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Auslander–Reiten Formulas and Almost-Split Sequences

Vårdstedt Persson, Bror LU (2024) In Bachelor's Theses in Mathematical Sciences MATK11 20241
Mathematics (Faculty of Engineering)
Mathematics (Faculty of Sciences)
Centre for Mathematical Sciences
Abstract
We aim to introduce to the concepts and constructions in Auslander–
Reiten theory, and to prove the Auslander–Reiten Theorem.
This involves introducing certain concepts, such as the Yoneda
extension groups, which we equip with a vector space structure. We
define projective resolutions and presentations, and their dual con-
cepts. We prove that the extension group can be constructed using
a projective resolution.
We introduce the Nakayama functor, the morphism category,
the Auslander–Reiten translate and the Auslander–Reiten sequences.
We conclude by showing the Auslander–Reiten Theorem.
Popular Abstract
Representation theory is a branch of mathematics that studies algebraic
structures by writing the objects in other ways, often as vector spaces and
matrices. This allows us to use our knowledge of matrices to understand
these more difficult algebraic structures. We will use vector spaces and
matrices to represent the structure of algebras.
Many matrix problems from linear algebra, such as writing matrices or
matrix pairs in normal form, can be generalized by using the language of
quiver representations. A quiver is a set of points and a collection of arrows
between them. To construct a representation of a quiver we assign to each
point, a vector space, and to each arrow, a matrix which describes a map
between the two vector... (More)
Representation theory is a branch of mathematics that studies algebraic
structures by writing the objects in other ways, often as vector spaces and
matrices. This allows us to use our knowledge of matrices to understand
these more difficult algebraic structures. We will use vector spaces and
matrices to represent the structure of algebras.
Many matrix problems from linear algebra, such as writing matrices or
matrix pairs in normal form, can be generalized by using the language of
quiver representations. A quiver is a set of points and a collection of arrows
between them. To construct a representation of a quiver we assign to each
point, a vector space, and to each arrow, a matrix which describes a map
between the two vector spaces.
It turns out that this construction has equivalent properties to an-
other algebraic structure called a module. This means that working with
quiver representations and certain modules is completely equivalent and
that mathematical results in one field can be transferred to the other. Nat-
urally this leads us to study the theory of modules.
We will be interested in modules over other algebras as well. Not all
modules are equivalent to a representation of a quiver. An algebra is a
type of general algebraic structure where some addition and multiplication
operations must be defined. For example, we can write the elements of
a certain algebra as upper triangular 2x2 matrices. This lets us use our
knowledge of linear algebra and upper triangular matrices to understand
modules over this algebra. (Less)
Please use this url to cite or link to this publication:
author
Vårdstedt Persson, Bror LU
supervisor
organization
course
MATK11 20241
year
type
M2 - Bachelor Degree
subject
keywords
Auslander Reiten Quiver Representation Algebra Module Theory Almost Split Sequences
publication/series
Bachelor's Theses in Mathematical Sciences
report number
LUNFMA-4172-2024
ISSN
1654-6229
other publication id
2024:K23
language
English
id
9177294
date added to LUP
2024-11-13 15:30:46
date last changed
2024-11-13 15:30:46
@misc{9177294,
  abstract     = {{We aim to introduce to the concepts and constructions in Auslander–
Reiten theory, and to prove the Auslander–Reiten Theorem.
This involves introducing certain concepts, such as the Yoneda
extension groups, which we equip with a vector space structure. We
define projective resolutions and presentations, and their dual con-
cepts. We prove that the extension group can be constructed using
a projective resolution.
We introduce the Nakayama functor, the morphism category,
the Auslander–Reiten translate and the Auslander–Reiten sequences.
We conclude by showing the Auslander–Reiten Theorem.}},
  author       = {{Vårdstedt Persson, Bror}},
  issn         = {{1654-6229}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Bachelor's Theses in Mathematical Sciences}},
  title        = {{Auslander–Reiten Formulas and Almost-Split Sequences}},
  year         = {{2024}},
}