scholarly journals The category of coherent sheaves over a weighted projective line and loop algebras

2018 ◽  
Vol 48 (11) ◽  
pp. 1551
Author(s):  
Dou Rujing ◽  
Xu Fan ◽  
Xiao Jie ◽  
Ruan Shiquan
Keyword(s):  
Projective Line ◽  
Weighted Projective Line ◽  
Coherent Sheaves ◽  
Loop Algebras

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

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 ${\mathfrak g}_{Q}$.

The composition Hall algebra of a weighted projective line

2013 ◽  
Vol 2013 (679) ◽  
pp. 75-124 ◽  
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.

Flat Covers in the Category of Quasi-coherent Sheaves Over the Projective Line

2004 ◽  
Vol 32 (4) ◽  
pp. 1497-1508 ◽  
Author(s):  
Edgar Enochs ◽  
S. Estrada ◽  
J. R. García Rozas ◽  
L. Oyonarte
Keyword(s):  
Projective Line ◽  
Coherent Sheaves

scholarly journals THE GROUP OF COVERING AUTOMORPHISMS OF A QUASI-COHERENT SHEAF ON $\bm{P}^1(K)$

2007 ◽  
Vol 50 (2) ◽  
pp. 325-341
Author(s):  
E. Enochs ◽  
S. Estrada ◽  
J. R. García Rozas ◽  
L. Oyonarte
Keyword(s):  
Commutative Ring ◽  
Wide Class ◽  
Vector Bundles ◽  
Universal Covering ◽  
Coherent Sheaf ◽  
Projective Line ◽  
Topological Properties ◽  
Coherent Sheaves ◽  
Flat Cover ◽  
Natural Way

AbstractCoGalois groups appear in a natural way in the study of covers. They generalize the well-known group of covering automorphisms associated with universal covering spaces. Recently, it has been proved that each quasi-coherent sheaf over the projective line $\bm{P}^1(R)$ ($R$ is a commutative ring) admits a flat cover and so we have the associated coGalois group of the cover. In general the problem of computing coGalois groups is difficult. We study a wide class of quasi-coherent sheaves whose associated coGalois groups admit a very accurate description in terms of topological properties. This class includes finitely generated and cogenerated sheaves and therefore, in particular, vector bundles.

scholarly journals Vector-valued modular forms and the modular orbifold of elliptic curves

2016 ◽  
Vol 13 (01) ◽  
pp. 39-63 ◽  
Author(s):  
Luca Candelori ◽  
Cameron Franc
Keyword(s):  
Algebraic Geometry ◽  
Elliptic Curves ◽  
Modular Forms ◽  
Vector Bundles ◽  
Projective Line ◽  
Free Module ◽  
Weighted Projective Line ◽  
Holomorphic Vector Bundles ◽  
Vector Valued ◽  
Standard Techniques

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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