Representations of trivial extensions of hereditary algebras

Tachikawa, Hiroyuki

doi:10.1007/bfb0088483

Cited by 42 publications

(30 citation statements)

References 7 publications

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“…The shapes of the connected components of the stable Auslander-Reiten quivers of the tame infinitesimal groups of height ≤ 1 and the tame semisimple infinitesimal groups of characteristic p ≥ 3 were described in [17, (7.2) (H, k). Thus, statements (1) and (2) follow from [36, (3.6)] and [42]. Moreover, for n ≥ 1 and r = 1, statement (3) follows from [17, (7.2)].…”

Section: Given M N ∈ Mod(g χ) the Hopf Algebra H(m(g)) Acts Triviamentioning

confidence: 97%

“…Since n ≥ 1, the bound quiver (∆ r , J r,n ) has infinitely many primitive walks (primitive V -sequences in the sense of [43, (2. 1 (withÃ 1 =Ã 12 ). By results of Tachikawa [42], the latter algebra is 2-parametric, so that H(G) is domestic with µ H(G) = 2. …”

Section: Given M N ∈ Mod(g χ) the Hopf Algebra H(m(g)) Acts Triviamentioning

confidence: 99%

“…Alternatively, we have G/M(G) ∼ = Q [r] , and then the proof of Theorem 6.1 shows that H(G) is a domestic algebra whose tame blocks are Morita equivalent to N 2 (r, 0). Since N 2 (r, 0) is the trivial extension H D(H) of a hereditary algebra H of Euclidean typeÃ 2p r−1 −1 , an application of [24, (3.9)] and [42] gives KG(N 2 (r, 0)) = 2. Since all other blocks are representation-finite algebras, H(G) has Krull-Gabriel dimension 2.…”

Section: ) H(g) Is Domestic With µ H(g) = 2 (4) H(g) Is Tame and γ mentioning

confidence: 99%

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Galois actions and blocks of tame infinitesimal group schemes

Farnsteiner

Skowroński

2007

Trans. Amer. Math. Soc.

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Abstract. Given an infinitesimal group G, that is defined over an algebraically closed field of characteristic p ≥ 3, we determine the block structure of the algebra of measures H(G) in case its principal block B 0 (G) is tame and the height of the factor group G/M(G) of G by its multiplicative center M(G) is at least two. Our results yield a complete description of the stable Auslander-Reiten quiver of H(G) along with a criterion for the domesticity of H(G).

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Section: Given M N ∈ Mod(g χ) the Hopf Algebra H(m(g)) Acts Triviamentioning

confidence: 97%

Section: Given M N ∈ Mod(g χ) the Hopf Algebra H(m(g)) Acts Triviamentioning

confidence: 99%

Section: ) H(g) Is Domestic With µ H(g) = 2 (4) H(g) Is Tame and γ mentioning

confidence: 99%

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Galois actions and blocks of tame infinitesimal group schemes

Farnsteiner

Skowroński

2007

Trans. Amer. Math. Soc.

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“…The algebra B 0 (SL(2) 1 T r ) is Morita equivalent to the trivial extension of tame hereditary radical square zero algebra of typeÃ 2p r−1 −1 . By work of Tachikawa [45], such algebras are 2-parametric, that is, there are in each dimension at most two continuous 1-parameter families of indecomposable modules.…”

Section: Domestic Infinitesimal Groupsmentioning

confidence: 99%

Group-Graded Algebras, Extensions of Infinitesimal Groups, and Applications

Farnsteiner

2009

Transformation Groups

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Abstract. Using results on algebras that are graded by p-groups, we study representations of infinitesimal groups G that possess a normal subgroup N ¢ G with a diagonalizable factor group G/N . When combined with rank varieties, Auslander-Reiten theory and Premet's work on SL(2)1-modules, these techniques lead to the determination of the indecomposable modules of the infinitesimal groups of domestic representation type.

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“…The representation theory of the trivial extension algebras has been extensively developed (see [2], [3], [5]- [8], [14], [16], [17], [22], [25], [26], [28]- [30], [34]- [36], [38]- [40] for some research in this direction).…”

Section: T (B) = B D(b)mentioning

confidence: 99%