Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 095, 24 pages      arXiv:1906.03055      https://doi.org/10.3842/SIGMA.2023.095

DG-Enhanced Hecke and KLR Algebras

Ruslan Maksimau a and Pedro Vaz b
a)  Laboratoire Analyse Géométrie Modélisation, CY Cergy Paris Université, 2 av. Adolphe Chauvin (Bat. E, 5ème étage), 95302 Cergy-Pontoise, France
b)  Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium

Received March 30, 2023, in final form November 15, 2023; Published online November 22, 2023

Abstract
We construct DG-enhanced versions of the degenerate affine Hecke algebra and of the affine Hecke algebra. We extend Brundan-Kleshchev and Rouquier's isomorphism and prove that after completion DG-enhanced versions of affine Hecke algebras (degenerate or nondegenerate) are isomorphic to completed DG-enhanced versions of KLR algebras for suitably defined quivers. As a byproduct, we deduce that these DG-algebras have homologies concentrated in degree zero. These homologies are isomorphic respectively to the degenerate cyclotomic Hecke algebra and the cyclotomic Hecke algebra.

Key words: Hecke algebra; KLR algebra; DG-algebra.

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References

  1. Brundan J., On the definition of Kac-Moody 2-category, Math. Ann. 364 (2016), 353-372, arXiv:1501.00350.
  2. Brundan J., Kleshchev A., Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras, Invent. Math. 178 (2009), 451-484, arXiv:0808.2032.
  3. Brundan J., Kleshchev A., Graded decomposition numbers for cyclotomic Hecke algebras, Adv. Math. 222 (2009), 1883-1942, arXiv:0901.4450.
  4. Kang S.-J., Kashiwara M., Categorification of highest weight modules via Khovanov-Lauda-Rouquier algebras, Invent. Math. 190 (2012), 699-742, arXiv:1102.4677.
  5. Khovanov M., Heisenberg algebra and a graphical calculus, Fund. Math. 225 (2014), 169-210, arXiv:1009.3295.
  6. Khovanov M., Lauda A.D., A diagrammatic approach to categorification of quantum groups I, Represent. Theory 13 (2009), 309-347, arXiv:0803.4121.
  7. Khovanov M., Lauda A.D., A categorification of quantum ${\rm sl}(n)$, Quantum Topol. 1 (2010), 1-92, arXiv:0807.3250.
  8. Khovanov M., Lauda A.D., A diagrammatic approach to categorification of quantum groups II, Trans. Amer. Math. Soc. 363 (2011), 2685-2700, arXiv:0804.2080.
  9. Kleshchev A., Representation theory of symmetric groups and related Hecke algebras, Bull. Amer. Math. Soc. (N. S.) 47 (2010), 419-481, arXiv:0909.4844.
  10. Lauda A.D., A categorification of quantum ${\rm sl}(2)$, Adv. Math. 225 (2010), 3327-3424, arXiv:0803.3652.
  11. Licata A., Savage A., Hecke algebras, finite general linear groups, and Heisenberg categorification, Quantum Topol. 4 (2013), 125-185, arXiv:1101.0420.
  12. Maksimau R., Stroppel C., Higher level affine Schur and Hecke algebras, J. Pure Appl. Algebra 225 (2021), 106442, 44 pages, arXiv:1805.02425.
  13. Miemietz V., Stroppel C., Affine quiver Schur algebras and $p$-adic ${\rm GL}_n$, Selecta Math. (N. S.) 25 (2019), 32, 66 pages, arXiv:1601.07323.
  14. Naisse G., Vaz P., An approach to categorification of Verma modules, Proc. Lond. Math. Soc. 117 (2018), 1181-1241, arXiv:1603.01555.
  15. Naisse G., Vaz P., On 2-Verma modules for quantum $\mathfrak{sl}_2$, Selecta Math. (N. S.) 24 (2018), 3763-3821, arXiv:1704.08205.
  16. Naisse G., Vaz P., 2-Verma modules, J. Reine Angew. Math. 782 (2022), 43-108, arXiv:1710.06293.
  17. Poulain d'Andecy L., Walker R., Affine Hecke algebras and generalizations of quiver Hecke algebras of type $B$, Proc. Edinb. Math. Soc. 63 (2020), 531-578, arXiv:1712.05592.
  18. Rostam S., Cyclotomic Yokonuma-Hecke algebras are cyclotomic quiver Hecke algebras, Adv. Math. 311 (2017), 662-729, arXiv:1603.03901.
  19. Rouquier R., Derived equivalences and finite dimensional algebras, in International Congress of Mathematicians, Vol. 2, European Mathematical Society, Zürich, 2006, 191-221, arXiv:math.RT/0603356.
  20. Rouquier R., 2-Kac-Moody algebras, arXiv:0812.5023.
  21. Stroppel C., Webster B., Quiver Schur algebras and $q$-Fock space, arXiv:1110.1115.
  22. Webster B., Knot invariants and higher representation theory, Mem. Amer. Math. Soc. 250 (2017), v+141 pages, arXiv:1309.3796.
  23. Webster B., On graded presentations of Hecke algebras and their generalizations, Algebr. Comb. 3 (2020), 1-38, arXiv:1305.0599.

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