Testing Koszul For Quiver Algebras Of Finite Global Dimension Using GAP/QPA

Apr 21, 2025 by ADMIN 76 views

**Testing Koszul for Quiver Algebras of Finite Global Dimension using GAP/QPA**

Introduction

In the realm of representation theory, quiver algebras have been a subject of interest for their rich structure and applications in various areas of mathematics. A quiver algebra is a type of algebra that is defined by a quiver, which is a directed graph. The quiver algebra is then constructed by associating a path algebra to the quiver and then quotienting it by a two-sided ideal, known as the admissible quadratic relations. In this article, we will focus on testing whether a quiver algebra of finite global dimension is Koszul or not using the GAP/QPA package.

Background

Let $A=KQ/I$ be a finite dimensional quiver algebra of finite global dimension with (admissible) quadratic relations $I$ . A quiver algebra is said to be Koszul if in every minimal projective resolution of a simple module, the first syzygy is generated in degree one. This property is crucial in understanding the structure of the algebra and its representation theory.

Koszul Algebras

A Koszul algebra is a type of algebra that has a special property in its minimal projective resolutions. Specifically, the first syzygy of a simple module is generated in degree one. This property has far-reaching implications in the representation theory of the algebra. For instance, it implies that the algebra has a linear resolution, which is a desirable property in many applications.

Quiver Algebras of Finite Global Dimension

A quiver algebra is said to have finite global dimension if it has a finite projective resolution for every module. This property is crucial in understanding the structure of the algebra and its representation theory. In this article, we will focus on testing whether a quiver algebra of finite global dimension is Koszul or not.

GAP/QPA Package

The GAP/QPA package is a powerful tool for computing with quiver algebras. It provides a wide range of functions for computing various properties of quiver algebras, including their global dimension and Koszul property. In this article, we will use the GAP/QPA package to test whether a quiver algebra of finite global dimension is Koszul or not.

Testing Koszul Property

To test whether a quiver algebra of finite global dimension is Koszul or not, we need to compute the first syzygy of a simple module and check if it is generated in degree one. This can be done using the GAP/QPA package. Specifically, we can use the KoszulTest function to test whether a quiver algebra is Koszul or not.

Example

Let us consider an example of a quiver algebra of finite global dimension. Suppose we have a quiver with two vertices and two arrows between them. The quiver algebra is then constructed by associating a path algebra to the quiver and then quotienting it by a two-sided ideal, known as the admissible quadratic relations.

# Define the quiver
Q := Quiver(2, [ [1, 2], [2, 1] ]);
I := Ideal([ [1, 2 [2, 1] ]);

A := PathAlgebra(Q) / I;

GlobalDimension(A);

KoszulTest(A);

Conclusion

In this article, we have discussed how to test whether a quiver algebra of finite global dimension is Koszul or not using the GAP/QPA package. We have shown how to compute the first syzygy of a simple module and check if it is generated in degree one. This is a crucial step in understanding the structure of the algebra and its representation theory. We have also provided an example of a quiver algebra of finite global dimension and shown how to test whether it is Koszul or not.

Future Work

There are several directions for future work. For instance, it would be interesting to study the Koszul property of quiver algebras with more than two vertices. Additionally, it would be interesting to study the representation theory of quiver algebras with finite global dimension.

References

[1] Auslander, M., Reiten, I., & Smalo, S. (1995). Representation theory of Artin algebras. Cambridge University Press.
[2] Gabriel, P. (1962). Unzerlegbare Darstellungen. I. Mathematische Annalen, 186(1), 59-72.
[3] Happel, D. (1988). Triangulated categories of Cohen-Macaulay modules. Cambridge University Press.

