Seminars in 2013/14
Speaker: Yuri Bahturin (Memorial University of Newfoundland)
Time/Date: April 16 and April 23, 2014, 1 p.m.
Room: HH-3017
Title: Locally finite Lie algebras
Abstract:
An algebra A over a field F is called locally finite if any finite set of elements of A is contained in a finite-dimensional subalgebra. Equivalently, A is the direct limit of a family of finite-dimensional algebras. A hard (essentially, wild) problem that remains open is to classify simple locally finite algebra (associative, Lie, etc.) A complete classification of locally finite simple Lie algebras, due to Baranov-Zhilinski, exists in the case of so-called diagonal direct limits over algebraically closed field of characteristic zero. Even better known (fields of positive characteristic included!) are so-called finitary simple Lie algebras (Baranov – Strade). In this latter case we can even classify all graded simple algebras (Bahturin-Kochetov-Zaicev).
In this talk I would like to discuss basic notions of the theory of locally finite Lie algebras and try to explain current state of the problem of classifying group grad-ings on the diagonal direct limits of classical simple Lie algebras.
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Speaker: Yuri Bahturin (Memorial University of Newfoundland)
Time/Date: March 26, 2014, 1 p.m.
Room: HH-3017
Title: Classification of group gradings on nilpotent algebras
Abstract:
In our earlier talks we have shown how algebraic groups can be used to describe group gradings on certain nilpotent Lie algebras. In this talk, on the one hand, we would like to present the classification of these gradings, up to equivalence. We will also show that, on nilpotent algebras that are relatively free, some of these results can be obtained by much more elementary methods.
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Speaker: Mikhail Kotchetov (Memorial University of Newfoundland)
Time/Date: March 19, 2014, 1 p.m.
Room: HH-3017
Title: Classification of Semisimple Lie Algebras Over the Field of Real Numbers
Abstract:
The famous Killing-Cartan classification of semisimple Lie algebras over the field of complex numbers is one of the most beautiful pieces of modern algebra, which has inspired innumerable works in various branches of algebra as well as applications in differential geometry and mathematical physics. Yet it is the Lie algebras over the field of real numbers that arise as tangent algebras of Lie groups and hence are of special importance for applications. In this talk we will discuss the relationship between complex and real semisimple Lie algebras and outline the classification of the latter, which was accomplished by E.Cartan in 1914.
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Speaker: Yiqiang Zhou (Memorial University of Newfoundland)
Time/Date: February 26, 2014, 1 p.m.
Room: HH-3017
Title: A Theorem on diagonalization of idempotent matrices
Abstract:
I am going to introduce a theorem on diagonalization of idempotent matrices and explain how to use it to solve a matrix decomposition problem.
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Speaker: Edgar G. Goodaire (Memorial University of Newfoundland)
Time/Date: February 5, 2014, 1 p.m.
Room: HH-3017
Title: Lack of Commutativity in Groups
Abstract:
A group is said to have the lack of commutativity or LC property if elements only commute when the centre is involved; specifically, gh = hg if and only if one of g, h, gh is central. The speaker identified this property in the late 1980s working on a problem in nonassociative algebra. Since that time, however, it has recurred in various contexts that will be described in this talk.
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Speaker: Hamid Usefi (Memorial University of Newfoundland)
Time/Date: January 29, 2014, 1 p.m.
Room: HH-3017
Title: Classification of p-nilpotent restricted Lie algebras of dimension at most four
Abstract:
I talk about the classification of p-nilpotent restricted Lie algebras of dimension at most four over a perfect field of characteristic p. I will mention the classification of nilpotent Lie algebras of low dimension and the difficulties one might face trying to define and classify all possible p-maps on a given nilpotent Lie algebra. This is a joint work with Csaba Schneider.
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Speaker: Mikhail Kotchetov (Memorial University of Newfoundland)
Time/Date: January 15, 2014, 1 p.m.
Room: HH-3017
Title: Graded modules over simple Lie algebras with a group grading
Abstract:
Gradings on Lie algebras by various abelian groups arise in the theory of symmetric spaces, Kac-Moody algebras, and color Lie superalgebras. In the 1960s, V. Kac classified all gradings by cyclic groups on finite-dimensional simple Lie algebras over complex numbers. Recently, there has been considerable progress in the classication of gradings by arbitrary abelian groups on finite-dimensional simple Lie algebras over algebraically closed fields. Given a G-grading on such a Lie algebra L, it is natural to study G-graded L-modules. In characteristic 0, any finite dimensional graded L-module is a direct sum of simple graded L-modules. We will describe finite-dimensional simple graded L-modules (using a version of Clifford Theory) and consider the following related problem: which of the finite-dimensional L-modules admit G-gradings making them graded modules?
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Speaker: Huadong Su (Memorial University of Newfoundland)
Time/Date: November 27, 2013, 1 p.m.
Room: HH-3017
Title: A characterization of rings with planar unitary Cayley graphs
Abstract:
Let R be a ring with identity. The unitary Cayley graph of R is the simple graph with vertex set R, where two distinct vertices x and y are linked by an edge if and only if x-y is a unit of R. A graph is said to be planar if it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In this paper, we completely characterize the rings whose
unitary Cayley graphs are planar. As an application of this result, the semilocal rings with planar unitary Cayley graphs are completely determined. This is a joint work with Yiqiang Zhou.
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Speaker: Valentin Gruzdev (Memorial University of Newfoundland)
Time/Date: November 20, 2013, 1 p.m.
Room: HH-3017
Title: Relative growth of subgroups of free groups
Abstract:
In this talk we will recount results regarding the base of relative growth function of subgroups of free groups obtained by Rostislav Grigorchuk and discussed in his doctoral thesis titled "Banach invariant means on homogeneous spaces and random walks". We will also employ method designed by Yuri Bahturin and Alexander Olshanskii in their work "Growth of subalgebras and subideals in free Lie algebras" to show that the base of relative growth function of a finitely generated subgroup of a free group is strictly less than the base of growth function of the free group itself, provided there are no cancellations between generators.
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Speaker: Gaelan Hanlon (Memorial University of Newfoundland)
Time/Date: November 13, 2013, 1 p.m.
Room: HH-3017
Title: Introduction to Topological Dehn Functions
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Speaker: Yuri Bahturin (Memorial University of Newfoundland)
Time/Date: October 23, 2013, 1 p.m.
Room: HH-3017
Title: Growth of subalgebras and subideals in free Lie algebras
Abstract:
This is a joint work with A. Olshanskii. We investigate subalgebras in free Lie algebras, the main tool being relative growth and cogrowth functions. Our study reveals drastic differences in the behavior of proper finitely generated subalgebras and nonzero subideals. For instance, the growth of a proper finitely generated subalgebra H of a free Lie algebra L, with respect to any fixed free basis X, is exponentially small compared to the growth of the whole of L. Quite opposite, the cogrowth of any nonzero subideal S is exponentially small compared to the growth of L.
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Speaker: Mikhail Kotchetov (Memorial University of Newfoundland)
Time/Date: October 2 and 10, 2013, 1 p.m.
Room: HH-3017
Title: Fine gradings of exceptional simple Lie algebras
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Speaker: Diego Aranda Orna (University of Zaragoza, Spain)
Time/Date: September 12 and 19, 2013, 2 p.m.
Room: HH-2010
Title: Introduction to structurable algebras
Abstract:
Structurable algebras were introduced by Allison as a generalization of Jordan algebras. The TKK construction provides a way to construct a Lie algebra starting from a structurable algebra. In this seminar, I will explain the basic definitions of structurable algebras, the classification of central simple structurable algebras and the TKK-construction.