in the early 1980's, are among the most significant recent developments in operator algebras. Subfactor theory leads to new representations of the braid group and knot invariants, and
finds numerous applications in the field of quantum statistical physics. Free probability theory originates from the study of free group von
Neumann algebras and is of growing interest for researchers coming from various different fields in mathematics, in particular random matrix
theory and the representation theory of symmetric groups.
I will show that braid group representations lead to noncommutative factorizations in the spirit of commuting squares, as they are familiar
in subfactor theory. On the other hand these factorizations provide a noncommutative version of independence which is much more general than
free independence. My results indicate that there is a deeper connection between subfactor theory and free probability. I will discuss this in
the example of the braid group von Neumann algebra with infinitely many canonical generators. This is in parts joint work with R. Gohm.
Friday,
March 30 at Carleton University
Speaker:
Alexei Myasnikov (McGill University)
Title: Generic complexity of undecidable problems
Abstract: In this talk I will discuss generic complexity of undecidable problems.
It turns out that some classical undecidable problems are, in fact, strongly
undecidable, i.e., they are undecidable on every strongly generic subset of
inputs. For instance, the classical Halting Problem for Turing machines is strongly undecidable. Also, the famous Rice theorem holds
for strongly undecidable problems. This
provides plenty of examples of strongly undecidable problems. Furthermore,
there are natural super-undecidable problems, i.e., problems which are
undecidable on every generic (not only strongly generic) subset of inputs. In
particular, there are finitely presented semigroups with super-undecidable word
problem. Finally, I am going to touch
on absolutely undecidable problems,
which are undecidable on every non-negligible set of inputs.
The
talk is based on a mixture of classical algebra, logic, and modern complexity
theory, but it is elementary and should be accessible to graduate students.
Saturday
March 31 Algebra Day at the University of Ottawa
Friday,
April 20, at the University of Ottawa
13:30 in KED B005
Speaker: Oliver Villa (ETH Zurich)
Title: Decomposable Forms and Involutions
Abstract: In his book on compositions of quadratic forms, Shapiro asks whether a quadratic form decomposes as a tensor product of quadratic
forms when its adjoint involution decomposes as a tensor product of involutions on central-simple algebras. We give a positive answer for quadratic forms
defined over a linked field and produce counterexamples.
