2023 – Page 3 – Dokuz Eylül University Faculty of Science Department of Mathematics

On Rings Whose Finite Length Indecomposable Modules Are Completely Determined by Their Composition Factors III

March 14 2023March 14 2023 by Zübeyir Türkoğlu

Victor Blasco Jimenez, Dokuz Eylül University.

Date: 15th of March, 2023, Wednesday, Time: 10.30 – 12.00.

Place:Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings).

Abstract: Today we will continue the discussion we started the other day. In particular, we will prove that a commutative Artinian Ring $R$ is of finite representation type if and only if it satisfies $\mathfrak{X}$ , that is, if and only if finitely generated indecomposable $R$ -modules are completely determined by their composition factors.

On Rings Whose Finite Length Indecomposable Modules Are Completely Determined by Their Composition Factors II

March 7 2023 by Zübeyir Türkoğlu

Victor Blasco Jimenez, Dokuz Eylül University. Date: 8th of March, 2023, Wednesday, Time: 10.30 – 12.00. Place:Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings). Abstract: Today we will continue the discussion we started the other day. In particular, we will show that a commutative Artinian Ring $R$ of finite representation type satisfies $\mathfrak{X}$ , that is, finitely generated indecomposable $R$ -modules are completely determined by their composition factors.

On Rings Whose Finite Length Indecomposable Modules Are Completely Determined by Their Composition Factors

February 28 2023February 28 2023 by Zübeyir Türkoğlu

Victor Blasco Jimenez, Dokuz Eylül University.

Date: 1st of March, 2023, Wednesday, Time: 10.30 – 12.00.

Place:Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings).

Abstract: From the Fundamental Theorem of Abelian Groups we can deduce that every finite length indecomposable module over $\mathbb{Z}$ is uniquely determined by its composition factors set, meaning that if we have two indecomposable finite length abelian groups $N^1$ and $N^2$ with the same length and, if for any $0\subseteq N_{1}^j\subseteq ... \subseteq N_{r}^j=N^j$ composition series for $N^j$ , $j=1,2$ , we get $\{N_{i+1}^1/N_{i}^1\}_{i=0}^{r-1}=\{N_{i+1}^2/N_{i}^2\}_{i=0}^{r-1}$ , then we must have $N^1\cong N^2$ . In this series of talks , we will study this property about the finite length indecomposable abelian groups in a more general way. We will start by focusing on the class of commutative rings $R$ which satisfy it, showing that it contains the class of Dedekind Domains. If time permits, we will see that if $R$ is any unital ring (not necessarily commutative) satisfying this property, which we will call “Property $\mathfrak{X}$ “, and $I$ is an ideal of $R$ , then also $R/I$ satisfies it. This work is part of my ongoing master thesis “Some methods of Category Theory in the Representation Theory of Artin Algebras”.

The Auslander-Reiten Quiver of the 2-Kronecker Algebra III

February 21 2023February 21 2023 by Zübeyir Türkoğlu

İrem Yıldız, Dokuz Eylül University.
Date: 22nd of February, 2023, Wednesday, Time: 10.30 – 12.00.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings).

Abstract: The Auslander-Reiten quiver of a finite dimensional $k$ -algebra $A$ is the quiver whose vertices are the isoclasses of indecomposable modules over $A$ , and the number of arrows between two vertices $[M]$ , $[N]$ is given by the $k$ -dimension of Irr $_{A}(M,N)$ .
In this talk, we will review some basic facts about almost split sequences and explain briefly how to construct them. Our aim is to construct the Auslander-Reiten quiver of the path algebra associated to the 2-Kronecker quiver.

Some Geometry for Robot Kinematics

January 17 2023January 17 2023 by volkan.oger

J.M. Selig, London South Bank University.
DAte: 26th January, 2023, Time: 14:00.
Place: Dokuz Eylül Üniv., Tınaztepe Yerleşkesi, Fen Fak. Matematik Böl.

