Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

$\imath$Hall algebras of weighted projective lines and quantum symmetric pairs III: quasi-split type

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Quasi-split ıquantum loop algebras appear inside geometrically defined ıHall algebras of weighted projective lines.

desk verdict A genuinely new ρ-complex framework and a plausible realization of quasi-split ıquantum loop algebras, but the load-bearing non-invariant relation is deferred to an omitted cancellation that the split-case citations do not obviously cover. read the letter →

arxiv 2411.13078 v1 pith:OFFELKJP submitted 2024-11-20 math.QA math.RT

Ming Lu , Shiquan Ruan This is my paper · ORCID
classification math.QAmath.RT MSC 17B3716E6018G80
keywords HallalgebrasıQuantumgroupsQuantumsymmetricpairsρ-complexesweightedprojectivelinesDrinfeldtypepresentationsemi-derivedRingel-Hall
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces ρ-complexes, a generalization of periodic complexes and ıquiver modules, for any abelian category with an involution, and uses them to build an ıHall algebra from the coherent sheaves of a weighted projective line. It then proves that the quasi-split ıquantum loop algebra in its Drinfeld-type presentation maps homomorphically into this ıHall algebra by sending every Drinfeld generator to an explicit sheaf-theoretic element. If the construction is correct, the quasi-split ıquantum groups of affine ADE and star-shaped type acquire a geometric realization as subalgebras of a Hall algebra, extending the classical Hall-algebra realizations of quantum groups to the coideal setting.

What carries the argument

The load-bearing object is the category C_ρ(A) of ρ-complexes: a pair (M, d) with M ∈ A and d: M → ρ(M) satisfying ρ(d)∘d = 0. For A = coh(X), the category of coherent sheaves on a weighted projective line with an involution ρ, the paper proves C_ρ(A) has the homological finiteness needed to support the machinery of semi-derived Ringel-Hall algebras: acyclic objects have projective and injective dimension at most one, and a relative derived category equivalence D_ρ(A) ≅ D^b(A)/(Σ∘ρ̂) holds. The ıHall algebra is the twisted semi-derived Ringel-Hall algebra of C_ρ(A). The homomorphism Ω is assembled from three ingredients: the embeddings ψ_{C_{p_i}} of the ıquantum group of sl_{p_i} into the ıHall algebra of the cyclic quiver at each invariant tube, the Drinfeld–Beck isomorphism Φ from the Drinfeld presentation to the usual presentation of the affine quantum group, and the natural embedding of the projective-line ıHall algebra into that of the weighted projective line. The root vectors in the tubes are defined as images under these embeddings, and the relations between ⋆ and [i,1] are checked by explicit computations with torsion sheaves.

What would settle it

Compute directly in the ıHall algebra of a small weighted projective line (for example, weight type (2,2), where the involution swaps the two branches) the image under Ω of the Drinfeld relation (5.33) between B⋆,k and B[i,1],l for q=2 and small k,l; if the two sides do not coincide as elements of the Hall algebra, Theorem 7.4 is false. A more targeted check is whether the composition Ω_{C_p} preserves the relation [B_{i,k}, B_{i,l+1}]_{$v^{{-2}}$} − $v^{{-2}}$[B_{i,k+1}, B_{i,l}]_{$v^{2}$} inside the Hall algebra of the cyclic quiver C_p.

Watch

Extended reading notes

Core claim

The central result is Theorem 7.4: for a star-shaped graph Γ with an involution ρ that lifts to an involution of a weighted projective line X, there exists a Q(v)-algebra homomorphism Ω from the quasi-split ıquantum loop algebra Dr ~Uı_v to the ıHall algebra ı ~H(X_k, ρ). The map sends the Drinfeld generators K⋆, K[i,j], C, B⋆,l, Θ⋆,r, H⋆,r, B[i,j],l, Θ[i,j],r, H[i,j],r to explicit elements of the ıHall algebra: the structure sheaf and its shifts, torsion sheaves arising from the tubes, and the universal series built from torsion sheaves supported at points. The bulk of the proof consists in verifying the Drinfeld relations (5.29)–(5.37) inside the ıHall algebra, with the genuinely new computations concentrated in the relations between the star vertex and the first vertex of each branch (Sections 9–10).

