We show that the morphism Ω from the ıquantum loop algebra Dr U ı (Lg) of split type to the ıHall... more We show that the morphism Ω from the ıquantum loop algebra Dr U ı (Lg) of split type to the ıHall algebra of the weighted projective line is injective if g is of finite or affine type. As a byproduct, we use the whole ıHall algebra of the cyclic quiver C n to realise the ıquantum loop algebra of affine gl n .
The ıHall algebra of a weighted projective line is defined to be the semi-derived Ringel-Hall alg... more The ıHall algebra of a weighted projective line is defined to be the semi-derived Ringel-Hall algebra of the category of 1-periodic complexes of coherent sheaves on the weighted projective line over a finite field. We show that this Hall algebra provides a realization of the ıquantum loop algebra, which is a generalization of the ıquantum group arising from the quantum symmetric pair of split affine type ADE in its Drinfeld type presentation. The ıHall algebra of the ıquiver algebra of split affine type A was known earlier to realize the same algebra in its Serre presentation. We then establish a derived equivalence which induces an isomorphism of these two ıHall algebras, explaining the isomorphism of the ıquantum group of split affine type A under the two presentations. Contents 147 8. Relations between star point and [i, 1], I 152 9. Relations between star point and [i, 1], II 156 10. Derived equivalence and two presentations of affine type A 159 Appendix A. Proofs of results in Section 8 162 Appendix B. Proofs of results in Section 9 170 Acknowledgments 186 References 186
The present paper focuses on the study of the stable category of vector bundles for the weighted ... more The present paper focuses on the study of the stable category of vector bundles for the weighted projective lines of weight triple. We find some important triangles in this category and use them to construct tilting objects with tubular endomorphism algebras for the case of genus one via cluster tilting theory.
We investigate group actions on the category of coherent sheaves over weighted projective lines. ... more We investigate group actions on the category of coherent sheaves over weighted projective lines. We show that the equivariant category with respect to certain finite group action is equivalent to the category of coherent sheaves over a weighted projective curve, where the genus of the underlying smooth projective curve and the weight data are given by explicit formulas. Moreover, we obtain a trichotomy result for all such equivariant relations. This equivariant approach provides an effective way to investigate smooth projective curves via weighted projective lines. As an application, we give a description for the category of coherent sheaves over a hyperelliptic curve of arbitrary genus. Links to smooth projective curves of genus 2 and also to Arnold's exceptional unimodal singularities are also established.
This present paper is devoted to the study of a class of Nakayama algebras Nn(r) given by the pat... more This present paper is devoted to the study of a class of Nakayama algebras Nn(r) given by the path algebra of the equioriented quiver An subject to the nilpotency degree r for each sequence of r consecutive arrows. We show that the Nakayama algebras Nn(r) for certain pairs (n, r) can be realized as endomorphism algebras of tilting objects in the bounded derived category of coherent sheaves over a weighted projective line, or in its stable category of vector bundles. Moreover, we classify all the Nakayama algebras Nn(r) of Fuchsian type, that is, derived equivalent to the bounded derived categories of extended canonical algebras. We also provide a new way to prove the classiőcation result on Nakayama algebras of piecewise hereditary type, which have been done by Happelś Seidel before.
This paper classifies all the tilting bundles in the category of coherent sheaves on the weighted... more This paper classifies all the tilting bundles in the category of coherent sheaves on the weighted projective line of weight type (2, 2, n), and investigates the abelianness of the "missing part" from the category of coherent sheaves to the category of finitely generated right modules on the associated tilted algebra for each tilting bundle.
Let coh X be the category of coherent sheaves over a weighted projective line X and let D b (coh ... more Let coh X be the category of coherent sheaves over a weighted projective line X and let D b (coh X) be its bounded derived category. The present paper focuses on the study of the right and left mutation functors arising in D b (coh X) attached to certain line bundles. As applications, we first show that these mutation functors give rise to simple reflections for the Weyl group of the star shaped quiver Q associated with X. By further dealing with the Ringel-Hall algebra of X, we show that these functors provide a realization for Tits' automorphisms of the Kac-Moody algebra gQ associated with Q, as well as for Lusztig's symmetries of the quantum enveloping algebra of gQ.
Using cluster tilting theory, we investigate tilting objects in the stable category of vector bun... more Using cluster tilting theory, we investigate tilting objects in the stable category of vector bundles on a weighted projective line of weight type (2, 2, 2, 2). More precisely, a tilting object consisting of rank-two bundles is constructed via cluster tilting mutation. Moreover, the cluster tilting approach also provides a new method to classify the endomorphism algebras of tilting objects in the category of coherent sheaves and the associated bounded derived category.
