  The aim of this paper is mainly to build a new representation-theoretic
realization of finite root systems through the so-called Frobenius-type
triangular matrix algebras by the method of reflection functors over any field.
Finally, we give an analog of APR-tilting module for this class of algebras.
  The major conclusions contains the known results as special cases, e.g. that
for path algebras over an algebraically closed field and for path algebras with
relations from symmetrizable cartan matrices. Meanwhile, it means the
corresponding results for some other important classes of algebras, that is,
the path algebras of quivers over Frobenius algebras and the generalized path
algebras endowed by Frobenius algebras at vertices.
