The Picard group of a structural matrix algebra
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.
scientific article; zbMATH DE number 1420032
scientific article