Some extension algebras for standard modules over KLR algebras of type A
Pith reviewed 2026-05-25 14:46 UTC · model grok-4.3
The pith
For positive roots in type A and Lie type A2, the Yoneda algebra of standard modules over KLR algebras is torsion free and intrinsically formal.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
When θ is a positive root in type A or of Lie type A2, the Yoneda algebra E_θ is torsion free and intrinsically formal. In these cases the algebra can be described explicitly.
What carries the argument
The Yoneda algebra E_θ = Ext^*_{R_θ}(Δ, Δ), which carries an A_∞-algebra structure used to reconstruct the category of standardly filtered modules.
If this is right
- The category of standardly filtered R_θ-modules is reconstructed from the A_∞-algebra E_θ in these special cases.
- E_θ has no torsion elements.
- The A_∞-structure on E_θ is formal, so higher products vanish up to homotopy.
- E_θ can be presented explicitly as an ordinary algebra without A_∞ operations in these cases.
Where Pith is reading between the lines
- If E_θ is intrinsically formal, then the Ext groups between standards determine the filtered module category without additional data.
- The explicit descriptions may allow computation of higher Ext groups in these root systems.
- Non-formality in general suggests that A_∞ structures are essential outside these cases.
Load-bearing premise
The A_∞-algebra structure on E_θ reconstructs the category of standardly filtered R_θ-modules.
What would settle it
A computation showing torsion in E_θ or a non-vanishing higher A_∞ product for a positive root θ in type A would falsify the claim.
read the original abstract
Khovanov-Lauda-Rouquier algebras $R_\theta$ of finite Lie type are affine quasihereditary with standard modules $\Delta(\pi)$ labeled by Kostant partitions of $\theta$. Let $\Delta$ be the direct sum of all standard modules. It is known that the Yoneda algebra $\mathcal{E}_\theta:=\operatorname{Ext}_{R_\theta}^*(\Delta, \Delta)$ carries a structure of an $A_\infty$-algebra which can be used to reconstruct the category of standardly filtered $R_\theta$-modules. In this paper, we explicitly describe $\mathcal{E}_\theta$ in two special cases: (1) when $\theta$ is a positive root in type $\mathtt{A}$, and (2) when $\theta$ is of Lie type $\mathtt{A_2}$. In these cases, $\mathcal{E}_\theta$ turns out to be torsion free and intrinsically formal. We provide an example to show that the $A_\infty$-algebra $\mathcal{E}_\theta$ is non-formal in general.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript explicitly computes the Yoneda algebra E_θ = Ext^*_{R_θ}(Δ, Δ) over KLR algebras R_θ of finite Lie type A in two special cases: (1) when θ is a positive root, and (2) when θ is of Lie type A2. In both cases it establishes that E_θ is torsion-free and intrinsically formal. It also supplies an explicit counter-example demonstrating that E_θ fails to be formal for general θ.
Significance. The explicit descriptions furnish concrete, checkable instances in which the A_∞-structure on the extension algebra is formal, thereby simplifying the reconstruction of the standardly filtered category in these cases. The counter-example delineates the boundary of the phenomenon and supplies a useful negative result. The work rests on previously established facts about KLR algebras and A_∞-structures rather than introducing new unverified assumptions.
minor comments (2)
- Abstract: the phrase 'it is known that the Yoneda algebra carries an A_∞-structure which can be used to reconstruct the category' would benefit from a specific citation to the prior literature establishing this reconstruction.
- The manuscript would be improved by a short table or diagram summarizing the explicit generators and relations obtained for E_θ in the two special cases.
Simulated Author's Rebuttal
We thank the referee for their positive assessment of the manuscript and for recommending acceptance.
Circularity Check
No significant circularity detected
full rationale
The paper's derivation consists of explicit computation of the Yoneda algebra E_θ (including A_∞ operations) in the two special cases, followed by direct verification that the resulting algebra is torsion-free and intrinsically formal. The background statement that the A_∞-structure reconstructs the standardly filtered category is cited as prior knowledge and is not invoked to establish the new claims about torsion-freeness or formality. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear in the argument chain; the counter-example for non-formality is likewise external to the positive claims.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption KLR algebras of finite Lie type are affine quasihereditary with standard modules labeled by Kostant partitions of θ.
- domain assumption The Yoneda algebra E_θ carries an A_∞-algebra structure that reconstructs the category of standardly filtered modules.
Reference graph
Works this paper leans on
-
[1]
J. Brundan, A. Kleshchev, and P. J. McNamara. Homologica l properties of finite type Khovanov-Lauda-Rouquier algebras. Duke Math. J. , 163:1353–1404, 2014
work page 2014
-
[2]
D. Buursma, A. Kleshchev, and D. J. Steinberg. Resolutio ns of standard modules over KLR algebras in type A. 2019
work page 2019
-
[3]
T. Kadeishvili. On the homology theory of fiber spaces. Uspekhi Mat. Nauk , 35:183–188, 1980. Translated in Russ. Math. Surv. 35:231-238, 1980
work page 1980
-
[4]
S. Kato. Poincar´ e-Birkhoff-Witt bases and Khovanov-La uda-Rouquier algebras. Duke Math. J., 163:619–663, 2014
work page 2014
-
[5]
B. Keller. Introduction to A-infinity algebras and modules. Homology, Homotopy Appl. , 3(1):1– 35, 2001
work page 2001
-
[6]
B. Keller. a-infinity algebras in representation theory. Representations of Algebras , I:64–86, 2002
work page 2002
-
[7]
M. Khovanov and A. Lauda. A diagrammatic approach to cate gorification of quantum groups I. Represent. Theory, 13:309–347, 2009
work page 2009
-
[8]
M. Khovanov, A. Lauda, M. Mackaay, and M. Stoˇ si´ c. Extended graphical calculus for categori- fied quantum sl(2). Mem. Amer. Math. Soc. , 219(1029), 2012
work page 2012
- [9]
-
[10]
A. Kleshchev and J. Loubert. Affine cellularity of Khovan ov-Lauda-Rouquier algebras in finite types. Int. Math. Res. Not. , 14:5659–5709, 2015
work page 2015
-
[11]
A. Kleshchev and D. J. Steinberg. Homomorphisms betwee n standard modules over finite-type KLR algebras. Compos. Math. , 153:621–646, 2017
work page 2017
- [12]
- [13]
-
[14]
I. G. Macdonald. Symmetric Functions and Hall Polynomials . Clarendon Press, 1995
work page 1995
-
[15]
L. Manivel. Symmetric Functions, Schubert Polynomials and Degeneracy Loci. Soci´ et´ e Math´ ematique de France, 1998
work page 1998
- [16]
-
[17]
C. Weibel. An Introduction to Homological Algebra . Cambridge University Press, 1994
work page 1994
-
[18]
G. Williamson. On an analogue of the James conjecture. Represent. Theory, 18:15–27, 2014. Department of Mathematics, University of Oregon, Eugene, O R 97403, USA E-mail address : dbuursma@uoregon.edu Department of Mathematics, University of Oregon, Eugene, O R 97403, USA E-mail address : klesh@uoregon.edu Department of Mathematics, University of Oregon, E...
work page 2014
discussion (0)
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