In this paper, we construct recollements and ladders for exceptional curves by using reduction/in... more In this paper, we construct recollements and ladders for exceptional curves by using reduction/insertion functors due to p-cycle construction. As applications to weighted projective lines, we classify recollements for the category of coherent sheaves over a weighted projective line, and give an explicit description of ladders in two different levels: the bounded derived category of coherent sheaves and the stable category of vector bundles. Theorem 1.1. (See Theorem 4.1) Up to equivalences, each nontrivial recollement of coh X(p, λ) has the following form for some sequences q and q: mod A(q) / / coh X(p, λ) ψ q / / v v h h coh X(p − q, λ).
In the present paper we prove that Hall polynomial exists for each triple of decomposition sequen... more In the present paper we prove that Hall polynomial exists for each triple of decomposition sequences which parameterize isomorphism classes of coherent sheaves of a domestic weighted projective line X over finite fields. These polynomials are then used to define the generic Ringel-Hall algebra of X as well as its Drinfeld double. Combining this construction with a result of Cramer, we show that Hall polynomials exist for tame quivers, which not only refines a result of Hubery, but also confirms a conjecture of Berenstein and Greenstein.
We use semi-derived Ringel-Hall algebras of quivers with loops to realize the whole quantum Borch... more We use semi-derived Ringel-Hall algebras of quivers with loops to realize the whole quantum Borcherds-Bozec algebras and quantum generalized Kac-Moody algebras.
International Mathematics Research Notices, Feb 4, 2023
The string group acts on the category of coherent sheaves over a weighted projective line by degr... more The string group acts on the category of coherent sheaves over a weighted projective line by degree-shift actions. We study the equivariant equivalence relations induced by degree-shift actions between weighted projective lines. We prove that such an equivariant equivalence is characterized by an admissible homomorphism between the associated string groups. We classify all these equivariant equivalences for the weighted projective lines of domestic and tubular types.
The ıHall algebra of the projective line is by definition the twisted semi-derived Ringel-Hall al... more The ıHall algebra of the projective line is by definition the twisted semi-derived Ringel-Hall algebra of the category of 1-periodic complexes of coherent sheaves on the projective line. This ıHall algebra is shown to realize the universal q-Onsager algebra (i.e., ıquantum group of split affine A 1 type) in its Drinfeld type presentation. The ıHall algebra of the Kronecker quiver was known earlier to realize the same algebra in its Serre type presentation. We then establish a derived equivalence which induces an isomorphism of these two ıHall algebras, explaining the isomorphism of the q-Onsager algebra under the two presentations. Contents 1. Introduction 1 2. ıHall algebra of the projective line 4 3. The q-Onsager algebra and its current presentation 11 4. ıHall algebra and q-Onsager algebra 13 5. Derived equivalence and two presentations 22 6. A Hall algebra realization of imaginary root vectors 27 References 29
In the present paper we study the derived Hall algebra for the bounded derived category of the ni... more In the present paper we study the derived Hall algebra for the bounded derived category of the nilpotent representations of a tame quiver over a finite field. We show that for any three given objects in the bounded derived category, the associated derived Hall numbers are given by a rational function in the cardinalities of ground fields.
Let A be a finitary hereditary abelian category and D(A) be its reduced Drinfeld double Hall alge... more Let A be a finitary hereditary abelian category and D(A) be its reduced Drinfeld double Hall algebra. By giving explicit formulas in D(A) for left and right mutations, we show that the subalgebras of D(A) generated by exceptional sequences are invariant under mutation equivalences. As an application, we obtain that if A is the category of finite dimensional modules over a finite dimensional hereditary algebra, or the category of coherent sheaves on a weighted projective line, the double composition algebra of A is generated by any complete exceptional sequence. Moreover, for the Lie algebra case, we also have paralleled results.
Les Annales de l'institut Fourier sont membres du Centre Mersenne pour l'édition scienti que ouve... more Les Annales de l'institut Fourier sont membres du Centre Mersenne pour l'édition scienti que ouverte www.centre-mersenne.org
In this paper, weinvestigate tilting objects and cluster-tilting objects in the categories associ... more In this paper, weinvestigate tilting objects and cluster-tilting objects in the categories associated with weighted projective lines. For the case of genus one, tilting objects in the stable category of vector bundles are proved to be precisely cluster-tilting objects in the associated cluster category. Particularly, cluster-tilting objects in the associated cluster category are induced by tilting sheaves in the category of coherent sheaves.
