The Hall algebra approach to Drinfeld’s presentation of quantum loop algebras

Dou, Rujing; Jiang, Yong; Xiao, Jie

doi:10.1016/j.aim.2012.07.026

Cited by 26 publications

(31 citation statements)

References 19 publications

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“…They showed that U (X) is a topological bialgebra and then defined DU (X) as its reduced Drinfeld double. Meanwhile, Dou, Jiang and Xiao [9] defined the "double composition algebra" DC(X) of coh-X as the subalgebra of D(coh-X) generated by two copies of C(X), say C ± (X), together with the group algebra K. Proof. It has been mentioned in [9] that DC(X) = DU (X).…”

Section: 2mentioning

confidence: 99%

“…Meanwhile, Dou, Jiang and Xiao [9] defined the "double composition algebra" DC(X) of coh-X as the subalgebra of D(coh-X) generated by two copies of C(X), say C ± (X), together with the group algebra K. Proof. It has been mentioned in [9] that DC(X) = DU (X). By [5,Corollary 5.23], the subalgebra DU (X) is generated by u ± O( x) , x ∈ L together with the torus K. Note that each line bundle O( x) is exceptional in coh-X.…”

Section: 2mentioning

confidence: 99%

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Exceptional sequences and Drinfeld double Hall algebras

Ruan

Zhang

2020

Journal of Pure and Applied Algebra

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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.2000 Mathematics Subject Classification. 17B37, 16G20.

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Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Exceptional sequences and Drinfeld double Hall algebras

Ruan

Zhang

2020

Journal of Pure and Applied Algebra

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“…where the sum is taken over all the isoclasses [L] in coh X, and F L M,N denotes the number of subsheaves X of L such that L ∼ = N and L/X ∼ = M . Following [34] (see also [8]), we can define the (reduced Drinfeld) double Ringel-Hall algebra DH(X) of X which admits a triangular decomposition…”

Section: Double Ringel-hall Algebra Of X and Lusztig's Symmetriesmentioning

confidence: 99%

“…where DC(coh X) denotes the composition subalgebra of DH(X). We refer to [8,Thm. 5.5] for the precise definition of Ξ.…”

Section: 3mentioning

confidence: 99%

“…By further dealing with properties of the reduced Drinfeld double DC(coh X) of the composition subalgebra of H(X), Burban and Schiffmann [4] obtained a new realization of the quantum enveloping algebras of affine Lie algebras of simply-laced types as well as some new embeddings between them, and derived new results on the 1 structure of the quantized enveloping algebras of the elliptic Lie algebras. Based on Schiffmann's work [31], Dou, Jiang and Xiao [8] explicitly constructed a collection of generators of DC(coh X) and verified that they satisfy all the Drinfeld relations of the corresponding quantum loop algebra.…”

Section: Introductionmentioning

confidence: 99%

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

Deng

Ruan

Xiao

2018

International Mathematics Research Notices

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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.

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Cohomological Hall algebras and affine quantum groups

Yang

Zhao

2017

Sel. Math. New Ser.

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Abstract. We study the preprojective cohomological Hall algebra (CoHA) introduced by the authors in [YZ1] for any quiver Q and any one-parameter formal group G. In this paper, we construct a comultiplication on the CoHA, making it a bialgebra. We also construct the Drinfeld double of the CoHA. The Drinfeld double is a quantum affine algebra of the Lie algebra gQ associated to Q, whose quantization comes from the formal group G. We prove, when the group G is the additive group, the Drinfeld double of the CoHA is isomorphic to the Yangian. IntroductionIn this paper, we construct comultiplications on certain cohomological Hall algebras, making them bialgebras. We also construct the Drinfeld double of these bialgebras. This gives a uniform way to construct both old and new affine-type quantum groups.The cohomological Hall algebra involved, called the preprojective cohomological Hall algebra (CoHA for short) and denoted by P(Q, A) or simply P, is associated to a quiver Q and an algebraic oriented cohomology theory A. The construction of the preprojective CoHA is given in [YZ1], as a generalization of the K-theoretic Hall algebra of commuting varieties studied by SchiffmannVasserot in [SV12]. The preprojective CoHA is defined to be the A-homology of the moduli of representations of the preprojective algebra of Q. It has the same flavor as the cohomological Hall algebra associated to quiver with potential defined by . The authors also construct an action of P(Q, A) on the A-homology of Nakajima quiver varieties associated toLet g Q be the corresponding symmetric Kac-Moody Lie algebra of Q and b Q ⊂ g Q be the Borel subalgebra. As is shown in [YZ1], a certain spherical subalgebra in an extension of P(Q, A), denoted by P s,e (Q, A) or P s,e , is a quantization of (a central extension of), where the quantization depends on the underlying formal group law of A. As in the case of quantized enveloping algebra of g Q , the Drinfeld double of the Borel subalgebra should be the entire quantum group. This is the subject of the present paper. We construct a coproduct on P(Q, A), and define the Drinfeld double, which is a quantization offor some 1-dimensional algebraic group or formal group G. When G is an affine algebraic group, the Drinfeld double of P(Q, A) recovers the Drinfeld realization of the quantization of the Manin triple associated to P 1 as described in [D86, § 4]. When G is a formal group which does not come from an algebraic group, this gives new affine quantum groups which have not been studied in literature. In the case when G is the elliptic curve, the method here gives a Drinfeld realization of the elliptic quantum group of [Fed94] which is previously unknown.In this paper, we focus on the purely algebraic description of P(Q, A) in terms of shuffle algebra without explicit reference to Nakajima quiver varieties. The shuffle algebra is reviewed in detail in § 1. The algebra P s,e has two deformation parameters t 1 , t 2 , coming from a 2-dimensional torus action. Let P s,e be the quotient of P s,e by the torsion...

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The Hall algebra approach to Drinfeld’s presentation of quantum loop algebras

Cited by 26 publications

References 19 publications

Exceptional sequences and Drinfeld double Hall algebras

Exceptional sequences and Drinfeld double Hall algebras

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

Cohomological Hall algebras and affine quantum groups

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