Seminars 09/10
Fall 2009 Seminars
September 23, 2009
Eugene Chibrikov (MUN) “Linear Bases of Free Lie Algebras and their Applications”
Abstract:
In this talk we will present a short survey on linear bases of free Lie algebras and their applications for combinatorial problems, embedding theorems, Groebner-Shirshov basis theory, etc. We will construct a right-normed basis of a free Lie algebra, the basis that consists of right-normed monomials, and show its connection with the well-known Lyndon-Shirshov basis.
October 21, 2009
Ken Price (University of Wisconsin- Oshkosh, USA) "Matrix
Structures from Directed Graphs"
Abstract:
The speaker will explain how directed graphs are used to construct blocked and group-graded matrices. The approach is based on laying a foundation in directed graph theory. The talk includes background on directed graphs and blocked matrices. The directed graphs we consider have a finite number of vertices and no multiple arrows. Loops are allowed. The vertex set and the arrow set of a directed graph D are denoted by V( D) and A( D), respectively. If there are n vertices then we may assume they are numbered so that V( D) = {1,...,n} and A( D) is a subset of V( D) x V(D). We drop the parentheses and comma for any arrow (v,w) and denote it simply by vw.
We use undirected paths to formulate definitions of independence
and spanning for sets of arrows. An independent spanning set of
arrows is a basis. Many familiar basis properties are established.
This is related to blocked and group-graded matrix algebras.
If ab is in A( D) then we let E_{ab} denote the standard unit
matrix, that is, E_{ab} is the n x n matrix whose ab-entry is 1 and
all of its other entries are 0. A matrix is blocked by D if it is a
linear combination of standard matrix units which are indexed by
arrows of D. If there is a function from a basis to an abelian
group then it can be extended to the entire directed graph. This
motivates studying functions from the basis of a directed graph to
abelian groups. The homomorphism places a grading on a subalgebra
of blocked matrices. We show that, in many cases, all gradings on
the blocked matrix subalgebra are determined in this way.
November 4,
2009
Yuri Bahturin (MUN) "Filtrations
and Distortion of Embeddings in Algebras"
Abstract:
Let R be an algebra over a field F, generated by a finite set X.
There
is a natural ascending filtration on R whose nth term consists of
the
values of polynomials of degree at most n in X. Such filtration
is
called the degree filtration on R. The growth of the sequence
of
dimensions of the degree filtrations is always majorated by an
exponential function. All degree filtrations on the same algebra
are
pairwise equivalent, in a natural sense. Let us call a filtration
with
the growth condition as just above a D-filtration. On any
infinite-dimensional algebra there are uncountably many
pairwise
non-equivalent D-filtrations. If R is a subalgebra in an algebra S
and
there is a D-filtration on S then its restriction is a D-filtration
on R.
Theorem. Any D-filtration on a countably dimensional associative or
Lie
algebra R is the restriction of a degree filtration on a
finitely
generated algebra S, where R is embedded as a subalgebra.
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This is joint work with Professor Alexander Olshanskii of
Vanderbilt
University.
November 18, 2009
Alon Regev (Northern Illinois University, USA)
“Noncommutative algebras over uncountable fields”
Abstract:
Questions about nilpotency and about algebraicity in algebras are
of
fundamental importance in the theory of non-commutative rings. In
this
talk I will discuss some of these questions, beginning from the
classical
problems of Koethe and Kurosh, and up to more recent developments
and
open questions in the field. In particular, I will discuss some
special
results for algebras over uncountable fields.
Winter 2010 Seminars
February 3, 2010
Mikhail Kotchetov (MUN) “Group gradings on simple Lie algebras of Cartan type”
Abstract:
February 17, 2010
Mikhail Kotchetov (MUN) “Group gradings on simple Lie algebras of Cartan type, Part 2”
March 17, 2010
Tom Baird (MUN) “An approach to equivariant cohomology using algebraic combinatorics”
Abstract:
In
their 1997 paper studying equivariant algebraic varieties,
Goresky-Kottwitz-MacPherson observed that in a range of interesting
examples, including toric varieties and flag manifolds, the
calculation of equivariant cohomology reduces to analyzing an
associated combinatorial object now called a GKM-graph. This is a
graph whose vertices correspond to fixed points of the action, and
whose edges are labeled by characters of the symmetry group. Their
paper inspired a great deal of subsequent work, collectively known
as "GKM theory", linking combinatorial algebra with equivariant
topology. In two lectures aimed at non-experts, I will survey some
of this theory.
The first lecture will be a gentle introduction to equivariant
topology and equivariant cohomology, establishing some background
and motivation.
The second lecture will be a survey of GKM theory including my
recent work on GKM-sheaves over hypergraphs.
March 24, 2010
Tom Baird (MUN) “An approach to equivariant cohomology using algebraic combinatorics, Part 2”
April 14, 2010
Zhuang Niu (MUN) "The classification of AH-algebras"
Abstract:
An AH-algebra is an
inductive limit of homogeneous C*-algebras. This class of
C*-algebras contains many naturally arising C*-algebras, for
instance, UHF-algebras, irrational rotation algebras, and
C*-algebra associated to certain minimal dynamical systems. The
class of simple AH-algebras with a certain restriction on dimension
growth were classified using the ordered K-group together with the
pairing with the tracial simplex.
In this talk, I will give a review on this classification
theorem.
April 21, 2010
Zhuang Niu (MUN) "The classification of AH-algebras, Part 2"
Summer 2010 Seminars
May 12, 2010
Eugene Chibrikov (MUN) “On free Sabinin algebras”
Abstract:
Sabinin algebras are algebraic objects that capture the local structure of analytic loops in the same way in which Lie algebras capture the local structure of Lie groups. They were introduced by L. Sabinin and P. Miheev in 1987. In this report we discuss some recent results concerning free Sabinin algebras and construct a linear basis of a free Sabinin algebra which is a generalization of Shirshov's scheme for choosing bases of free Lie algebras.