Load-bearing premise

Everything rests on two previously established external results being valid in exactly the form used: the embedding of the ıquantum group of sl_p into the ıHall algebra of a cyclic quiver, and the Drinfeld–Beck isomorphism identifying the two presentations of the affine quantum group; if either fails inside the Hall algebra context, the definition of the root vectors in the tubes collapses and the asserted homomorphism may not preserve the Drinfeld relations.

Editorial extensions

If this is right

  • The Drinfeld-type presentation of the quasi-split ıquantum loop algebra of star-shaped type, including quasi-split affine ADE, is realized inside the ıHall algebra of a weighted projective line with involution.
  • The relations (5.29)–(5.37) of the quasi-split ıquantum loop algebra are verified inside the ıHall algebra, so the latter contains the full Drinfeld-type algebra as a subalgebra if injectivity holds.
  • The ρ-complex formalism provides a uniform framework covering 1-periodic, 2-periodic, and ıquiver Hall algebras; in particular the split-type realizations are recovered as the ρ = Id case.
  • The paper states that following the arguments of [LR23] one can prove Ω is injective for g of finite or affine type, which would make the realization a genuine embedding for the star-shaped ADE cases.
  • The Euler form computations (Theorem C) give a systematic way to compute Hall products in the ıHall algebra, reducing them to Euler forms and extension counts in the underlying category of coherent sheaves.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If injectivity is established for finite and affine type (as the authors indicate), the ıHall algebra would yield a PBW-type basis of the quasi-split ıquantum loop algebra indexed by coherent sheaves, mirroring Schiffmann's basis for quantum loop algebras.
  • The ρ-complex category C_ρ(A) may be useful beyond weighted projective lines: any hereditary category with an involution satisfying the homological propositions would produce an ıHall algebra, so the same construction could be tested on other categories such as modules over canonical algebras or higher genus curves (a direction the paper itself flags).
  • A direct testable consequence of the construction is that the images of the Drinfeld generators in the ıHall algebra should satisfy the same integrality and positivity constraints as their counterparts in the split case; checking these for small q and small weights could provide independent evidence for the homomorphism's correctness.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper introduces, for a hereditary k-linear abelian category with an involution ρ, the category C_ρ(A) of ρ-complexes and the associated twisted semi-derived Ringel-Hall algebra, called the ıHall algebra. For a weighted projective line X with an involution induced by an involution of the associated star-shaped graph Γ, the authors define explicit elements in the ıHall algebra corresponding to the generators of the quasi-split ıquantum loop algebra Dr~Uı in its Drinfeld-type presentation, and claim in Theorem 7.4 that these assignments define a Q(v)-algebra homomorphism. The proof is organized as a verification of the Drinfeld relations (5.29)–(5.37), using prior results for cyclic quivers and for the split case, with the new quasi-split non-invariant tube case treated in Sections 9–10. The main new conceptual ingredients are the category of ρ-complexes, the derived equivalence D_ρ(A) ≃ D^b(A)/Σ∘ρ̂ (Theorem A), and the Euler-form identities in Theorem C.

Significance. If Theorem 7.4 is correct, the paper provides a geometric realization of quasi-split ıquantum loop algebras of star-shaped type—covering quasi-split affine ADE types—as subalgebras of a twisted semi-derived Hall algebra. This is a substantial extension of the split-type realizations in [LR24] and gives independent motivation for the new framework of ρ-complexes. The paper contains a large amount of explicit computation, including detailed root-vector formulas in tubes, and it is careful about the distinction between invariant and non-invariant tubes. No machine-checked proofs or code are provided; the contribution is theoretical.