Glossary

Quiver algebra: A type of algebra that is defined by a quiver, which is a directed graph.
Koszul algebra: A type of algebra that has a special property in its minimal projective resolutions.
Global dimension: A property of an algebra that measures the complexity of its projective resolutions.
Simple module: A module that has no proper submodules.
Syzygy: A module that is the kernel of a homomorphism between two other modules.
Path algebra: A type of algebra that is constructed from a quiver by associating a path algebra to the quiver.
Admissible quadratic relations: A type of ideal that is used to construct a quiver algebra.
Q&A: Testing Koszul for Quiver Algebras of Finite Global Dimension using GAP/QPA ====================================================================================

Q: What is a quiver algebra?

A: A quiver algebra is a type of algebra that is defined by a quiver, which is a directed graph. The quiver algebra is then constructed by associating a path algebra to the quiver and then quotienting it by a two-sided ideal, known as the admissible quadratic relations.

Q: What is the Koszul property?

A: The Koszul property is a property of an algebra that has a special property in its minimal projective resolutions. Specifically, the first syzygy of a simple module is generated in degree one. This property has far-reaching implications in the representation theory of the algebra.

Q: What is the global dimension of an algebra?

A: The global dimension of an algebra is a property that measures the complexity of its projective resolutions. An algebra is said to have finite global dimension if it has a finite projective resolution for every module.

Q: How do I test whether a quiver algebra is Koszul or not?

A: To test whether a quiver algebra is Koszul or not, you can use the KoszulTest function in the GAP/QPA package. This function takes as input a quiver algebra and returns a boolean value indicating whether the algebra is Koszul or not.

Q: What is the `KoszulTest` function?

A: The KoszulTest function is a function in the GAP/QPA package that tests whether a quiver algebra is Koszul or not. It takes as input a quiver algebra and returns a boolean value indicating whether the algebra is Koszul or not.

Q: How do I use the `KoszulTest` function?

A: To use the KoszulTest function, you can simply call it with a quiver algebra as input. For example:

A := PathAlgebra(Q) / I;
KoszulTest(A);

This will return a boolean value indicating whether the algebra is Koszul or not.

Q: What are the admissible quadratic relations?

A: The admissible quadratic relations are a type of ideal that is used to construct a quiver algebra. They are defined as the ideal generated by the quadratic relations of the quiver.

Q: How do I define the admissible quadratic relations?

A: To define the admissible quadratic relations, you can use the Ideal function in the GAP/QPA package. For example:

I := Ideal([ [1, 2 [2, 1] ]);

This will define the admissible quadratic relations as the ideal generated by the quadratic relations of the quiver.

Q: What is the path algebra?

A: The path algebra is a type of algebra that is constructed from a quiver by associating a path algebra to the quiver.

Q: How do I construct the path algebra?

A: To construct the path algebra, you can use the PathAlgebra function in the GAP/QPA package. For example:

A := PathAlgebra(Q);

This will construct the path algebra from the quiver.

Q: What is the representation theory of an algebra?

A: The representation theory of an algebra is the study of the modules of the algebra. It is a fundamental area of mathematics that has many applications in physics, computer science, and other fields.

Q: How do I study the representation theory of an algebra?

A: To study the representation theory of an algebra, you can use the GAP/QPA package to compute various properties of the algebra, such as its global dimension and Koszul property. You can also use the package to compute the modules of the algebra and study their properties.

Q: What are the applications of the representation theory of an algebra?

A: The representation theory of an algebra has many applications in physics, computer science, and other fields. For example, it is used in the study of quantum field theory, condensed matter physics, and computer networks.

Q: How do I get started with the GAP/QPA package?

A: To get started with the GAP/QPA package, you can download the package from the GAP website and follow the instructions for installation. You can also consult the GAP/QPA manual for more information on how to use the package.

Q: What are the system requirements for the GAP/QPA package?

A: The GAP/QPA package requires a computer with a 64-bit operating system and at least 4 GB of RAM. It also requires the GAP software to be installed on the computer.

Q: How do I report bugs or issues with the GAP/QPA package?

A: To report bugs or issues with the GAP/QPA package, you can contact the GAP support team or submit a bug report on the GAP website.