Cellular algebras and Cartan matrices

Abstract

Let A be a finite dimensional algebra over a field k, and let C be the Cartan matrix of A. Usually, the eigenvalues of C being integers do not imply the semisimplicity of A. However, we prove that a cellular algebra A is semisimple if and only if det(C)=1 and all eigenvalues of C are integers. Moreover, we use Cartan matrices to classify the cellular algebras with the property that the determinant of the Cartan matrix equals a given prime p and all eigenvalues are integers. We also give a classification of cellular Nakayama algebras with integral eigenvalues of their Cartan matrices. Finally, we show that if A is a cellular algebra then its trivial extension T(A) is also a cellular algebra. In particular, if a non-simple connected cellular algebra A is quasi-hereditary, then the Cartan matrix of T(A) has at least one non-integral eigenvalue. The main tool used in this paper is the well-known Perron–Frobenius theory on non-negative matrices.