major comments (3)
  1. [§10.1, Proposition 10.3] The proof of (10.3) is the linchpin of the relation (10.1) between H_[i,1] and [O(l⃗ c)], and hence of Theorem 7.4 through relation (5.31). The proof is reduced to Lemmas 10.1 and 10.2, but Lemma 10.1 is dismissed with 'the proof is the same as [LR24, Lemma 9.3]', and Lemma 10.2 ends with 'the remaining proof is completely the same as [LR24, Lemma 9.4]'. The present quasi-split non-invariant case (ρ(i)≠i) is not the split case: the category is C_ρ(coh(X)) with ρ swapping two tubes, and the root vectors are images of the full quantum group via the Drinfeld–Beck isomorphism, not of the ıquantum group. In particular, the morphism counts in the displayed part of Lemma 10.2 (Im(f) ≅ S^{(kp_1)}_{2,0} or S^{(kp_1-1)}_{2,0}) have no direct counterpart in [LR24]. Therefore the identity (10.3) is not actually verified in the manuscript. The authors should either supply the full cancellation or provide a detailed derivation of (10.3) adapted to the non-invariant situation; this is necessary to establish the central claim.
  2. [§9.4, Propositions 9.7 and 9.8] The proofs of relations (9.8) and (9.9) are declared 'completely same' to [LR24, Propositions 8.5 and 8.6] at the end of a section whose standing assumption is ρ(i)≠i. Since [LR24] concerns the split/invariant case, this citation is not automatically sufficient for the quasi-split non-invariant case. The same objection as for Proposition 10.3 applies: the omitted computations involve products of line-bundle classes with tube root vectors under the involution swapping two tubes. Please spell out the modifications needed for the quasi-split non-invariant case, or give the computations explicitly.
  3. [§7.1.2, equations (7.6)–(7.8)] In the non-invariant case the root vectors are defined through the composition ι_i∘Ω_{C_{p_i}}, where ι_i : SDH_{Z2}(kC_{p_i}) → ıH(X,ρ) is not injective; the paper itself notes after (7.5) that ι_i([K_δ]) = [K_δ] = ι_i([K*_δ]). The paper does not discuss whether the kernel of ι_i can interfere with the Drinfeld relations verified in the target algebra. This is not necessarily an error, but since all tube relations in Section 8 are proved by passing through this non-injective map, the authors should state explicitly why the identified elements are harmless for the specific combinations appearing in (7.6)–(7.8).
minor comments (5)
  1. [§2.7, Theorem 2.16] In the proof of Theorem 2.16 the identity ⟨K,A⟩ = ⟨Im(d),A⟩ + ⟨ρ(Im(d)),A⟩ is derived twice, with a repeated sentence beginning 'Using (2.10), we have'; please remove the duplication.
  2. [Abstract and Proposition 3.1] There are minor typographical errors, for example 'an d' in the abstract and 'folllowing' in Proposition 3.1; a careful proofreading pass is recommended.
  3. [§4.4, proof of Proposition 4.5] The action of G = diag(1,-1) ∈ PGL(2,k) and the condition λ_{2i-1} = -λ_{2i} should be stated in an explicit affine chart, especially because the base field is finite.
  4. [§6.3, Proposition 6.3] The passage from identities in the composition subalgebra of ıH(kC_n, Id) to identities in the associated graded algebra ~H(kC_n) is only sketched in one sentence; please expand the argument and state the filtration compatibility explicitly.
  5. [Remark 7.5] Remark 7.5 asserts that injectivity 'can be proved' by following [LR23, Theorem 3.2], but no proof is included; since Theorem 7.4 only asserts a homomorphism, please rephrase this either as a conjecture or provide the proof.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the homomorphism Ω is defined by explicit Hall algebra elements and the Drinfeld relations are verified by direct computation, with cited prior theorems serving as independent support.

full rationale

The construction is not circular. Theorem 7.4 defines Ω by explicit assignments: K⋆ ↦ [K_O], B⋆,l ↦ -(q-1)^{-1}[O(l⃗ c)], and branch generators as images of root vectors in cyclic quiver Hall algebras via embeddings ψ (from [LW23, Theorem 9.6] and [LW22a, Theorem H]) composed with Drinfeld–Beck isomorphisms Φ (from [Be94] and [LW21b]). The proof then verifies relations (5.29)–(5.37) as identities inside the Hall algebra; for branch/branch and branch/central relations the computations are performed directly or reduced to published theorems with independent proofs. No relation is imposed by definition of the target algebra; rather, the Hall algebra is proven to satisfy the same relations. The heavy self-citation (e.g., [LR24], [LW23], [LRW23]) is citation of prior published results whose assumptions do not include Theorem 7.4, so it does not reduce the central claim to its own inputs. The one in-text gap is Proposition 10.3: relation (10.3) is said to follow from Lemmas 10.1 and 10.2 by 'easy (but tedious) cancellations, which we omit here', and Lemmas 10.1–10.2 refer to [LR24] for analogous split-case computations. This is a proof-completeness and correctness concern, not circularity: the non-invariant case genuinely involves C_Z2(rep kC_pi) with swapped tubes, and the cited split-case arguments do not automatically force the identity. If the omitted cancellation failed, the homomorphism theorem would be unproved, but that would be a gap, not a circular reduction. Accordingly no circular step is identified.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces ρ-complexes as a mathematical definition, not as an empirically postulated entity. The axioms listed are the external theorems and domain conditions that the central homomorphism depends on. There are no fitted parameters in this pure mathematics setting.

