Algebraseminariet

Torsdag 19 december 2024, kl 10.15-12.00, Jakob Palmkvist, Chalmers

Titel: Non-associative structures in extended geometry

Sammanfattning: I will present a generalisation of vector fields on a vector space, where the vector space is generalised to a highest-weight module over a Kac-Moody algebra. The generalised vector field is an element in a non-associative superalgebra defined by the module and the Kac-Moody algebra. Also the Lie derivative of a vector field with respect to another is generalised and expressed in a simple way in terms of this superalgebra. It reproduces the generalised Lie derivative in the general framework of extended geometry, developed in collaboration with Martin Cederwall (arXiv:1711.07694). In special cases it reduces to the generalised Lie derivative in double and exceptional field theory, which unify diffeomorphisms with gauge transformations in supergravity theories.

Torsdag 5 december 2024, kl 10.15-12.00, Joakim Arnlind, Linköping

Titel: Projective modules and noncommutative vector bundles

Sammanfattning: Projective modules play a central role in noncommutative (and commutative) algebraic formulations of geometry. I will recall the classical result by R. Swan (1962), showing that vector bundles over topological spaces are in one-to-one correspondence with projective modules over the ring of continuous functions, and describe how this results motivates the concept of noncommutative vector bundles.

Torsdag 21 november 2024, kl 10.15-12.00, Axel Hultman, Linköping

Titel: Graph complexes and the evasiveness conjecture

Sammanfattning: The evasiveness conjecture has its origins in computer science. It roughly asserts that every monotone graph property is difficult to recognize by inspecting an unknown graph edge by edge. In 1984, Kahn, Saks, and Sturtevant beautifully applied fixed point theory for group actions on topological spaces in order to prove the conjecture for graphs with a prime power number of vertices. The general case is still open.

I will discuss the evasiveness conjecture and sketch the argument of Kahn, Saks, and Sturtevant. Following a tangent, I will also discuss some other aspects of simplicial complexes of graphs.

Torsdag 10 oktober 2024, kl 10.15-12.00, Helena Jonsson, Uppsala Universitet

Titel: J-equivalence for associative algebras

Sammanfattning: The question at the heart of this seminar is the following: given associative algebras A and B, does there exist an A-B-bimodule M, and a B-A-bimodule N, such that the regular A-A-bimodule appears as a direct summand of the tensor product of M and N over B?

Our interest in this question comes from the representation theory of the bicategory (or monoidal category) of A-A-bimodules. Expresssing the tensor combinatorics of A-A-bimodules in terms of left-, right-, and two-sided relations, inspired by Green's relations for semigroups, is a key to understanding the "simple" representations. However, A-mod-A is of wild type for most algebras A, making the tensor combinatorics very difficult to study explicitly. We therefore introduce the J-relation for algebras as a means to compare the tensor combinatorics of bimodules over different algebras.

In this talk, I will discuss constructions that result in J-related algebras, as well as properties that are perserved by the relation. I will also give many examples of J-related algebras, both equivalent and non-equivalent.

Torsdag 12 september 2024, kl. 10.15-12.00, Álvaro Lozano, Universidad de Zaragoza

Titel: Two Dynamical Questions and One Negative Answer

Sammanfattning: One of the most fundamental results on measurable orbit equivalence is the Connes-Feldman-Weiss theorem, which states that any amenable, non-singular, countable equivalence relation can be generated by a single measurable transformation. In the topological setting, Giordano, Putnam, Skau, and Matui showed that any minimal $\mathbb{Z}^n$ action on the Cantor set is orbit equivalent to a minimal action of the integers; that is, the equivalence relation given by the orbits can be generated by a single homeomorphism. Therefore, it is natural to ask whether a result similar to the Connes-Feldman-Weiss theorem exists in the topological setting.

On the other hand, Hector asked whether generic leaves of laminations are quasi-isometric to discrete groups, as in the case of two-ended generic leaf laminations (this is discussed in Blanc's PhD thesis).

Here, we aim to present an example that serves as a counterexample to both of these questions. If time permits, we will also discuss the remaining open questions related to the previous ones.

This is a joint work with F. Alcalde and M. Martinez.

Tisdag 11 juni 2024, kl. 10.15-11.15, Anita Rojas, Universidad de Chile

Titel: Decomposing abelian varieties: algorithms and applications

Sammanfattning: In this talk, we will present an effective procedure to explicitly find the decomposition of a polarized abelian variety into its simple factors if a period matrix is known. In addition, we will show two algorithms to compute the period matrix for an abelian variety, depending on the given geometric information about it. The goal is to fully decompose an abelian variety with a non-trivial automorphism group by successively decomposing their factor subvarieties arising from the group action, even when these no longer have a group action. We will also illustrate how to use our algorithms showing a completely decomposable Jacobian variety of dimension 101, which fills the Ekedahl-Serre gap.

Torsdag 16 maj 2024, kl. 10.15-12.00, Vladimir Tkachev (Tkatjev), Matematiska institutionen, Linköpings universitet

Titel: 'Nonassociative calculus' on metrized algebras, II

Sammanfattning: I will explain how the Freudenthal-Springer construction of Jordan algebras works in a classification of Cartan isoparametric hypersurfaces (in particular for the eiconal equation) and give an outline how it can be helpful in a general context for discovering of `hidden' Jordan algebra structures. I will also discuss the class of metrized algebras over reals and explain why the Peirce value 1/2 (coming from the Jordan algebra theory) is exceptional. For example, for in any metrized algebra all idempotents are primitive and their nontrivial Peirce eigenvalues are at most 1/2.

Torsdag 25 april 2024, kl. 10:15-12:00, Vladimir Tkachev (Tkatjev), Matematiska institutionen, Linköpings universitet

Titel: 'Nonassociative calculus' on metrized algebras, I

Sammanfattning: I will start with an introduction to commutative non-associative algebras and in particular explain both the definition of a metrized algebra and their basic properties (in general noncommutative case). I will also give an outline of the Freudenthal-Springer construction of Jordan algebras arising from cubic forms and its application to a classification of the eiconal equation Some further related concepts and results will also be discussed.

Torsdag 7 mars 2024 - Higher-dimensional Auslander–Reiten theory

Talare: Erik Darpö

Språk: Svenska

Sammanfattning: 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.

Torsdag 1 februari 2024, Representations of finite-dimensional algebras

Talare: Erik Darpö

Sammanfattning: 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.

Torsdag 7 december 2023, Jan Snellman, Matematiska institutionen, Linköpings universitet

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

Plats: Kompakta rummet
Sammanfattning: Vi redogör för resultat av Kruskal-Katona, Linusson, Kozlov med flera om antal och möjligt utseende för så kallad f-vektorer till simpliciella komplex. Vi diskuterar sambandet mellan dessa och Hilbertfunktioner för homogena kvoter av den yttre algebran. Relaterade spörsmål om Hilbertfunktioner till homogena kvoter av den symmetriska algebran beskrivs av f-vektorer till s.k. multikomplex, vilka är standardmonomen till artinska monomideal.

Torsdag 16 november 2023, Jonathan Nilsson, Matematiska institutionen, Linköpings universitet

Titel: Representation theory for Lie algebras (part 2)

Plats: Kompakta rummet
Sammanfattning: 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.

Torsdag 2 november 2023, Jonathan Nilsson, Matematiska institutionen, Linköpings universitet

Titel: Representation theory for Lie algebras (part 1)

Plats: Kompakta rummet,
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 2023, 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 2023, 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 2023, 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 2023, 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 2023, 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 2023, 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 2023, 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 2023, 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.