Cellular algebras and Cartan matrices.
type: Item
instance of: scholarly article
title: Cellular algebras and Cartan matrices.
zbMATH Open document ID: 1067.16023
DOI: 10.1016/S0024-3795(02)00411-1
author: Dajing Xiang
author: Changchang Xi
published in: Linear Algebra and its Applications
publication date: 2003-05-25T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 16E40
zbMATH DE Number: 1915501
zbMATH Keywords: cellular algebras
zbMATH Keywords: Cartan matrices
zbMATH Keywords: eigenvalues
zbMATH Keywords: semisimple algebras
reviewed by: Eduardo N. Marcos
MaRDI profile type: Publication
cites work: Identities in combinatorics. III: Further aspects of ordered set sorting
cites work: Q4101884
cites work: Q4318498
cites work: Q3943082
cites work: On Quasihereditary Rings
cites work: Q3804882
cites work: Q4693152
cites work: Cellular algebras
cites work: Matrix Analysis
cites work: Q4217072
cites work: On the number of cells of a cellular algebra
cites work: Q3222981
cites work: The structure of the partition algebras
cites work: On an algebraic approach to higher dimensional statistical mechanics
cites work: QUANTUM INVARIANTS OF 3-MANIFOLDS ASSOCIATED WITH CLASSICAL SIMPLE LIE ALGEBRAS
cites work: Semisimple and modular categories from link invariants
cites work: The representation theory of the Temperley-Lieb algebras
cites work: On the quasi-heredity of Birman-Wenzl algebras
cites work: Q4700357
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description: scientific article; zbMATH DE number 1915501