assumptions (5)
  • domain assumption char(k) ≠ 2
    Assumed throughout (Section 2, start). Used in Proposition 2.6 to show K• is a direct summand of π2F(K•); the splitting argument may fail in characteristic 2.
  • domain assumption Weighted projective line geometry
    Section 4 uses the Geigle-Lenzing classification of torsion sheaves and the Euler form on K0(coh(X)) from [GL87] and [Sch04].
  • standard math Drinfeld-Beck isomorphism for U_v(sl_pi)
    Used in Section 6 to transfer the Drinfeld presentation into the Hall algebra of cyclic quivers via Ω_{C_pi}; this is Beck's theorem [Be94].
  • domain assumption Semi-derived Ringel-Hall algebra machinery for C_ρ(A)
    The paper proves the needed homological properties (Props 2.6, 2.13, Thm 2.16) but relies on the general framework of [LP21] and [LW22a, Appendix A] to define the localization and twisted multiplication.
  • domain assumption Lifting of graph involution to coh(X)
    Proposition 4.5 asserts existence of the lift when the number of fixed branches is at most 2; the main theorem assumes such a lift.

how reviews work

0 comments
Cite this review

Pith. "Pith review of $\imath$Hall algebras of weighted projective lines and quantum symmetric pairs III: quasi-split type." pith.science (2026). https://pith.science/paper/OFFELKJP

@misc{pith2026241113078,
  author       = {Pith},
  title        = {Pith review of: $\imath$Hall algebras of weighted projective lines and quantum symmetric pairs III: quasi-split type},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OFFELKJP}},
  note         = {Machine review of arXiv:2411.13078}
}
abstract

From a category $\mathcal{A}$ with an involution $\varrho$, we introduce $\varrho$-complexes, which are a generalization of (bounded) complexes, periodic complexes and modules of $\imath$quiver algebras. The homological properties of the category $\mathcal{C}_\varrho(\mathcal{A})$ of $\varrho$-complexes are given to make the machinery of semi-derived Ringel-Hall algebras applicable. The $\imath$Hall algebra of the weighted projective line $\mathbb{X}$ is the twisted semi-derived Ringel-Hall algebra of $\mathcal{C}_\varrho({\rm coh}(\mathbb{X}))$, where $\varrho$ is an involution of ${\rm coh}(\mathbb{X})$. This $\imath$Hall algebra is used to realize the quasi-split $\imath$quantum loop algebra, which is a generalization of the $\imath$quantum group arising from the quantum symmetric pair of quasi-split affine type ADE in its Drinfeld type presentation.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. GIM and Elliptic Lie algebras via Ringel--Hall Lie algebras

    math.RT 2026-08 conditional novelty 6.0 of 10

    Every symmetrizable GIM algebra is isomorphic to the integral Ringel-Hall Lie algebra of a 2-periodic orbit category built from a valued quiver with an involution.

Reference graph

Works this paper leans on

60 extracted references · 48 canonical work pages · cited by 1 Pith paper

  1. [1]

    Auslander and R

    M. Auslander and R. Buchweitz, The homological theory of maximal Cohen-Macaulay approximation , Colloque en l'honneur de Pierre Samuel (Orsay, 1987). M\' e m. Soc. Math. France (N.S.) No. 38 (1989), 5--37

    work page 1989

  2. [2]

    Baseilhac and S

    P. Baseilhac and S. Kolb,

    work page

  3. [3]

    Baumann and C

    P. Baumann and C. Kassel, The Hall algebra of the category of coherent sheaves on the projective line , J. reine angew. Math. 533 (2001), 207--233

    work page 2001

  4. [4]