Abstract:The talk will begin with a brief review of dual quaternions and the realisation of the group of rigid-body displacements by the Study quadric. Next we look at some linear subspaces of the Study quadric and their interpretation as sets of displacements. Following this we will describe some sets of displacements that are intersections of the Study quadric with linear subspaces of the surrounding P^7. Then we will discuss some Segre varieties. These can be realised by simple serial linkages. A final extended example shows how some of these ideas can be used to solve problems in the theory of mechanisms.

Quaternions, Dual Quaternions and Clifford algebras

January 17 2023January 17 2023 by volkan.oger

J.M. Selig, London South Bank University.
Date: 24th January, 2023, Time: 14:00.
Place: Dokuz Eylül Üniv., Tınaztepe Yerleşkesi, Fen Fak. Matematik Böl.

Abstract:After a brief review of Hamilton’s quaternions and how they can be used to represent rotations, Clifford’s dual quaternions will be discussed. The use of this algebra to represent rigid-body displacements will be explained. As will the relation to the Study quadric. The representation of twists, infinitesimal rigid-body displacements, will also be considered. Finally, the notion of Clifford algebras will be introduced and various examples will be considered. In particular, examples rep-resenting the algebra of 3-dimensional Euclidean geometry will be outlined.

The Auslander-Reiten Quiver of the 2-Kronecker Algebra II

January 16 2023January 16 2023 by Zübeyir Türkoğlu

İrem Yıldız, Dokuz Eylül University.
Date: 17th of January, 2023, Tuesday, Time: 13:30 – 15:30.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings).

The Auslander-Reiten Quiver of the 2-Kronecker Algebra

January 10 2023January 10 2023 by Zübeyir Türkoğlu

İrem Yıldız, Dokuz Eylül University.
Date: 10th of January, 2023, Tuesday, Time: 13:30 – 15:30.
Place: Dokuz Eylül Univ., Tınaztepe Campus, Faculty of Science, Department of Mathematics, Room B206 (Online-Sakai-Graduate Meetings).

Torsion and Torsion-Free Classes Arising Through Objects Of Finite Type in a Grothendieck Category

January 9 2023January 9 2023 by Zübeyir Türkoğlu

Sinem Odabaşı, Universidad de Murcia.
Date: 3th of January, 2023, Tuesday, Time: 13.00 - 14.30.
Place: Online-Join Zoom Meeting (https://zoom.us/j/99412179515?pwd=bUlLdStFaVpjMHhWb21LaFlRMWI1Zz09) 
Meeting ID: 994 1217 9515 Passcode: 553266

Abstract: In a module category, the class FPn(R) of the so-called finitely n-presented left R-modules induces two crucial classes In(R) and Fn(Rop) of left FPn-injective and right FPn-flat Rmodules, respectively. It is known that certain homological properties of In(R) and Fn(Rop) and closure properties of FPn(R) are closely related, and these properties determine ring theoretic properties of R. The authors in [1] introduce the class FPn(G) of objects of type FPn in a Grothendieck category G as a generalization of the class FPn(R), and study certain aspects of the associated class In(G) of FPn-injective objects in G. In this talk, further homological aspects of the class In(G) will be presented. We show that the projective dimension of the class FPn(G) controls how far is In(G) from being a torsion class. Under mild conditions on G, which permit us to have ‘the external tensor product functor’, we introduce the class Fn(G) of FPn-flat objects in G. We will exhibit a close interaction between the classes In(G) and Fn(G). Further applications will be presented in Ab-valued functor categories showing n-coherency in terms of the domain category This is a joint work with Daniel Bravo, Carlos Parra and Marco Perez; see [2].

References

[1] Bravo, D., Gillespie, J. & Perez, M. A. (2019). Locally typeFPn and n-coherent categories. arXiv:1908.10987

[2] Bravo, D., Odabas¸ı, S., Parra, C & Perez, M. A. (2022). Torsion and Torsion-free classes from objects of finite type in Grothendieck categories. Journal of Algebra, 608, 412-444. DOI: 10.1016/j.jalgebra.2022.05.029.