Frobenius-Perron theory of the bound quiver algebras containing loops

J. M. Chen; J. Y. Chen

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arXiv · 2204.05458

Frobenius-Perron theory of the bound quiver algebras containing loops

J. M. Chen·J. Y. Chen

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Abstract

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.

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BibTeX RIS

J. M. Chen, J. Y. Chen. 2022-04-12. Frobenius-Perron theory of the bound quiver algebras containing loops. https://arxiv.org/abs/2204.05458

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