### Thursday 7 March - Higher-dimensional Auslander–Reiten theory

**Time: **10:15-12:00

**Speaker**: Erik Darpö

**Language: **Swedish

**Abstract:** In this second talk, we shall introduce some basic concepts of, and ideas behind, higher-dimensional Auslander–Reiten theory. While a complete understanding of the category of finite-dimensional modules is beyond reach for most algebras, some of them come with a so-called d-cluster-tilting subcategory. Such subcategories furnish an analogue of the classical Auslander–Reiten theory; concepts such as almost split sequence and Auslander–Reiten quiver have “higher-dimensional” analogues in these categories. There mere existence of a cluster-tilting subcategory also has significant consequences for the homological properties of the entire module category.

Thursday 1 February, Representations of finite-dimensional algebras

**Speaker**: Erik Darpö

**Summary**: In this talk, the first of two, I shall give a short introduction to the representation theory of finite-dimensional algebras in general, and Auslander-Reiten theory in particular.

Thursday 7 December, Jan Snellman, Matematiska institutionen, Linköpings universitet

#### Title: f-vektorer till simpliciella komplex, resultat av Kruskal-Katona, Kozlov, Avramova-Herzog-Hibi, samt Nicklasson

Location: Kompakta rummet

Abstract: TBA

### Thursday 16 November, Jonathan Nilsson, Departement of Mathematics, Linköping University

#### Title: Representation theory for Lie algebras (part 2)

Place: Kompakta rummet

Abstract: In this second half of the seminar, I will continue our exploration of some well-known classes of Lie algebra modules, including weight- and Whittaker modules. I will then shift focus to my own contributions to the field, particularly the construction and classification of certain families of Lie algebra modules where an abelian subalgebra acts freely. Additionally, I will present a selection of open problems in this area of research that I find particularly interesting.

### Thursday 2 November, Jonathan Nilsson, Departement of Mathematics, Linköping University

#### Title: Title: Representation theory for Lie algebras (part 1)

**Place: Kompakta rummet, please note**

Abstract: 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.

### Thursday 28 September, Victor Hildebrandsson, Departement of Mathematics, Linköping University

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

Location: Hopningspunkten

Abstract: 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.

### Thursday 21 September, Mika Norlén Jäderberg, Departement of Mathematics, Linköping University

#### Title: Twisted Dual Extension Algebras

Location: Hopningspunkten

Abstract: 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.

### Thursday 15 June Axel Tiger Norkvist, Departement of Mathematics, Linköping University

#### Title: The Noncommutative Geometry of Real Calculi

Abstract: 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.