    Beck, Braid group actions and quantum affine algebras , Commun

    J. Beck, Braid group actions and quantum affine algebras , Commun. Math. Phys. 165 (1994), 555--568

    work page 1994

  5. [5]

    Bridgeland, Quantum groups via Hall algebras of complexes , Ann

    T. Bridgeland, Quantum groups via Hall algebras of complexes , Ann. Math. 177 (2)(2013), 739--759

    work page 2013

  6. [6]

    B\" u hler, Exact categories , Expo

    T. B\" u hler, Exact categories , Expo. Math. 28 (1)(2010), 1--69

    work page 2010

  7. [7]

    Burban and O

    I. Burban and O. Schiffmann,

    work page

  8. [8]

    Burban and O

    I. Burban and O. Schiffmann, Two descriptions of the quantum affine algebra U_v( sl _2) via Hall algebra approach , Glasgow Math. J. 54 (2012), 283--307

    work page 2012

Show all 60 references
  1. [9]

    Burban and O

    I. Burban and O. Schiffmann, The composition Hall algebra of a weighted projective line , J. reine angew. Math. 679 (2013), 75--124

    2013

  2. [10]

    Cibils and E

    C. Cibils and E. Marcos,

  3. [11]

    Damiani, From the Drinfeld realization to the Drinfeld-Jimbo presentation of affine quantum algebras: injectivity , Publ

    I. Damiani, From the Drinfeld realization to the Drinfeld-Jimbo presentation of affine quantum algebras: injectivity , Publ. Res. Inst. Math. Sci. 51 (2015), 131--171

    2015

  4. [12]

    S. E. Dickson, A torsion theory for abelian categories , Trans. Amer. Math

  5. [13]

    B. Deng, J. Du and Q. Fu, A double Hall algebra approach to affine quantum Schur-Weyl theory, LMS Lect. Notes Ser. 401 , 2012, Cambridge University Press, Cambridge

    2012

  6. [14]

    R. Dou, Y. Jiang and J. Xiao, Hall algebra approach to Drinfeld's presentation of quantum loop algebras , Adv. Math. 231 (2012), 2593--2625

    2012

  7. [15]

    Drinfeld, A new realization of Yangians and quantized affine algebras , Soviet Math

    V. Drinfeld, A new realization of Yangians and quantized affine algebras , Soviet Math. Dokl. 36 (1988), 212--216

    1988

  8. [16]

    E. E. Enochs and O. M. G. Jenda, Relative homological algebra , de Gruyter Exp. Math. 30, Walter de Gruyter Co., 2000

    2000

  9. [17]

    Geigle and H

    W. Geigle and H. Lenzing, A class of weighted projective curves arising in representation theory of finite dimensional algebras , Singularities, representation of algebras, and vector bundles, Lect. Notes Math. 1273 (1987), 265--297

    1987

  10. [18]

    Gorsky, Semi-derived and derived Hall algebras for stable categories , IMRN, Vol

    M. Gorsky, Semi-derived and derived Hall algebras for stable categories , IMRN, Vol. 2018 , No. 1, 138--159

    2018

  11. [19]

    Green, Hall algebras, hereditary algebras and quantum groups , Invent

    J.A. Green, Hall algebras, hereditary algebras and quantum groups , Invent. Math. 120 (1995), 361--377

    1995

  12. [20]

    Happel, On the deived category of a finite-dimesional algebra , Comment

    D. Happel, On the deived category of a finite-dimesional algebra , Comment. Math. Helv. 62 (3)(1987), 339--389

    1987

  13. [21]

    Happel, Triangulated Categories in the Representation Theory of Finite Dimensional Algebras

    D. Happel, Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. London Math. Soc. Lecture Notes Ser. 119 , Cambridge Univ. Press, Cambridge, 1988

    1988

  14. [22]

    Happel, On Gorenstein algebras , In: Progress in Math

    D. Happel, On Gorenstein algebras , In: Progress in Math. 95, Birkh\" a user Verlag, Basel, 1991, 389--404

    1991

  15. [23]

    Happel, I

    D. Happel, I. Reiten and S. O. Smal , Tilting in abelian categories and quasitilted algebras , Mem. Amer. Math. Soc. 120 (1996), no. 575

