Algebraseminariet

Torsdag 2 november, Jonathan Nilsson, Matematiska institutionen, Linköpings universitet

Titel: Representation theory for Lie algebras (part 1)

Plats: Kompakta rummet, obs!
Sammanfattning: In this first segment of the seminar, I will focus on the basic theory of Lie algebras and their representations. I will give an outline of the classical theory for finite-dimensional simple complex Lie algebras, starting with Killing-Cartan theory and leading up to the classification of finite-dimensional simple modules. Key topics covered include Cartan subalgebras, root spaces, weights, enveloping algebras and central characters. Additionally, a glimpse into the realm beyond finite-dimensional modules will be provided, touching upon Cuspidal modules, Whittaker-modules and Gelfand-Zetlin modules.

Torsdag 28 september, Victor Hildebrandsson, Matematiska institutionen, Linköpings universitet

Titel: Octonion algebras over schemes and the equivalence of isotopes and isometric quadratic forms

Plats: Hopningspunkten

Sammanfattning: Octonion algebras are certain algebras with a multiplicative quadratic form. In 2019, Alsaody and Gille show that, for octonion algebras over unital commutative rings, there is an equivalence between isotopes and isometric quadratic forms. The contravariant equivalence from unital commutative rings to affine schemes, sending a ring to its spectrum, leads us to a question: can the equivalence of isometry and isotopy be generalized to octonion algebras over a (not necessarily affine) scheme? We will begin the presentation with some necessary background in algebraic geometry and octonion algebras over rings. Then we will give the basic definitions of octonion algebras over schemes. We show that an isotope of an octonion algebra C over a scheme is isomorphic to a twist by the Aut(C)–torsor. Then we conclude the thesis by giving an affirmative answer to our question.

Torsdag 21 september, Mika Norlén Jäderberg, Matematiska institutionen, Linköpings universitet

Titel: Twisted Dual Extension Algebras

Plats: Hopningspunkten

Sammanfattning: Quasi-hereditary algebras first arose from the study of the highest weight categories that appear in the representation theory of semisimple complex Lie-algebras. Since then they have found many applications, such as the representation theory of Schur-algebras and algebraic groups. Central to the theory of quasi-hereditary algebras is a collection of modules called standard modules. Of special interest is the category of modules that admit filtrations by standard modules. A useful tool for studying this category is the notion of an exact Borel subalgebra, which are subalgebras that satisfy formal properties similar to universal enveloping algebras of Borel subalgebras from Lie theory.

In this seminar, we will consider a special class of algebras called twisted dual extension algebras. Informally, given two directed algebras B and A and a set M of constants, one can construct an algebra A(B,A,M). This algebra contains both B and A and the constants in M govern how elements in B and A interact. Under certain technical assumptions on B and A, we provide necessary and sufficient conditions for this algebra to be quasi-hereditary whenever B and A are monomial algebras. We also prove that B is an exact Borel subalgebra of A(B,A,M) whenever these conditions are satisfied. Finally, we compute the Ext-algebra of the standard modules of A(B,A,M) in terms of the Ext-algebra of the simple B-modules.

Torsdag 15 juni Axel Tiger Norkvist, Matematiska institutionen, Linköpings universitet

Titel: The Noncommutative Geometry of Real Calculi

Sammanfattning: The profound connection observed between algebra and geometry during the 1940s and beyond has transformed both fields, with extensive exploration of commutative algebras and their relationship to topological spaces. In the 1980s, mathematicians recognized the value of applying a geometric perspective to noncommutative algebras, leading to remarkable advancements in the field of noncommutative geometry. This talk focuses on Real Calculi, a derivation-based approach that generalizes aspects of Riemannian Geometry to noncommutative algebras, and explores some of the challenges that arise in the process.

