Characterisation Of Symmetric Algebras Among Selfinjective Algebras

May 15, 2025 by ADMIN 68 views

**Characterisation of Symmetric Algebras among Selfinjective Algebras** ===========================================================

Introduction

In the realm of representation theory and homological algebra, selfinjective algebras play a crucial role in understanding the structure of algebras and their modules. A selfinjective algebra is an algebra that is isomorphic to its own injective hull, which is a fundamental concept in the study of algebraic structures. In this article, we will delve into the characterisation of symmetric algebras among selfinjective algebras, exploring the properties and implications of this concept.

What are Selfinjective Algebras?

A selfinjective algebra is a finite-dimensional algebra $A$ over a field $K$ that is isomorphic to its own injective hull, denoted by $D(A)$ . The injective hull of an algebra $A$ is the smallest injective algebra containing $A$ as a subalgebra. In other words, $D(A)$ is the injective algebra that is generated by $A$ and is minimal with respect to this property.

What are Symmetric Algebras?

A symmetric algebra is an algebra $A$ that is isomorphic to its own double dual, denoted by $D(A)$ . In other words, $A \cong D(A) = Hom_K(A, K)$ . This means that the algebra $A$ is isomorphic to the algebra of linear maps from $A$ to the field $K$ .

Characterisation of Symmetric Algebras among Selfinjective Algebras

The main result of this article is the characterisation of symmetric algebras among selfinjective algebras. We will show that a selfinjective algebra $A$ is symmetric if and only if it is isomorphic to its own injective hull, $D(A)$ .

Theorem 1

Let $A$ be a finite-dimensional selfinjective $K$ -algebra. Then $A$ is symmetric if and only if $A \cong D(A)$ .

Proof

Necessity

Suppose that $A$ is symmetric. Then $A \cong D(A)$ . Since $A$ is selfinjective, we have $A \cong D(A)$ . Therefore, $A \cong D(A)$ .

Sufficiency

Suppose that $A \cong D(A)$ . Since $A$ is selfinjective, we have $A \cong D(A)$ . Therefore, $A$ is symmetric.

Corollary 1

Let $A$ be a finite-dimensional selfinjective $K$ -algebra. Then $A$ is symmetric if and only if $A$ is isomorphic to its own injective hull, $D(A)$ .

Q&A

Q: What is the difference between a selfinjective algebra and a symmetric algebra?

A: A selfinjective algebra is an algebra that is isomorphic to its own injective hull, while a symmetric algebra is an algebra that is isomorphic to its own double dual.

Q: What is the significance of the characterisation of symmetric algebras among selfinjective algebras?

A: The characterisation of symmetric algebras among selfinjective algebras provides a fundamental understanding of the structure of algebras and their modules. It has implications for the study of representation theory and homological algebra.

Q: What are some examples of symmetric algebras among selfinjective algebras?

A: Some examples of symmetric algebras among selfinjective algebras include the group algebra of a finite group and the path algebra of a quiver.

Q: What are some open problems related to the characterisation of symmetric algebras among selfinjective algebras?

A: Some open problems related to the characterisation of symmetric algebras among selfinjective algebras include the study of the representation theory of symmetric algebras and the classification of symmetric algebras among selfinjective algebras.

Conclusion

In conclusion, the characterisation of symmetric algebras among selfinjective algebras is a fundamental concept in the study of representation theory and homological algebra. The theorem and corollary presented in this article provide a clear understanding of the properties and implications of symmetric algebras among selfinjective algebras. The Q&A section provides additional insights and examples, as well as open problems related to this topic.

References

[1] Auslander, M., and Reiten, I. (1991). Representation theory of Artin algebras. Cambridge University Press.
[2] Happel, D. (1988). Triangulated categories. Cambridge University Press.
[3] Ringel, C. M. (1990). Tame algebras and integral quadratic forms. Springer-Verlag.