We show that the morphism Ω from the ıquantum loop algebra Dr U ı (Lg) of split type to the ıHall... more We show that the morphism Ω from the ıquantum loop algebra Dr U ı (Lg) of split type to the ıHall algebra of the weighted projective line is injective if g is of finite or affine type. As a byproduct, we use the whole ıHall algebra of the cyclic quiver C n to realise the ıquantum loop algebra of affine gl n .
The ıHall algebra of a weighted projective line is defined to be the semi-derived Ringel-Hall alg... more The ıHall algebra of a weighted projective line is defined to be the semi-derived Ringel-Hall algebra of the category of 1-periodic complexes of coherent sheaves on the weighted projective line over a finite field. We show that this Hall algebra provides a realization of the ıquantum loop algebra, which is a generalization of the ıquantum group arising from the quantum symmetric pair of split affine type ADE in its Drinfeld type presentation. The ıHall algebra of the ıquiver algebra of split affine type A was known earlier to realize the same algebra in its Serre presentation. We then establish a derived equivalence which induces an isomorphism of these two ıHall algebras, explaining the isomorphism of the ıquantum group of split affine type A under the two presentations. Contents 147 8. Relations between star point and [i, 1], I 152 9. Relations between star point and [i, 1], II 156 10. Derived equivalence and two presentations of affine type A 159 Appendix A. Proofs of results in Section 8 162 Appendix B. Proofs of results in Section 9 170 Acknowledgments 186 References 186
The present paper focuses on the study of the stable category of vector bundles for the weighted ... more The present paper focuses on the study of the stable category of vector bundles for the weighted projective lines of weight triple. We find some important triangles in this category and use them to construct tilting objects with tubular endomorphism algebras for the case of genus one via cluster tilting theory.
We investigate group actions on the category of coherent sheaves over weighted projective lines. ... more We investigate group actions on the category of coherent sheaves over weighted projective lines. We show that the equivariant category with respect to certain finite group action is equivalent to the category of coherent sheaves over a weighted projective curve, where the genus of the underlying smooth projective curve and the weight data are given by explicit formulas. Moreover, we obtain a trichotomy result for all such equivariant relations. This equivariant approach provides an effective way to investigate smooth projective curves via weighted projective lines. As an application, we give a description for the category of coherent sheaves over a hyperelliptic curve of arbitrary genus. Links to smooth projective curves of genus 2 and also to Arnold's exceptional unimodal singularities are also established.
This present paper is devoted to the study of a class of Nakayama algebras Nn(r) given by the pat... more This present paper is devoted to the study of a class of Nakayama algebras Nn(r) given by the path algebra of the equioriented quiver An subject to the nilpotency degree r for each sequence of r consecutive arrows. We show that the Nakayama algebras Nn(r) for certain pairs (n, r) can be realized as endomorphism algebras of tilting objects in the bounded derived category of coherent sheaves over a weighted projective line, or in its stable category of vector bundles. Moreover, we classify all the Nakayama algebras Nn(r) of Fuchsian type, that is, derived equivalent to the bounded derived categories of extended canonical algebras. We also provide a new way to prove the classiőcation result on Nakayama algebras of piecewise hereditary type, which have been done by Happelś Seidel before.
This paper classifies all the tilting bundles in the category of coherent sheaves on the weighted... more This paper classifies all the tilting bundles in the category of coherent sheaves on the weighted projective line of weight type (2, 2, n), and investigates the abelianness of the "missing part" from the category of coherent sheaves to the category of finitely generated right modules on the associated tilted algebra for each tilting bundle.
Let coh X be the category of coherent sheaves over a weighted projective line X and let D b (coh ... more Let coh X be the category of coherent sheaves over a weighted projective line X and let D b (coh X) be its bounded derived category. The present paper focuses on the study of the right and left mutation functors arising in D b (coh X) attached to certain line bundles. As applications, we first show that these mutation functors give rise to simple reflections for the Weyl group of the star shaped quiver Q associated with X. By further dealing with the Ringel-Hall algebra of X, we show that these functors provide a realization for Tits' automorphisms of the Kac-Moody algebra gQ associated with Q, as well as for Lusztig's symmetries of the quantum enveloping algebra of gQ.
Using cluster tilting theory, we investigate tilting objects in the stable category of vector bun... more Using cluster tilting theory, we investigate tilting objects in the stable category of vector bundles on a weighted projective line of weight type (2, 2, 2, 2). More precisely, a tilting object consisting of rank-two bundles is constructed via cluster tilting mutation. Moreover, the cluster tilting approach also provides a new method to classify the endomorphism algebras of tilting objects in the category of coherent sheaves and the associated bounded derived category.
