The Green rings of Taft algebras

Chen, Huixiang; Oystaeyen, Freddy Van; Zhang, Yinhuo

doi:10.1090/s0002-9939-2013-11823-x

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Introduction2

Proof 1) From the Assumption One Gets1

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Cited by 55 publications

(76 citation statements)

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“…Let L in (M R ) denote the set of right R-submodules which are indecomposable. The first assertion of the following result was already proved for Taft algebras in [CGL,Section 2], see also [CVZ,Theorem 2.5].…”

Section: Proof 1) From the Assumption One Getsmentioning

confidence: 67%

On Indecomposable Ideals Over Some Algebras

Ardizzoni

Stumbo

2019

Algebr Represent Theor

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Section: Proof 1) From the Assumption One Getsmentioning

confidence: 67%

On Indecomposable Ideals Over Some Algebras

Ardizzoni

Stumbo

2019

Algebr Represent Theor

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“…Let J be the ideal of Z[X]♯Ĝ generated by the following subset By Corollary 4.14 (2), the image of the above set under φ is a Z-basis of r(W). This implies that φ is an injection, and so it is a ring isomorphism.…”

Section: 1mentioning

confidence: 99%

Green ring of the category of weight modules over the Hopf–Ore extensions of group algebras

Sun

Chen

2019

Communications in Algebra

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In this paper, we continue our study of the tensor product structure of category W of weight modules over the Hopf-Ore extensions kG(χ −1 , a, 0) of group algebras kG, where k is an algebraically closed field of characteristic zero. We first describe the tensor product decomposition rules for all indecomposable weight modules under the assumption that the orders of χ and χ(a) are different. Then we describe the Green ring r(W) of the tensor category W. It is shown that r(W) is isomorphic to the polynomial algebra over the group ring ZĜ in one variable when |χ(a)| = |χ| = ∞, and that r(W) is isomorphic to the quotient ring of the polynomial algebra over the group ring ZĜ in two variables modulo a principle ideal when |χ(a)| < |χ| = ∞. When |χ(a)| |χ| < ∞, r(W) is isomorphic to the quotient ring of a skew group ring Z[X]♯Ĝ modulo some ideal, where Z[X] is a polynomial algebra over Z in infinitely many variables.

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“…The study of representation rings (or Green rings) of Hopf algebras has been revived recently. In [11] and [16], the authors investigated respectively the representation rings of Taft algebras and generalized Taft algebras based on the decomposition rules of tensor product modules given by Cibils [12]. The representation ring r(H n (q)) of a Taft algebra H n (q) is isomorphic to a polynomial ring in two variables modulo two relations.…”

Section: Introductionmentioning

confidence: 99%