Speaker: Helen Dos Santos, Memorial University of Newfoundland
Room: HH-3017
Date/Time: Wednesday, March 15, 2017 at 1pm
Title: Group Gradings on the Associative Superalgebra and on the Lie superalgebras

Abstract:
We classify gradings by arbitrary abelian groups on the classical simple Lie superalgebras P (n), n ≥ 2, and on the simple associative superal- gebras M (m, n), m, n ≥ 1, over an algebraically closed field: fine gradings up to equivalence and G-gradings, for a fixed group G, up to isomorphism. As a corollary, we also classify up to isomorphism the G-gradings on the classical Lie superalgebra A(m, n) that are induced from G-gradings on M (m + 1, n + 1). In the case of Lie superalgebras, the characteristic is assumed to be 0.

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Speaker: Caio De Naday Horndardt, Memorial University of Newfoundland
Date/Time: Wednesday, March 1, 2017 at 1pm
Room: HH3017
Title: Group Gradings on Matrix Superalgebras

Abstract:
The investigation of the group gradings on Lie superalgebras led us to revisit the group gradings on (associative) matrix superalgebras. A description of all possible group gradings on matrix superalgebras was given by Bahturin and Shestakov in 2006, but with no classification results. In this talk, we will present an alternative description and a classification up to isomorphism. Work together with Helen Samara Dos Santos and Mikhail Kochetov.

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Speaker: Svea Mierach, Memorial University of Newfoundland
Date/Time: Wednesday, Feb 1, 2017, 1:00p.m.
Room: HH-3017
Title: Hopf Algebras, Cohomology and the Modular Group II

Abstract:
As discussed in the first talk, the modular group acts on the center of a finite-dimensional factorizable Hopf algebra, which is the zeroth Hochschild cohomology of this algebra. In this talk, we will present an extension of this action to the entire Hochschild cocomplex. After briefly recalling the action of the modular group on the center of a finite-dimensional factorizable Hopf algebra, the definition of the Hochschild cocomplex and the lifts of the Radford and Drinfeld map presenting last time, we will lift the antipode to an isomorphism of cocomplexes. Using these maps we are able to define an action of the modular group on the Hochschild cohomology of a factorizable Hopf algebra and to show that the relations are satisfied. A key point in the verification of the relations is the fact that the left and the right action of central elements on a bimodule induce the same map on the Hochschild cohomology groups.

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Speaker: Svea Mierach, Memorial University of Newfoundland
Date/Time: Wednesday, January 25, 2017 at 1pm
Room: HH-3017
Title: Hopf Algebras, Cohomology and the Modular Group I

Abstract:
The modular group acts on the center of a factorizable Hopf algebra, which is the zeroth Hochschild cohomology group of this algebra. In this talk, we will present an extension of this action to the entire Hochschild cocomplex. In the talk, we will first recall the definition of a factorizable Hopf algebra. Then we will present the well-known action of the modular group on its center. For this purpose, we will introduce the Drinfeld map and the Radford map. After that, we will discuss the definition of the Hochschild cocomplex and introduce some related cocomplexes. These allow us to find generalizations of the Drinfeld and the Radford map. Using these maps we are able to define an action of the modular group on the Hochschild cohomology of a factorizable Hopf algebra and to show that the relations are satisfied.

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Speaker: Yorck Sommerhäuser, Memorial University of Newfoundland
Date/Time: Wednesday, January 18, 2016
Room: HH-3017
Title:The Modular Group and its Presentations

Abstract:
The homogeneous modular group is the group of two times two matrices with integer coefficients. It can be presented in various ways by using a set of two generators. In the talk, we describe three such presentations, exhibit the relations between the generators in each of the cases, and prove that these relations are defining.

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Speaker: Mikhail Kotchetov, Memorial University of Newfoundland
Date/Time: November 30, 2016 at 1pm
Room: HH-3017
Title: Graded-simple algebras and modules via the loop construction

Abstract:
The construction of (twisted) loop and multiloop algebras plays an important role in the theory of infinite-dimensional Lie algebras. Given a grading by the cyclic group Z/mZ on a semisimple Lie algebra, the loop construction produces a Z-graded infinite-dimensional Lie algebra. This was generalized by Allison, Berman, Faulkner and Pianzola to arbitrary nonassociative algebras and arbitrary quotients of abelian groups. In view of their results, the recent classification of gradings by arbitrary abelian groups on finite-dimensional simple Lie algebras (over an algebraically closed field of characteristic 0) yields a classification of finite-dimensional graded-simple Lie algebras. Mazorchuk and Zhao have recently applied the loop construction to modules. In this talk, we will show how this leads to a classification of finite-dimensional graded-simple modules over simple Lie algebras with a grading.

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Speaker: Yorck Sommerhäuser, Memorial University of Newfoundland
Date/Time: November 23, 2016 at 1pm
Room: HH-3017
Title: Yetter-Drinfel'd Hopf Algebras and Their Associated Algebras II

Abstract:
As explained in the first part, every left Yetter-Drinfel'd Hopf algebra can also be viewed as a right YetterDrinfel'd Hopf algebra, but over the dual Hopf algebra. Therefore, this algebra gives rise to two different Radford biproducts. In the talk, we describe two algebras that contain both of these Radford biproducts as subalgebras. Both algebras admit a triangular decomposition, but only one of them is a Hopf algebra. We also compare these algebras to the ones that appear in the construction of deformed enveloping algebras.

