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- W597856747 abstract "The important notion of reflection functors was introduced into the representation theory of algebras by Bernstein-Gelfand-Ponomarev [8]. Those functors were defined only for hereditary tensor algebras given by quivers and species [12]. Then Auslander-Platzeck-Reiten [7] arranged the notion by non-diagramatic treatment so that it is possible to apply the concept for any algebras. Brenner-Butler [10] extended the Auslander-Platzeck-Reiten partial Coxeter functor and defined the tilting theory. Further, Happel-Ringel [15] generalized the Brenner-Butler tiltingtheory and studied tiltedalgebras. We regard the tiltingtheory as a powerful method of deforming algebras and their module categories. A tiltingfunctor, hower, is nothing but a Morita equivalence, for any self-injectivealgebra. Hence, it is natural to search for a way of applying the tiltingtheory to the study of self-injectivealgebras. Let A be a basic indecomposable artin algebra. Denote by mod-A (resp. Amod) the category of all finitelygenerated right (resp. left) A-modules. Let D: mod-A^yl-mod be the ordinary duality functor. In the following, we shall consider the trivialextension self-injectivealgebra R=AxDA defined as follows: R is A@DA as an additive group and its multiplication is given by (a,q)-(a',qf)= (a-a',a-q'+q-ar) for any (a,q),(a',q')eARDA = R. In the paper [19], Tachikawa started in the study of self-injectivealgebras R, and in [20], he has proved that mod-i? is equivalent to mod-S (S―BxDB) if A is hereditary tensor algebra and B is given by reflection procedure from A. Here mod-R is the projectively(= injectively)stable category of mod-i? in the sense of Auslander. Let ezA be a primitive idempotent such that eA is simple non-injective and r~leA(S)ADA = 0, where t~l(resp. r) denotes the Auslander-Reiten translation TrD (resp. DTr). By putting TA = 0―e)A@T~1eA and B=~End{TA), the AuslanderPlatzeck-Reiten partial Coxeter functor is defined to be the functor Hoiru (T, ?):" @default.
- W597856747 created "2016-06-24" @default.
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- W597856747 date "1985-06-01" @default.
- W597856747 modified "2023-10-18" @default.
- W597856747 title "Partial Coxeter functors and stable equivalences for self-injective algebras" @default.
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- W597856747 doi "https://doi.org/10.21099/tkbjm/1496160201" @default.
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