In this paper, we construct recollements and ladders for exceptional curves by using reduction/in... more In this paper, we construct recollements and ladders for exceptional curves by using reduction/insertion functors due to p-cycle construction. As applications to weighted projective lines, we classify recollements for the category of coherent sheaves over a weighted projective line, and give an explicit description of ladders in two different levels: the bounded derived category of coherent sheaves and the stable category of vector bundles. Theorem 1.1. (See Theorem 4.1) Up to equivalences, each nontrivial recollement of coh X(p, λ) has the following form for some sequences q and q: mod A(q) / / coh X(p, λ) ψ q / / v v h h coh X(p − q, λ).
In the present paper we prove that Hall polynomial exists for each triple of decomposition sequen... more In the present paper we prove that Hall polynomial exists for each triple of decomposition sequences which parameterize isomorphism classes of coherent sheaves of a domestic weighted projective line X over finite fields. These polynomials are then used to define the generic Ringel-Hall algebra of X as well as its Drinfeld double. Combining this construction with a result of Cramer, we show that Hall polynomials exist for tame quivers, which not only refines a result of Hubery, but also confirms a conjecture of Berenstein and Greenstein.
We use semi-derived Ringel-Hall algebras of quivers with loops to realize the whole quantum Borch... more We use semi-derived Ringel-Hall algebras of quivers with loops to realize the whole quantum Borcherds-Bozec algebras and quantum generalized Kac-Moody algebras.
International Mathematics Research Notices, Feb 4, 2023
The string group acts on the category of coherent sheaves over a weighted projective line by degr... more The string group acts on the category of coherent sheaves over a weighted projective line by degree-shift actions. We study the equivariant equivalence relations induced by degree-shift actions between weighted projective lines. We prove that such an equivariant equivalence is characterized by an admissible homomorphism between the associated string groups. We classify all these equivariant equivalences for the weighted projective lines of domestic and tubular types.
The ıHall algebra of the projective line is by definition the twisted semi-derived Ringel-Hall al... more The ıHall algebra of the projective line is by definition the twisted semi-derived Ringel-Hall algebra of the category of 1-periodic complexes of coherent sheaves on the projective line. This ıHall algebra is shown to realize the universal q-Onsager algebra (i.e., ıquantum group of split affine A 1 type) in its Drinfeld type presentation. The ıHall algebra of the Kronecker quiver was known earlier to realize the same algebra in its Serre type presentation. We then establish a derived equivalence which induces an isomorphism of these two ıHall algebras, explaining the isomorphism of the q-Onsager algebra under the two presentations. Contents 1. Introduction 1 2. ıHall algebra of the projective line 4 3. The q-Onsager algebra and its current presentation 11 4. ıHall algebra and q-Onsager algebra 13 5. Derived equivalence and two presentations 22 6. A Hall algebra realization of imaginary root vectors 27 References 29
In the present paper we study the derived Hall algebra for the bounded derived category of the ni... more In the present paper we study the derived Hall algebra for the bounded derived category of the nilpotent representations of a tame quiver over a finite field. We show that for any three given objects in the bounded derived category, the associated derived Hall numbers are given by a rational function in the cardinalities of ground fields.
Let A be a finitary hereditary abelian category and D(A) be its reduced Drinfeld double Hall alge... more Let A be a finitary hereditary abelian category and D(A) be its reduced Drinfeld double Hall algebra. By giving explicit formulas in D(A) for left and right mutations, we show that the subalgebras of D(A) generated by exceptional sequences are invariant under mutation equivalences. As an application, we obtain that if A is the category of finite dimensional modules over a finite dimensional hereditary algebra, or the category of coherent sheaves on a weighted projective line, the double composition algebra of A is generated by any complete exceptional sequence. Moreover, for the Lie algebra case, we also have paralleled results.
Les Annales de l'institut Fourier sont membres du Centre Mersenne pour l'édition scienti que ouve... more Les Annales de l'institut Fourier sont membres du Centre Mersenne pour l'édition scienti que ouverte www.centre-mersenne.org
In this paper, weinvestigate tilting objects and cluster-tilting objects in the categories associ... more In this paper, weinvestigate tilting objects and cluster-tilting objects in the categories associated with weighted projective lines. For the case of genus one, tilting objects in the stable category of vector bundles are proved to be precisely cluster-tilting objects in the associated cluster category. Particularly, cluster-tilting objects in the associated cluster category are induced by tilting sheaves in the category of coherent sheaves.
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