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Speaker: Yorck Sommerhäuser, Memorial University of Newfoundland
Date/Time: November 16, 2016 at 1pm
Room: HH-3017
Title: Yetter-Drinfel'd Hopf Algebras and Their Associated Algebras I

Abstract:
With each Yetter-Drinfel'd Hopf algebra, one can associate a variety of algebras, most prominently the Radford biproduct, but sometimes also a Hopf algebra with triangular decomposition, which appears in the construction of deformed enveloping algebras, its dual, and similar algebras. In the talk, we introduce new algebras of this kind and discuss their interrelation with the old ones.

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Speaker: Dr. Yuri Bahturin, Memorial University of Newfoundland
Date/Time: Wednesday, November 2, 2016, 21:00p.m.
Room: HH-3017
Title: Classification of Real Graded Division Algebras II (Joint with Mikhail Zaicev)

Abstract:
This is the second part of the talk with this title. An associative algebra A graded by a group G is called a graded division algebra if every nonzero homogeneous element is invertible. We give a complete classification, up to equivalence, of real graded division algebras A, in the case where A is finite-dimensional and the group G is abelian. As a consequence, we can describe all real finite-dimensional graded simple algebras.

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Speaker: Dr. Yuri Bahturin, Memorial University of Newfoundland
Date/Time: Wednesday, October 19, 2016, 1:00p.m.
Room: HH-3017
Title: Classification of Real Graded Division Algebras (Joint with Mikhail Zaicev)

Abstract:
An associative algebra A graded by a group G is called a graded division algebra if every nonzero homogeneous element is invertible. We give a complete classification, up to equivalence, of real graded division algebras A, in the case where A is finite- dimensional and the group G is abelian. As a consequence, we can describe all real finite-dimensional graded simple algebras.

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Speaker: Svea Mierach, Memorial University of Newfoundland
Date/Time: Wednesday, October 5, 2016 at 1pm
Room: HH-3017
Title: On the Hochschild Cocomplex of a Factorizable Hopf Algebra II

Abstract:
As discussed in the first talk, the modular group acts on the center of a finite-dimensional factorizable Hopf algebra, which is the zeroth Hochschild cohomology of this algebra. The question we would like to answer is whether it is possible to extend this action to an action on the entire Hochschild cocomplex. After briefly recalling the action of the modular group on the center of a finite-dimensional factorizable Hopf algebra and the definition of the Hochschild complex and cocomplex, we will generalize the Drinfeld map, the Radford map, and the antipode to isomorphisms of cocomplexes. We will then explain how these maps might be used to introduce an action of the modular group on the Hochschild cocomplex, although the defining relations remain to be established.

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Speaker: Svea Mierach, Memorial University of Newfoundland
Date/Time: Wednesday, September 28, 2016 at 1pm
Room: HH-3017
Title: On the Hochschild Cocomplex of a Factorizable Hopf Algebra I

Abstract:
The modular group acts on the center of a finite-dimensional factorizable Hopf algebra, which is the zeroth Hochschild cohomology of this algebra. The question discussed in the talk is whether it is possible to extend this action to an action on the whole Hochschild cocomplex. In the talk, we will first recall the definition of a factorizable Hopf algebra. Then we will present the well-known action of the modular group on the center of a finite-dimensional factorizable Hopf algebra. For this purpose, we will introduce the Drinfeld map and the Radford map. After that, we will discuss the definition of the Hochschild complex and cocomplex and generalize the Drinfeld map, the Radford map, and the antipode to isomorphisms of cocomplexes. We will explain how these maps might be used to define an action of the modular group on the Hochschild cocomplex, although the defining relations still remain to be established. The talk is based on my master’s thesis, written at the University of Hamburg under the supervision of Christoph Schweigert and Simon Lentner.

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Speaker: Dr. Mark Penney
Date/Time: Wednesday, September 21, 2016 at 1pm
Room: HH-3017
Title: Categorification of Zelevinsky's PSH algebras

 Abstract:
In his 1981 text “Representations of finite classical groups: a Hopf algebra approach” Zelevinsky introduced positive, self-adjoint Hopf (PSH) algebras. Using general structural results of PSH algebras Zelevinsky provided a uniform and elementary treatment of many results in the representation theory of symmetric groups, wreath product groups, and general linear groups over
finite fields.

Categorification is the name of a general program attempting to realise structures on sets and vector spaces as `shadows' of structures on categories. A simple example is that the representation ring of a group is a `shadow' of the full category of representations of that group.

In this talk I will begin by reviewing Zelevinsky's PSH algebras. I will then spend some time discussing categorification with the aim of describing the categorical analogues of bialgebras. Finally, I will explain how to construct categorifications of the main examples of PSH algebras. Time permitting, I may also discuss what new perspectives categorification provides for PSH algebras.

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Speaker: Mike Watson, Memorial University of Nwfoundland
Date/Time: Wednesday, July 6, 2016, 1:00p.m.
Room: HH-3017
Title: Growth Functions for Commutator Subgroups

Abstract:
In this work, we compare growth functions of different bases of commutator subgroups of free groups. Of the bases that we consider, the geodesic Schreier bases appear to be the fastest. In connection with groups, we consider a basis for the free Lie commutator subalgebra. In contrast with groups, the growth of a homogeneous basis for a free Lie subalgebra doesnot depend on the choice of basis.