2022/04/12 by J. M. Chen, Chen, J. M., J. Y. Chen +1
Chemistry · Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Molecular spectroscopy and chirality #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2204.05458
openalex publication_date 2022/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The spectral radius of matrix, also known as Frobenius-Perron dimension, is a useful tool for studying linear algebras and plays an important role in the classification of the representation categories of algebras. In this paper, we study the Frobenius-Perron theory of the representation categories of bound quiver algebras containing loops, find a way to calculate the Frobenius-Perron dimension of these algebras when they satisfy the commutativity condition of loops. As an application, we prove that the Frobenius-Perron dimension of the representation category of a modified ADE bounded quiver algebra is equal to the maximum number of loops at a vertex. Finally, we point out that there also exists infinite dimensional algebras whose Frobenius-Perron dimension is equal to the maximal number of loops by giving an example.