    1996

  16. [24]

    Hubery, From triangulated categories to Lie algebras: A theorem of Peng and Xiao , Trends in representation theory of algebras and related topics, Contemp

    A. Hubery, From triangulated categories to Lie algebras: A theorem of Peng and Xiao , Trends in representation theory of algebras and related topics, Contemp. Math. 406 (2006), 51--66

    2006

  17. [25]

    Hubery, Three presentations of the Hopf algebra U_v( gl _n) , http://www.maths.leeds.ac.uk/

    A. Hubery, Three presentations of the Hopf algebra U_v( gl _n) , http://www.maths.leeds.ac.uk/

  18. [26]

    Kapranov, Eisenstein series and quantum affine algebras , J

    M. Kapranov, Eisenstein series and quantum affine algebras , J. Math. Sci. 84 (1997), 1311--1360

    1997

  19. [27]

    Iwanaga, On rings with finite self-injective dimension , Comm

    Y. Iwanaga, On rings with finite self-injective dimension , Comm. Algebra 7 (4)(1979), 393--414

    1979

  20. [28]

    Keller, Chain complexes and stable categories , Manus

    B. Keller, Chain complexes and stable categories , Manus. Math. 67 (1990), 379--417

    1990

  21. [29]

    Keller, On triangulated orbit categories , Doc

    B. Keller, On triangulated orbit categories , Doc. Math. 10 (2005), 551--581

    2005

  22. [30]

    Kolb, Quantum symmetric Kac-Moody pairs , Adv

    S. Kolb, Quantum symmetric Kac-Moody pairs , Adv. Math. 267 (2014), 395--469

    2014

  23. [31]

    Letzter, Symmetric pairs for quantized enveloping algebras , J

    G. Letzter, Symmetric pairs for quantized enveloping algebras , J. Algebra 220 (2)(1999), 729-767

    1999

  24. [32]

    Letzter, Coideal subalgebras and quantum symmetric pairs , New directions in Hopf algebras (Cambridge), MSRI publications, 43 , Cambridge Univ

    G. Letzter, Coideal subalgebras and quantum symmetric pairs , New directions in Hopf algebras (Cambridge), MSRI publications, 43 , Cambridge Univ. Press, 2002, pp. 117--166

    2002

  25. [33]

    Lengzing and H

    H. Lengzing and H. Meltzer, The automorphism group of the derived category for a weighted projective line , Comm. Algebra 28 (4)(2000), 1685--1700

    2000

  26. [34]

    Lin and L

    J. Lin and L. Peng, Semi-derived Ringel-Hall algebras and Hall algebras of odd-periodic relative derived categories , Sci. China Math. (2024), 67 (2024), no.8, 1735--1760

    2024

  27. [35]

    Lu and L

    M. Lu and L. Peng, Semi-derived Ringel-Hall algbras and Drinfeld doubles , Adv. Math. 383 (2021), 107668

    2021

  28. [36]

    Lu and S

    M. Lu and S. Ruan, Hall algebras of weighted projective lines and quantum symmetric pairs , Represent. Theory 28 (2024), 12--188

    2024

  29. [37]

    Lu and S

    M. Lu and S. Ruan, Hall algebras of weighted projective lines and quantum symmetric pairs II: injectivity , Math. Z. 307 (2024), no. 3, Paper No. 53

    2024

  30. [38]

    M. Lu, S. Ruan and W. Wang, Hall algebra of Jordan quiver and Hall-Littlewood functions , arXiv:2104.12336 http://arxiv.org/abs/2104.12336

  31. [39]

    M. Lu, S. Ruan and W. Wang, Hall algebra of the projective line and q -Onsager algebra , Trans. Amer. Math. Soc. 376 (2023), 1475--1505

    2023

  32. [40]

    Lu and W

    M. Lu and W. Wang, Hall algebras and quantum symmetric pairs II: reflection functors , Commun. Math. Phys. 381 (2021), 799--855

    2021

  33. [41]

    Lu and W

    M. Lu and W. Wang, A Drinfeld type presentation of affine ıquantum groups I: Split ADE type , Adv. Math. 393 (2021), 108111

    2021

  34. [42]

