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APPROXIMATIONS OF ALGEBRAS BY STANDARDLY STRATIFIED ALGEBRAS
 

Summary: APPROXIMATIONS OF ALGEBRAS
BY STANDARDLY STRATIFIED ALGEBRAS
Istv´an ´Agoston1
, Vlastimil Dlab2
and Erzs´ebet Luk´acs1
Abstract. The paper has its origin in an attempt to answer the following question:
Given an arbitrary finite dimensional associative K-algebra A, does there exist a quasi-
hereditary algebra B such that the subcategories of all A-modules and all B-modules,
filtered by the corresponding standard modules are equivalent. Such an algebra will
be called a quasi-hereditary approximation of A. The question is answered in the
appropriate language of standardly stratified algebras: For any K-algebra A, there
is a uniquely defined basic algebra B = (A) such that BB is -filtered and the
subcategories F(A) and F(B) of all -filtered modules are equivalent; similarly
there is a uniquely defined basic algebra C = (A) such that CC is -filtered and
the subcategories F(A) and F(C ) of all -filtered modules are equivalent. These
subcategories play a fundamental role in the theory of stratified algebras. Since, in
general, it is difficult to localize these subcategories in the category of all A-modules,
the construction of (A) and (A) often helps to describe them explicitly. By applying
consecutively the operators and for an algebra, we get a sequence of standardly
stratified algebras which, after a finite number of steps, stabilizes in a properly stratified

  

Source: Ágoston, István - Institute of Mathematics, Eötvös Loránd University

 

Collections: Mathematics