Cellular algebras and Cartan matrices.
This paper studies the cellular algebras with respect to some restriction on their Cartan matrices. The most strong restriction happens when the Cartan matrix has determinant one and all of its eigenvalues are integers. The authors prove that in this case the algebra is semisimple. Inspired by this result the authors study the cellular algebras whose Cartan matrices have the property of having all eigenvalues integers and having determinant \(p\) or \(p^2\) for a prime number \(p\). They show equality between the number of such algebras of dimension \(n\) and the number of partitions of \(n\) whose parts are coprime. They also give a list of all cellular algebras with the property above whose determinant \(d\) is such that \(2\leq d\leq 7\). In section 6, they have a theorem which gives a complete classification of the cellular Nakayama algebras. They also study, in general, the behavior of the property of having all eigenvalues integers for the Cartan matrix. In the last section they show that if an algebra \(A\) is cellular with respect to an involution \(i\) then the trivial extension \(A\oplus D(A)\) is cellular with respect to the involution \(a+f\to i(a)+i\circ f\). They also show that if the trivial extension of a non-simple indecomposable cellular algebra has the integral property then the global dimension of the original algebra is infinite.
scientific article; zbMATH DE number 1915501
scientific article