×

zbMATH — the first resource for mathematics

Matrices over upper triangular bimodules and \(\Delta\)-filtered modules over quasi-hereditary algebras. (English) Zbl 0978.16009
Let \(A\) be a finite dimensional \(K\)-algebra over a field \(K\), and let \(M\) be a finite dimensional \(A\)-\(A\)-bimodule. It is well-known that the category \(\text{Mat}(M)\) of matrices in the sense of Yu. Drozd is an additive Krull-Schmidt category and \(\text{Mat}(M)\) is equivalent with the category \(\text{prin}(\Lambda)\) of prinjective modules over the triangular matrix algebra \(\Lambda=\left(\begin{smallmatrix} A&M^*\\ 0&A\end{smallmatrix}\right)\). The authors show that if the algebra \(A\) is directed, the bimodule \(M\) is upper triangular, \(P\) is a projective generator of the category \(\text{Mat}(M)\) and \(R\) is the algebra opposite to the endomorphism algebra \(\text{End}(P)\) of \(P\), then \(R\) is quasi-hereditary and the functor \(\operatorname{Hom}(P,-)\) induced an equivalence between \(\text{Mat}(M)\) and the category \({\mathcal F}(\Delta)\) of \(\Delta\)-filtered \(R\)-modules. The algebra \(A\) is a quotient of a subalgebra of \(R\).
Reviewer: D.Simson (Toruń)

MSC:
16G20 Representations of quivers and partially ordered sets
16E60 Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc.
14L30 Group actions on varieties or schemes (quotients)
16D90 Module categories in associative algebras
16S50 Endomorphism rings; matrix rings
PDF BibTeX XML Cite
Full Text: DOI EuDML