Frobenius-Perron theory of the bound quiver algebras containing loops. (English) Zbl 1537.18016
The spectral radius, also known as the Frobenius-Perron dimension, of a matrix is a fundamental and highly practical property in various mathematical disciplines such as linear algebra, combinatorics, topology, probability, and statistics. Research has shown that the Frobenius-Perron dimension has strong connections with the representation type of a category. To gain a better understanding of the Frobenius-Perron dimension of an endofunctor, Wicks calculated the Frobenius-Perron dimension of the representation category of a modified ADE bounded quiver algebra with arrows in a specific direction. The calculation showed that the Frobenius-Perron dimension of this category was equal to the maximum number of loops at a vertex. Wicks then posed the question of what would happen if the directions of the arrows were changed.
In this paper, the authors examine the Frobenius-Perron theory of the representation categories of bound quiver algebras that have loops. It is known that a bound quiver algebra with loops has infinitely many representations, making it difficult to fully describe the homomorphism spaces (or extension spaces) between objects in the representation category. Hence they focus on these algebras when they satisfy the commutativity condition of loops. To be more specific, let \(A\) be a bound quiver algebras satisfying the commutativity condition of loops and \(B = A/J\) be its quotient algebra where \(J\) is the ideal generated by all the loops. They examine \(B\) as a representation-directed algebra and show that the Frobenius-Perron dimension of \(A\) equals the maximum number of loops at each vertex. As an application, they demonstrate that the Frobenius-Perron dimension of the representation category of a modified ADE bound quiver algebra is also equal to the maximum number of loops at each vertex, regardless of arrow directions selected. This provides a direct answer to the question asked by Wicks. Moreover, they show that there also exists infinite dimensional algebras whose Frobenius-Perron dimension is equal to the maximal number of loops by calculating the Frobenius-Perron dimension of the representation categories of polynomial algebras.
In this paper, the authors examine the Frobenius-Perron theory of the representation categories of bound quiver algebras that have loops. It is known that a bound quiver algebra with loops has infinitely many representations, making it difficult to fully describe the homomorphism spaces (or extension spaces) between objects in the representation category. Hence they focus on these algebras when they satisfy the commutativity condition of loops. To be more specific, let \(A\) be a bound quiver algebras satisfying the commutativity condition of loops and \(B = A/J\) be its quotient algebra where \(J\) is the ideal generated by all the loops. They examine \(B\) as a representation-directed algebra and show that the Frobenius-Perron dimension of \(A\) equals the maximum number of loops at each vertex. As an application, they demonstrate that the Frobenius-Perron dimension of the representation category of a modified ADE bound quiver algebra is also equal to the maximum number of loops at each vertex, regardless of arrow directions selected. This provides a direct answer to the question asked by Wicks. Moreover, they show that there also exists infinite dimensional algebras whose Frobenius-Perron dimension is equal to the maximal number of loops by calculating the Frobenius-Perron dimension of the representation categories of polynomial algebras.
Reviewer: Payam Bahiraei (Rasht)
MSC:
| 18G80 | Derived categories, triangulated categories |
| 16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |
| 16E10 | Homological dimension in associative algebras |
| 16E35 | Derived categories and associative algebras |
Keywords:
bound quiver algebras containing loops; canonical algebra; Frobenius-Perron dimension; representation-directed algebraReferences:
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