Torsdag 25 maj Axel Hultman, Matematiska institutionen, Linköpings universitet

Titel: Coxeter groups

Sammanfattning:  We shall get acquainted with Coxeter groups from a perspective heavily slanted toward combinatorics. Along the way, algebraic and geometric structures will show up. Objects like Schubert varieties, Hecke algebras, and Kazhdan-Lusztig polynomials may appear. We hope (sometimes perhaps too naïvely) that the combinatorics of Coxeter groups have interesting things to say about them. Some ideas in this direction will be presented.

Torsdag 23 mars Axel Hultman, Matematiska institutionen, Linköpings universitet

Titel: Coxeter groups

Sammanfattning: We shall get acquainted with Coxeter groups from a perspective heavily slanted toward combinatorics. Along the way, algebraic and geometric structures will show up. Objects like Schubert varieties, Hecke algebras, and Kazhdan-Lusztig polynomials may appear. We hope (sometimes perhaps too naïvely) that the combinatorics of Coxeter groups have interesting things to say about them. Some ideas in this direction will be presented.

Torsdag 9 mars Daniel Lännström, Matematiska institutionen, Linköpings universitet

Titel: Graded Ring Theory and Leavitt path algebras

Sammanfattning: In this talk I will discuss my research concerned with so-called epsilon-strongly group graded rings and Leavitt path algebras. Unlike in many common general reference works in algebra, we will NOT assume that our rings are equipped with a multiplicative identity element. I will begin my talk by presenting different ways to define local units for non-unital rings. Next, I will detail some background on Leavitt path algebras, which can be seen as algebraic analouge of certain graph C^*-algebras. This area of research has seen a lot of recent activity which prompted the inclusion of Leavitt path algebras in the 2020 Mathematics Subject Classification (MSC2020). I will also present some important work from the 1980s by Dade about strongly graded rings and how that relates to the recently introduce notion of epsilon-strongly graded rings. Finally, I will talk about some joint research with P. Lundström (previously known as P. Nystedt), J. Öinert and S. Wagner.

Torsdag 16 februari Joakim Arnlind, Matematiska institutionen, Linköpings universitet, del 2

Titel: Noncommutative Levi-Civita connections

Sammanfattning:  Over the last 40 years, it has become apparent that it is fruitful to extend geometric notions to the setting of noncommutative algebras; both from a mathematical point of view, when studying geometric spaces with few (or no) interesting functions, and from a physical point of view, where noncommutative geometry lies at the heart of constructing theories of quantum gravity.

While the topological aspects of noncommutative geometry are by now fairly well studied, the Riemannian aspects are far less understood, although a lot of progress has been made during the last decade. In particular, the role of the Levi-Civita connection, which is an important object in Riemannian geometry, as well as a fundamental one in general relativity, is not completely understood.

In this series of lectures, I will present an algebraic view on Riemannian geometry, illustrating how one can make sense of e.g. differential forms, vector bundles, metrics and connections in a noncommutative setting. Moreover, I will present material related to the work I've done on the existence and uniqueness of Levi-Civita connections for noncommutative vector bundles.

Torsdag 26 januari Joakim Arnlind, Matematiska institutionen, Linköpings universitet, del 1

Titel: Noncommutative Levi-Civita connections

Sammanfattning:  Over the last 40 years, it has become apparent that it is fruitful to extend geometric notions to the setting of noncommutative algebras; both from a mathematical point of view, when studying geometric spaces with few (or no) interesting functions, and from a physical point of view, where noncommutative geometry lies at the heart of constructing theories of quantum gravity.

While the topological aspects of noncommutative geometry are by now fairly well studied, the Riemannian aspects are far less understood, although a lot of progress has been made during the last decade. In particular, the role of the Levi-Civita connection, which is an important object in Riemannian geometry, as well as a fundamental one in general relativity, is not completely understood.

In this series of lectures, I will present an algebraic view on Riemannian geometry, illustrating how one can make sense of e.g. differential forms, vector bundles, metrics and connections in a noncommutative setting. Moreover, I will present material related to the work I've done on the existence and uniqueness of Levi-Civita connections for noncommutative vector bundles.