Endomorphism algebras of preprojective partial tilting modules (Q1209038)

Let \(\vec\Delta\) be a connected finite quiver without oriented cycle, \(A=k \vec\Delta\) the corresponding path algebra with \(k\) an algebraically closed field. If \(_ A T\) is a preprojective tilting module, then \(B= \text{End}(_ A T)\) is called a tame (wild) concealed algebra provided \(\vec \Delta\) is a Euclidean (wild) graph. The main result is the following: Let \(A\) be a concealed algebra, \(_ A M\) a preprojective partial tilting module, and \(B= \text{End}(_ A M)\) be connected. Then (1) \(B\) is either a tilted algebra of Dynkin type or a concealed algebra; (2) Let \(M(1),\dots, M(r)\) be all non-isomorphic indecomposable direct summands of \(M\), \(t_{ij}= \dim_ k \Hom_ A (M(i),M(j))\) with \(i,j= 1,\dots,n\). \(C=(t_{ij})_{r\times r}\), \(X(x)= xC^{-T} x^ T\) the corresponding quadratic form, then \(B\) is of Dynkin (tame; wild) type if and only if \(X(x)\) is positive definite (positive semi-definite with radical rank 1; indefinite).