Algebras of finite context type. I: Faithful representations and spectral Jordan canonical forms (Q1922906)

Fix an algebraically closed field \(K\) and consider a finite dimensional \(K\)-algebra \(A\). All the modules considered are left finitely generated. One can retrieve the algebra \(A\) from any faithful \(A\)-module. The author investigates properties of faithful modules in order to find data dependent on the algebra \(A\) rather than on a particular module. An important role is played by Jordan modules, that is, modules which are faithful direct sums of local uniform modules. It is proved that given an arbitrary finite dimensional algebra \(A\) there exists a Jordan module and the minimal number of direct summands in a Jordan module equals the \(K\)-dimension of the two-sided socle \(S\) of \(A\). The main part of the paper is devoted to algebras of finite context type, that is, the algebras having a faithful module with finite submodule lattice. From now on assume that \(A\) is such an algebra. The author proves that if \(M=\bigoplus_i M_i\), \(N=\bigoplus_i N_i\) are Jordan modules which are canonical (that is, they have the minimal number of local uniform summands, each of them with finite submodule lattice) then (after a suitable renumeration of \(N_i\)'s) \(M_i\) and \(N_i\) are isomorphic \(A\)-contexts for any \(i\), which implies in particular that they have isomorphic submodule lattices. However this does not mean that \(M\) and \(N\) are isomorphic \(A\)-modules. Finally to a faithful \(A\)-module \(M\) with a fixed \(K\)-basis the author associates a so called spectral matrix and proves that the spectral matrices of canonical Jordan modules are determined by \(A\) uniquely up to similarity by a diagonal matrix and a permutation matrix. It is also remarked that it is possible to determine effectively if two algebras have the same spectral matrices of canonical Jordan modules which provides an indicator for isomorphism of algebras of finite context type.