Quantum groups via Hall algebras of complexes

Bridgeland, Tom

doi:10.4007/annals.2013.177.2.9

Cited by 104 publications

(200 citation statements)

References 12 publications

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“…A lot of efforts have been paid on the progress (see [3,8,20,22]) and the most recent progress is given by Bridgeland in [1]. We hope that the main result in the present paper can provide a strong evidence for the connection between canonical bases and root categories.…”

Section: Introductionsupporting

confidence: 49%

A parameterization of the canonical bases of affine modified quantized enveloping algebras

Xiao

Zhao

2016

Chin. Ann. Math. Ser. B

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For a symmetrizable Kac-Moody Lie algebra g, Lusztig introduced the corresponding modified quantized enveloping algebraU and its canonical basisḂ given by Lusztig in 1992. In this paper, in the case that g is a symmetric Kac-Moody Lie algebra of finite or affine type, the authors define a set M which depends only on the root category R and prove that there is a bijection between M andḂ, where R is the T 2 -orbit category of the bounded derived category of the corresponding Dynkin or tame quiver. The method in this paper is based on a result of Lin, Xiao and Zhang in 2011, which gives a PBW-type basis of U + .

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Section: Introductionsupporting

confidence: 49%

A parameterization of the canonical bases of affine modified quantized enveloping algebras

Xiao

Zhao

2016

Chin. Ann. Math. Ser. B

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“…Third way was introduced by Toën in [23] who constructed the derived Ringel-Hall algebra from the derived category. Recently, Bridgeland [1] gave a more clever way to consider directly the Z 2 -graded complexes (or 2-cycle complexes as called in [13], or 2-periodic complexes as in some literature) of projective modules of an algebra with finite global dimension to construct the Ringel-Hall algebra. He showed that the full quantized enveloping algebra can be also realized by his Ringel-Hall algebra.…”

Section: Introductionmentioning

confidence: 99%

The Bridgeland's Ringel–Hall algebra associated to an algebra with global dimension at most two

Geng

Peng

2015

Journal of Algebra

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Communicated by Michel Van den Bergh MSC: 16G10 16W35 Keywords: Ringel-Hall algebras 2-periodic complexes Bridgeland's Ringel-Hall algebras Tilted algebras Canonical algebrasFor any finite dimensional associative algebra with global dimension ≤ 2, we show that there is an embedding from the twisted Ringel-Hall algebra to the Bridgeland's Ringel-Hall algebra. In particular, this result is true for tilted algebras and canonical algebras.

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“…So one considers the localization of H tw (C m (P)) with respect to the set S (cf. [1,2]). Definition 2.6.…”

Section: Hall Algebras Given A-modules L M N We Define Extmentioning

confidence: 99%

“…Since the uniqueness of the minimal projective resolution up to isomorphism, the complex C M is well-defined up to isomorphism. By [1,2], for each r ∈ Z m , we define…”

Section: Hall Algebras Given A-modules L M N We Define Extmentioning

confidence: 99%

“…In 2013, for each finite dimensional hereditary algebra A over a finite field, Bridgeland [1] defined an algebra associated to A, called Bridgeland's Hall algebra of A, which is the Ringel-Hall algebra of 2-cyclic complexes over projective A-modules with some localization and reduction. He proved that the quantized enveloping algebra associated to the hereditary algebra A can be embedded into Bridgeland's Hall algebra of A.…”

Section: Introductionmentioning

confidence: 99%

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Bridgeland’s Hall algebras and Heisenberg doubles

Zhang

2018

J. Algebra Appl.

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Abstract. Let A be a finite dimensional hereditary algebra over a finite field, and let m be a fixed integer such that m = 0 or m > 2. In the present paper, we first define an algebra L m (A) associated to A, called the m-periodic lattice algebra of A, and then prove that it is isomorphic to Bridgeland's Hall algebra DH m (A) of m-cyclic complexes over projective A-modules. Moreover, we show that there is an embedding of the Heisenberg double Hall algebra of A into DH m (A).

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Quantum groups via Hall algebras of complexes

Abstract: Abstract. We describe quantum enveloping algebras of symmetric Kac-Moody Lie algebras via a finite field Hall algebra construction involving Z 2 -graded complexes of quiver representations.

Cited by 104 publications

References 12 publications

A parameterization of the canonical bases of affine modified quantized enveloping algebras

A parameterization of the canonical bases of affine modified quantized enveloping algebras

The Bridgeland's Ringel–Hall algebra associated to an algebra with global dimension at most two

Bridgeland’s Hall algebras and Heisenberg doubles

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