# Seminars 13/14

**2013/2014 Seminars**

**April 16 and April 23, 2014**

Yuri Bahturin (MUN) "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.

*HH-3017 at 1:00 p.m.*

**March 26, 2014**

Yuri Bahturin (MUN) "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.

*HH-3017 at 1:00 p.m.*

**March 19, 2014**

Mikhail Kotchetov (MUN) "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.

*HH-3017 at 1:00 p.m.*

**February 26, 2014**

Yiqiang Zhou (MUN) "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.

*HH-3017 at 1:00 p.m.*

**February 5, 2014**

Edgar G. Goodaire (MUN) "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.

*HH-3017 at 1:00 p.m.*

**January 29, 2014**

Hamid Usefi (MUN) "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.

*HH-3017 at 1:00 p.m.*

**January 15, 2014**

Mikhail Kotchetov (MUN) "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?

*HH-3017 at 1:00 p.m.*

**November 27, 2013**

Huadong Su (MUN) "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.

*HH-3017 at 1:00 p.m.*

**November 20, 2013**

Valentin Gruzdev (MUN) "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.

*HH-3017 at 1:00 p.m.*

**November 13, 2013**

Gaelan Hanlon (MUN) "Introduction to Topological Dehn Functions".

*HH-3017 at 1:00 p.m.*

**October 23, 2013**

Yuri Bahturin (MUN) "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.

*HH-3017 at 1:00 p.m.*

**October 2 and 10, 2013**

Mikhail Kotchetov (MUN) "Fine gradings of exceptional simple Lie algebras".

*HH-3017 at 1:00 p.m.*

**September 12 and 19, 2013**

Diego Aranda Orna, University of Zaragoza, Spain "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.

*HH-2010 (AAC) at 2 p.m.*

**August 28, 2013**

Alexander Baranov (University of Leicester, UK) "Direct limits of involution simple associative algebras".

Abstract.We will discuss a classification of the (countable) direct limits of finite dimensional involution simple associative algebras over an algebraically closed field of arbitrary characteristic. This also classifies the corresponding dimension groups. The set of invariants consists of two supernatural numbers and two real parameters.

*H-3017 at 1 p.m.*