scholarly journals The Hall algebra of the category of coherent sheaves on the projective line

2001 ◽  
Vol 2001 (533) ◽  
Author(s):  
P. Baumann ◽  
C. Kassel
Keyword(s):  
Projective Line ◽  
Hall Algebra ◽  
Coherent Sheaves

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.

scholarly journals Spherical Hall Algebras of Weighted Projective Curves

2018 ◽  
Vol 2020 (15) ◽  
pp. 4721-4775
Author(s):  
Jyun-Ao Lin
Keyword(s):  
Vector Bundles ◽  
Characteristic Functions ◽  
Projective Curve ◽  
Hall Algebra ◽  
Smooth Projective Curve ◽  
Coherent Sheaves ◽  
Hall Algebras ◽  
Arbitrary Genus ◽  
Projective Curves ◽  
Fixed Rank

Abstract In this article, we deal with the structure of the spherical Hall algebra $\mathbf{U}$ of coherent sheaves with parabolic structures on a smooth projective curve $X$ of arbitrary genus $g$. We provide a shuffle-like presentation of the bundle part $\mathbf{U}^>$ and show the existence of generic spherical Hall algebra of genus $g$. We also prove that the algebra $\mathbf{U}$ contains the characteristic functions on all the Harder–Narasimhan strata. These results together imply Schiffmann’s theorem on the existence of Kac polynomials for parabolic vector bundles of fixed rank and multi-degree over $X$. On the other hand, the shuffle structure we obtain is new and we make links to the representations of quantum affine algebras of type $A$.

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 On the K-Theoretic Hall Algebra of a Surface

2020 ◽  
Author(s):  
Yu Zhao
Keyword(s):  
Projective Surface ◽  
Hall Algebra ◽  
Shuffle Algebra ◽  
Coherent Sheaves ◽  
Smooth Projective Surface

Abstract In this paper, we define the $K$-theoretic Hall algebra for dimension $0$ coherent sheaves on a smooth projective surface, prove that the algebra is associative, and construct a homomorphism to a shuffle algebra introduced by Negut [ 10].

scholarly journals TWO DESCRIPTIONS OF THE QUANTUM AFFINE ALGEBRA Uv() VIA HALL ALGEBRA APPROACH

2011 ◽  
Vol 54 (2) ◽  
pp. 283-307 ◽  
Author(s):  
IGOR BURBAN ◽  
OLIVIER SCHIFFMANN
Keyword(s):  
Integral Form ◽  
Derived Categories ◽  
Quantum Affine Algebra ◽  
Hall Algebra ◽  
Affine Algebra ◽  
Coherent Sheaves ◽  
Enveloping Algebra ◽  
Algebra Approach ◽  
Quantized Enveloping Algebra

AbstractWe compare the reduced Drinfeld doubles of the composition subalgebras of the category of representations of the Kronecker quiver and the category of coherent sheaves on ℙ1. Using this approach, we show that the Drinfeld–Beck isomorphism for the quantized enveloping algebra Uv() is a corollary of an equivalence between the derived categories Db(Rep()) and Db(Coh(ℙ1)). This technique allows to reprove several results on the integral form of Uv().

GENERALIZED QUASI-COHERENT SHEAVES

2003 ◽  
Vol 02 (01) ◽  
pp. 63-83 ◽  
Author(s):  
E. ENOCHS ◽  
S. ESTRADA ◽  
J. R. GARCÍA-ROZAS ◽  
L. OYONARTE
Keyword(s):  
Projective Line ◽  
Coherent Sheaves ◽  
Projective Representations ◽  
Line P ◽  
Torsion Free

The category of quasi-coherent sheaves on the projective line P1(k) (k is a field) is equivalent to certain representations of the quiver • → • ← •. Many of the techniques which are used to study these sheaves apply to more general categories. We give the definitions of these more general categories and then consider one particular such category in depth. In this particular category we prove that there are no (nonzero) projective representations but that the category admits flat covers (or, equivalently in this situation, torsion free covers) and cotorsion envelopes.

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