Tilting theory
which make use of the analogous transformations which we like to think of as a change of basis for a fixed root- system - a tilting of the axes relative to the roots which results in a different subset of roots lying in the positive cone. ... For this reason, and because the word
In mathematics, specifically representation theory, tilting theory describes a way to relate the module categories of two algebras using so-called tilting modules and associated tilting functors. Here, the second algebra is the endomorphism algebra of a tilting module over the first algebra.
Tilting theory was motivated by the introduction of reflection functors by Bernšteĭn, Gelfand & Ponomarev (1973); these functors were used to relate representations of two quivers. These functors were reformulated by Auslander, Platzeck & Reiten (1979), and generalized by Brenner & Butler (1980) who introduced tilting functors. Happel & Ringel (1982) defined tilted algebras and tilting modules as further generalizations of this.
Suppose that A is a finite-dimensional unital associative algebra over some field. A finitely-generated right A-module T is called a tilting module if it has the following three properties:
Given such a tilting module, we define the endomorphism algebra B = EndA(T). This is another finite-dimensional algebra, and T is a finitely-generated left B-module. The tilting functors HomA(T,−), Ext1
A(T,−), −⊗BT and TorB
1(−,T) relate the category mod-A of finitely-generated right A-modules to the category mod-B of finitely-generated right B-modules.
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