The Picard group of a structural matrix algebra
type: Item
instance of: scholarly article
title: The Picard group of a structural matrix algebra
zbMATH Open document ID: 0973.16012
DOI: 10.1016/S0024-3795(99)00200-1
author: Jeremy Haefner
author: Trae Holcomb
published in: Linear Algebra and its Applications
publication date: 2001-08-29T00:00:00Z
review text: Let \(K\) be a field, \(P\) a finite preordered set, and \(A=I(P)\) the incidence algebra for \(P\) over \(K\). \textit{S.~P.~ Coelho} [Linear Algebra Appl. 195, 35-58 (1993; Zbl 0810.16027)] completely determined the structure of the group \(\Aut_K(A)\) of \(K\)-automorphisms (those that fix \(K\) elementwise) of \(A\). The authors consider this problem from a broader perspective by looking at the Picard group of \(A\), \(\text{Pic}(A)\), formed by the bimodule isomorphism classes of invertible \(A\)-bimodules, with multiplication given by the tensor product. The connection with Coelho's paper comes from the fact that there exists a natural group homomorphism from the complete automorphism group \(\Aut(A)\) to \(\text{Pic}(A)\) whose kernel \(\text{Inn}(A)\) is the group of inner automorphisms of \(A\), so that the group of outer automorphisms \(\text{Out}(A)=\Aut(A)/\text{Inn}(A)\) can be viewed as a subgroup of \(\text{Pic}(A)\). Then, the goal of the paper is to determine the structure of \(\text{Pic}(A)\). The authors establish a connection between \(\text{Out}(A)\) and the group of outer automorphisms of the basic algebra \(\widetilde A\), which is an incidence algebra of the partially ordered set \(\widetilde P\) associated to \(P\). Among other results, the authors obtain necessary and sufficient conditions for \(\text{Out}(A)\) to be a natural invariant for the Morita equivalence class of \(A\), and also necessary and sufficient conditions for the ring \(M_n(K)\) to be strongly graded by a group \(G\) and coefficient ring \(A\) containing \(n\) primitive, orthogonal idempotents.
Mathematics Subject Classification ID: 16G30
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16S35
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16W50
zbMATH DE Number: 1420032
zbMATH Keywords: incidence algebras
zbMATH Keywords: structural matrix algebras
zbMATH Keywords: inner automorphisms
zbMATH Keywords: outer automorphisms
zbMATH Keywords: Picard groups
zbMATH Keywords: Morita equivalences
zbMATH Keywords: group graded rings
reviewed by: José L. Gómez-Pardo
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1420032