University of Ottawa, Carleton University and Centre de Recherches Mathématiques
54th ALGEBRA DAY
Saturday, May 1, and Sunday, May 2, 2004
Schedule for Saturday, May 1:
- 10:00 Coffee in room 104 (Lounge)
- 10:30 Vladimir Chernousov (University of Alberta):
Abelian extensions and algebraic groups
This is a joint result with P. Gille and Z. Reichstein. Let G be a linear algebraic group defined over a field k. We prove that (under mild assumptions on k and G) there exists a finite k-subgroup S of G such that the natural morphism
H^1(k,S) to H^1(k,G) is surjective for every field extension K/k. We give two applications of this result. First one is related to a question when cocycles with coefficients in G are strongly unramified and have trivial fixed point obstructions. The second application is related to a variant
of the algebraic form of Hilbert's 13th problem and the problem of cohomological dimension of a maximal abelian extension of the base field k.
- 11:45 Lunch in room 104 (Lounge)
- 13:15 Sergei Krutelevich (University of Ottawa): The Freudenthal construction and orbits of exceptional groups
We will begin with an elementary introduction to octonion algebras and matrices over them. Then we will give two (equivalent) constructions of certain algebraic groups and their irreducible representations: one from Jordan- and another from Lie-theoretic point of view. We will conclude with a discussion of integral orbits of these groups, and, time permitting, speak of their relation to higher composition laws in number theory.
- 14:45 Mikhail Kotchetov (Carleton University): Identities and coidentities of Hopf algebras
We consider the problem of finding necessary and sufficient conditions for a Hopf algebra to satisfy a polynomial identity as an algebra. Well-known results on group algebras (Passman), universal enveloping algebras (Latyshev and Bahturin) and restricted enveloping algebras (Passman and Petrogradsky) give the answer for specific examples of Hopf algebras. We generalize these results and solve the problem for cocommutative Hopf algebras - completely in characteristic zero and partially in prime characteristic. We also consider the dual problem of determining when a Hopf algebra satisfies a coidentity as a coalgebra. We solve this problem for commutative Hopf algebras.
- 15:45 Coffee in room 104 (Lounge)
- 16:15 John Faulkner (University of Virginia) : Hopf duals, algebraic groups, and Jordan pairs
Abstract : The Hom functor is used to construct various algebras and coalgebras including the continuous dual of a Hopf algebra with a residually finitely generated projective linear topology. The dual is used to construct a k-group scheme. The results are applied to the study of algebraic groups associated with Jordan pairs.
Schedule for Sunday, May 2:
- 9:30 Coffee in room 104 (Lounge)
- 10:00 Mohan Putcha (North
Carolina State University):
Combinatorics of reductive monoids
Reductive monoids are Zariski closures of reductive groups. The monoid M(n,k) of all n x n matrices over an algebraically closed field k is such an example, being the closure of the general linear group GL(n,k). We will
discuss the combinatorics of three posets associated with a reductive monoid M with unit group G. The first invariant of M is the cross-section lattice C. This is an idempotent cross-section of the G x G- orbits (J-classes) of M, preserving the order. For M(n,k), C consists of the
standard idempotents of different rank. We will discuss the combinatorics of C, in particular computing its Moebius function. The second invariant of M is the Renner monoid R. This is a finite inverse monoid whose unit group
is the Weyl group W and can be used to extend the Bruhat decomposition from G to M. For M(n,k), R just consists of the symmetric inverse semigroup of partial permutation matrices. We give a description of the Bruhat-Chevalley order on R via projection maps between the W x W- orbits (J-classes) of R
and use it to obtain some shellability properties. The third invariant of M is a poset R* associated with the conjugacy decomposition of M. For M(n,k), R* consists of partitions of m, where m ranges from 0 to n, and the order is an extension of the dominance order on the partitions of n. We will discuss some properties of R* and present some conjectures.
- 11:15 Ottmar Loos (Universität Innsbruck): Derivations and automorphisms of exceptional Jordan algebras
We study the derivation algebra of a reduced Albert algebra, i.e., a Jordan algebra J of 3 by 3 hermitian matrices over a Cayley algebra C. If the characteristic is 2, we show that the derivation algebra of J has a unique proper nonzero ideal I, with quotient Der(J)/I independent of C. On the group level, this corresponds to a special isogeny between Aut(J) and
Aut(J_s), the automorphism group of the split Albert algebra.
Location: All talks will be in the
Department of Mathematics and Statistics of the University of
Ottawa, 585 King Edward (KED on the campus map), room B5.
Parking: Participants will be given a parking
permit for the parking lot T behind the department. Please talk
to one of the local organizers as soon as possible after your arrival.
Financial support for
participating graduate students and postdoctoral fellows is
available. If you are interested please contact one of the local organizers as soon as possible, but before April 14, as soon as possible, and ask your supervisor to do the same.
Local organizers: Erhard Neher (firstname.lastname@example.org) and
Michel Racine (email@example.com).