The t-stabilities and t-structures for the Weighted Projective Line of Type (2,1)

This paper presents the theory of holomorphic vector-valued modular forms from a geometric perspective. More precisely, we define certain holomorphic vector bundles on the modular orbifold of generalized elliptic curves whose sections are vector-valued modular forms. This perspective simplifies the theory, and it clarifies the role that exponents of representations of [Formula: see text] play in the holomorphic theory of vector-valued modular forms. Further, it allows one to use standard techniques in algebraic geometry to deduce free-module theorems and dimension formulae (deduced previously by other authors using different techniques), by identifying the modular orbifold with the weighted projective line [Formula: see text].

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Applications of Mutations in the Derived Categories of Weighted Projective Lines to Lie and Quantum Algebras

International Mathematics Research Notices ◽

10.1093/imrn/rny175 ◽

2018 ◽

Vol 2020 (19) ◽

pp. 5814-5871

Author(s):

Bangming Deng ◽

Shiquan Ruan ◽

Jie Xiao

Keyword(s):

Projective Line ◽

Derived Categories ◽

Line Bundles ◽

Hall Algebra ◽

Moody Algebra ◽

Weighted Projective Line ◽

Coherent Sheaves ◽

Enveloping Algebra ◽

The Right ◽

Projective Lines

Abstract Let $\textrm{coh}\ \mathbb{X}$ be the category of coherent sheaves over a weighted projective line $\mathbb{X}$ and let $D^b(\textrm{coh}\ \mathbb{X})$ be its bounded derived category. The present paper focuses on the study of the right and left mutation functors arising in $D^b(\textrm{coh}\ \mathbb{X})$ 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 $\mathbb{X}$. By further dealing with the Ringel–Hall algebra of $\mathbb{X}$, we show that these functors provide a realization for Tits’ automorphisms of the Kac–Moody algebra ${\mathfrak g}_{Q}$ associated with ${Q}$, as well as for Lusztig’s symmetries of the quantum enveloping algebra of ${\mathfrak g}_{Q}$.

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Bounded t-Structures on the Bounded Derived Category of Coherent Sheaves over a Weighted Projective Line

Algebras and Representation Theory ◽

10.1007/s10468-019-09929-w ◽

2019 ◽

Vol 23 (6) ◽

pp. 2167-2235

Author(s):

Chao Sun

Keyword(s):

Projective Line ◽

Derived Category ◽

Weighted Projective Line ◽

Coherent Sheaves

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The composition Hall algebra of a weighted projective line

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle.2012.023 ◽

2013 ◽

Vol 2013 (679) ◽

pp. 75-124 ◽

Cited By ~ 4

Author(s):

Igor Burban ◽

Olivier Schiffmann

Keyword(s):

Lie Algebras ◽

Projective Line ◽

Affine Lie Algebras ◽

Hall Algebra ◽

Weighted Projective Line ◽

Drinfeld Double ◽

Coherent Sheaves ◽

Enveloping Algebras ◽

Quantized Enveloping Algebras

Abstract In this article, we deal with properties of the reduced Drinfeld double of the composition subalgebra of the Hall algebra of the category of coherent sheaves on a weighted projective line. This study is motivated by applications in the theory of quantized enveloping algebras of some Lie algebras. We obtain a new realization of the quantized enveloping algebras of affine Lie algebras of simply-laced types as well as some new embeddings between them. Moreover, our approach allows to derive new results on the structure of the quantized enveloping algebras of the toroidal algebras of types D4(1, 1), E6(1, 1), E7(1, 1) and E8(1, 1). In particular, our method leads to a construction of a modular action and allows to define a PBW-type basis for that classes of algebras.

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