Noncommutative curves of genus zero: related to finite dimensional algebras

Kussin, Dirk

doi:10.1090/memo/0942

Cited by 22 publications

(98 citation statements)

References 51 publications

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“…Indeed, we have the following commutative diagram: [19], the results of our paper extend to the case of an arbitrary base field as long as canonical algebras in the sense of Ringel [24] or weighted projective lines are concerned. In the more general situation, dealing with canonical algebras in the sense of [25], equivalently with categories of coherent sheaves on exceptional curves [16], the uniqueness result Proposition 6.3 does not carry over by [14], affecting our proofs of Theorem 6.4 and subsequent results (Theorem 7.1, Theorem 7.5).…”

Section: Proposition 74 Assume H Is Tubular Then There Is An Exactmentioning

confidence: 99%

The cluster category of a canonical algebra

Barot

Kussin

Lenzing

2010

Trans. Amer. Math. Soc.

109

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Abstract. We study the cluster category of a canonical algebra A in terms of the hereditary category of coherent sheaves over the corresponding weighted projective line X. As an application we determine the automorphism group of the cluster category and show that the cluster-tilting objects form a cluster structure in the sense of Buan, Iyama, Reiten and Scott. The tilting graph of the sheaf category always coincides with the tilting or exchange graph of the cluster category. We show that this graph is connected if the Euler characteristic of X is non-negative, or equivalently, if A is of tame (domestic or tubular) representation type.

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Section: Proposition 74 Assume H Is Tubular Then There Is An Exactmentioning

confidence: 99%

The cluster category of a canonical algebra

Barot

Kussin

Lenzing

2010

Trans. Amer. Math. Soc.

109

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“…Good, explicit knowledge of the ghosts, the members of the ghost group, combined with the combinatorial methods, is therefore important for exploring categories of finite dimensional modules. Several of the problems posed in [47] will be solved. Concerning the ghost group our main result is the following.…”

Section: S(h) = [K(h) : Z(k(h))]mentioning

confidence: 99%

“…Let H be of genus zero. Assume that there is a point x such that the tubular shift σ x is efficient in the sense of [47]. Then the ghost group G(H) is finite, generated by Picard-shifts σ x −d(y) • σ y , which are of order e τ (y), where y runs through the points y = x with e τ (y) > 1.…”

Section: S(h) = [K(h) : Z(k(h))]mentioning

confidence: 99%

“…For the generalization of Picard-shifts with respect to non-homogeneous tubes we refer to [55] and [47].…”

Section: (τ -Multiplicity)mentioning

confidence: 99%

“…If we have p(x) = 1 for all x, then we call H non-weighted (or homogeneous [47]); this can be also expressed as follows (NC 6') Ext 1 (S, S) = 0 (equivalently: τ S S) for each simple object S.…”

Section: Introductionmentioning

confidence: 99%

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Weighted noncommutative regular projective curves

Kussin¹

2017

J. Noncommut. Geom.

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Abstract. Let H be a noncommutative regular projective curve over a perfect field k. We study global and local properties of the Auslander-Reiten translation τ and give an explicit description of the complete local rings, with the involvement of τ . We introduce the τ -multiplicity eτ (x), the order of τ as a functor restricted to the tube concentrated in x. We obtain a local-global principle for the (global) skewness s(H), defined as the square root of the dimension of the function (skew-) field over its centre. In the case of genus zero we show how the ghost group, that is, the group of automorphisms of H which fix all objects, is determined by the points x with eτ (x) > 1. Based on work of Witt we describe the noncommutative regular (smooth) projective curves over the real numbers; those with s(H) = 2 we call Witt curves. In particular, we study noncommutative elliptic curves, and present an elliptic Witt curve which is a noncommutative Fourier-Mukai partner of the Klein bottle. If H is weighted, our main result will be formulae for the orbifold Euler characteristic, involving the weights and the τ -multiplicities. As an application we will classify the noncommutative 2-orbifolds of nonnegative Euler characteristic, that is, the real elliptic, domestic and tubular curves. Throughout, many explicit examples are discussed.

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Higher S-dualities and Shephard-Todd groups

Cecotti

Zotto

2015

J. High Energ. Phys.

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Seiberg and Witten have shown that in N = 2 SQCD with N f = 2N c = 4 the S-duality group PSL(2, Z) acts on the flavor charges, which are weights of Spin (8), by triality. There are other N = 2 SCFTs in which SU(2) SYM is coupled to stronglyinteracting non-Lagrangian matter: their matter charges are weights of E 6 , E 7 and E 8 instead of Spin(8). The S-duality group PSL(2, Z) acts on these weights: what replaces Spin(8) triality for the E 6 , E 7 , E 8 root lattices?In this paper we answer the question. The action on the matter charges of (a finite central extension of) PSL(2, Z) factorizes trough the action of the exceptional ShephardTodd groups G 4 and G 8 which should be seen as complex analogs of the usual triality group S 3 ≃ Weyl(A 2 ). Our analysis is based on the identification of S-duality for SU(2) gauge SCFTs with the group of automorphisms of the cluster category of weighted projective lines of tubular type.

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Noncommutative curves of genus zero: related to finite dimensional algebras

Cited by 22 publications

References 51 publications

The cluster category of a canonical algebra

The cluster category of a canonical algebra

Weighted noncommutative regular projective curves

Higher S-dualities and Shephard-Todd groups

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