    Lu and W

    M. Lu and W. Wang, Hall algebras and quantum symmetric pairs I: Foundations , Proc. London Math. Soc. (3) 124 (2022), 1--82

    2022

  35. [43]

    Lu and W

    M. Lu and W. Wang, Braid group symmetries on quasi-split quantum groups via Hall algebras, Selecta Math. 28 (2022), Article number 84

    2022

  36. [44]

    Lu and W

    M. Lu and W. Wang, Hall algebras and quantum symmetric pairs of Kac-Moody type , Adv. Math. 430 (2023), 109215

    2023

  37. [45]

    M. Lu, W. Wang and W. Zhang, Braid group actions and and quasi-split affine quantum groups I , Represent. Theory 27 (2023), 1000--1040, arXiv:2203.11286 https://arxiv.org/abs/2203.11286

    2023

  38. [46]

    M. Lu, W. Wang and W. Zhang, Braid group actions and and quasi-split affine quantum groups II: higher ranks , Commun. Math. Phys. 405 (2024), Paper No. 142,

    2024

  39. [47]

    Lusztig, Canonical bases arising from quantized enveloping algebras , J

    G. Lusztig, Canonical bases arising from quantized enveloping algebras , J. Amer. Math. Soc. 3 (1990), 447--498

    1990

  40. [48]

    Lusztig, Introduction to Quantum Groups, Birkh\" a user, Boston, 1993

    G. Lusztig, Introduction to Quantum Groups, Birkh\" a user, Boston, 1993

    1993

  41. [49]

    Peng and J

    L. Peng and J. Xiao, Root categories and simple Lie algebras , J. Algebra, 198 (1)(1997), 19-56

    1997

  42. [50]

    Riedtmann, Lie algebras generated by indecomposables , J

    Ch. Riedtmann, Lie algebras generated by indecomposables , J. Algebra 170 (1994),

    1994

  43. [51]

    Ringel, Tame algebras and integral quadratic forms, Lect

    C.M. Ringel, Tame algebras and integral quadratic forms, Lect. Notes Math. 1099 , Springer-Verlag, Berlin, 1984

    1984

  44. [52]

    Ringel, Hall algebras and quantum groups , Invent

    C.M. Ringel, Hall algebras and quantum groups , Invent. Math. 101 (1990), 583-591

    1990

  45. [53]

    Ringel and P

    C.M. Ringel and P. Zhang, Representations of quivers over the algebra of dual numbers , J. Algebra 475 (2017), 237--360

    2017

  46. [54]

    Schiffmann, The Hall algebra of a cyclic quiver and canonical bases of Fock spaces , Intern

    O. Schiffmann, The Hall algebra of a cyclic quiver and canonical bases of Fock spaces , Intern. Math. Res. Notices 8 (2002), 413--440

    2002

  47. [55]

    Schiffmann, Noncommutative projective curves and quantum loop algebras , Duke Math

    O. Schiffmann, Noncommutative projective curves and quantum loop algebras , Duke Math. J. 121 (2004), 113--168

    2004

  48. [56]

    Schiffmann, Lectures on Hall algebras , In: Geometric methods in representation theory

    O. Schiffmann, Lectures on Hall algebras , In: Geometric methods in representation theory. II, 1--141, S\' e min. Congr., 24-II, Soc. Math. France, Paris (2012)

    2012

  49. [57]

    Stai, The triangulated hull of periodic complexes , Math

    T. Stai, The triangulated hull of periodic complexes , Math. Res. Lett. 25 (2017), 199--236

    2017

  50. [58]

    Wang, Quantum symmetric pairs , Proceedings of ICM 2022, Volume 4 , 3080--3102

    W. Wang, Quantum symmetric pairs , Proceedings of ICM 2022, Volume 4 , 3080--3102

    2022

  51. [59]

    Wang and W

    W. Wang and W. Zhang, An intrinsic approach to relative braid group symmetries on quantum groups , Proc. London Math. Soc. 127 (2023), 1338--1423

    2023

  52. [60]

    Zhang, A Drinfeld type presentation of affine quantum groups II: split BCFG type , Lett

    W. Zhang, A Drinfeld type presentation of affine quantum groups II: split BCFG type , Lett. Math. Phys. 112 , 89 (2022)

    2022

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.