Deformed Cartan matrices and generalized preprojective algebras II: general type

1. Amiot, C., Iyama, O., Reiten, I., et al.: Preprojective algebras and c-sortable words. Proc. Lond. Math.

Soc. (3) 104(3), 513–539 (2012). https://doi.org/10.1112/plms/pdr020

2. Aoki, T., Higashitani, A., Iyama, O., et al.: Fans and polytopes in tilting theory I: foundations, preprint.

arXiv:2203.15213 v2 (2022)

3. Assem, I., Simson, D., Skowro´nski, A.: Elements of the representation theory of associative algebras, vol.

1, London Mathematical Society Student Texts, vol. 65. Cambridge University Press, Cambridge (2006).

https://doi.org/10.1017/CBO9780511614309

4. Bédard, R.: On commutation classes of reduced words in Weyl groups. Eur. J. Combin. 20(6), 483–505

(1999). https://doi.org/10.1006/eujc.1999.0296

5. Bouwknegt, P., Pilch, K.: On deformed W -algebras and quantum affine algebras. Adv. Theor. Math. Phys.

2(2), 357–397 (1998). https://doi.org/10.4310/ATMP.1998.v2.n2.a6

6. Brenner, S., Butler, M.C.R., King, A.D.: Periodic algebras which are almost Koszul. Algebr. Represent.

Theory 5(4), 331–367 (2002). https://doi.org/10.1023/A:1020146502185

7. Buan, A.B., Iyama, O., Reiten, I., et al.: Mutation of cluster-tilting objects and potentials. Amer. J. Math.

133(4), 835–887 (2011). https://doi.org/10.1353/ajm.2011.0031

8. Chari, V.: Braid group actions and tensor products. Int. Math. Res. Not. 7, 357–382 (2002). https://doi.

org/10.1155/S107379280210612X

9. Dlab, V., Ringel, C.M.: On algebras of finite representation type. J. Algebra 33, 306–394 (1975). https://

doi.org/10.1016/0021-8693(75)90125-8

10. Dlab, V., Ringel, C.M.: Indecomposable representations of graphs and algebras. Mem. Amer. Math. Soc.

6(173), v+57 (1976). https://doi.org/10.1090/memo/0173

11. Dlab, V., Ringel, C. M.: The preprojective algebra of a modulated graph. In: Representation theory, II

(Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., vol. 832.

Springer, Berlin, pp. 216–231 (1980)

12. Frenkel, E., Reshetikhin, N.: Deformations of W -algebras associated to simple Lie algebras. Commun.

Math. Phys. 197(1), 1–32 (1998). https://doi.org/10.1007/BF02099206

13. Fu, C., Geng, S.: Tilting modules and support τ -tilting modules over preprojective algebras associated

with symmetrizable Cartan matrices. Algebr. Represent. Theory 22(5), 1239–1260 (2019). https://doi.

org/10.1007/s10468-018-9819-z

14. Fujita, R.: Graded quiver varieties and singularities of normalized R-matrices for fundamental modules.

Selecta Math. (NS) 28(1), Paper No. 2, 45 (2022). https://doi.org/10.1007/s00029-021-00715-5

15. Fujita, R., Murakami, K.: Deformed Cartan matrices and generalized preprojective algebras I: finite type.

Int. Math. Res. Not. IMRN 8, 6924–6975 (2023). https://doi.org/10.1093/imrn/rnac054

16. Fujita, R., Sj, Oh.: Q-data and representation theory of untwisted quantum affine algebras. Commun.

Math. Phys. 384(2), 1351–1407 (2021). https://doi.org/10.1007/s00220-021-04028-8

17. Gabriel, P.: Indecomposable representations. II. In: Symposia Mathematica, vol. XI (Convegno di Algebra

Commutativa, INDAM, Rome, 1971 & Convegno di Geometria, INDAM, Rome, 1972). Academic Press,

London, pp. 81–104 (1973)

18. Gautam, S., Toledano Laredo, V.: Meromorphic tensor equivalence for Yangians and quantum loop

algebras. Publ. Math. Inst. Hautes Études Sci. 125, 267–337 (2017). https://doi.org/10.1007/s10240017-0089-9

19. Geiß, C., Leclerc, B., Schröer, J.: Quivers with relations for symmetrizable Cartan matrices III: convolution

algebras. Represent. Theory 20, 375–413 (2016). https://doi.org/10.1090/ert/487

20. Geiss, C., Leclerc, B., Schröer, J.: Quivers with relations for symmetrizable Cartan matrices I: foundations.

Invent. Math. 209(1), 61–158 (2017). https://doi.org/10.1007/s00222-016-0705-1

21. Geiss, C., Leclerc, B., Schröer, J.: Quivers with relations for symmetrizable Cartan matrices IV: crystal

graphs and semicanonical functions. Selecta Math. (NS) 24(4), 3283–3348 (2018). https://doi.org/10.

1007/s00029-018-0412-4

22. Geiß, C., Leclerc, B., Schröer, J.: Rigid modules and Schur roots. Math. Z. 295(3–4), 1245–1277 (2020).

https://doi.org/10.1007/s00209-019-02396-5

∞ ) and applications. J. Algebra 329, 147–162 (2011). https://doi.org/

23. Hernandez, D.: The algebra Uq (sl

10.1016/j.jalgebra.2010.04.002

24. Hernandez, D., Leclerc, B.: Quantum Grothendieck rings and derived Hall algebras. J. Reine Angew.

Math. 701, 77–126 (2015). https://doi.org/10.1515/crelle-2013-0020

25. Iyama, O., Reiten, I.: Fomin–Zelevinsky mutation and tilting modules over Calabi–Yau algebras. Amer.

J. Math. 130(4), 1087–1149 (2008). https://doi.org/10.1353/ajm.0.0011

26. Kashiwara, M., Oh, Sj.: t-quantized Cartan matrix and R-matrices for cuspidal modules over quiver Hecke

algebras, preprint. arXiv:2302.08700 (2023)

123

Deformed Cartan matrices and generalized preprojective...

Page 27 of 27

63

27. Kashiwara, M., Oh, Sj.: The (q, t)-Cartan matrix specialized at q = 1 and its applications. Math. Z.

303(42) (2023). https://doi.org/10.1007/s00209-022-03195-1

28. Keller, B.: Quantum Cartan matrices categorified. Online talk at the meeting “Categorifications in representation theory", Leicester, September 16, 2020 (2020)

29. Kimura, T., Pestun, V.: Fractional quiver W-algebras. Lett. Math. Phys. 108(11), 2425–2451 (2018).

https://doi.org/10.1007/s11005-018-1087-7

30. Kimura, T., Pestun, V.: Quiver W-algebras. Lett. Math. Phys. 108(6), 1351–1381 (2018). https://doi.org/

10.1007/s11005-018-1072-1

31. Marcus, A., Pan, S.: Tilting complexes for group graded self-injective algebras. Tsukuba J. Math. 43(2),

211–222 (2019). https://doi.org/10.21099/tkbjm/1585706452

32. Mizuno, Y.: Classifying τ -tilting modules over preprojective algebras of Dynkin type. Math. Z. 277(3–4),

665–690 (2014). https://doi.org/10.1007/s00209-013-1271-5

33. Murakami, K.: On the module categories of generalized preprojective algebras of Dynkin type. Osaka J.

Math. 59(2), 387–402 (2022)

34. Murakami, K.: PBW parametrizations and generalized preprojective algebras. Adv. Math. 395, Paper No.

108, 144 (2022). https://doi.org/10.1016/j.aim.2021.108144

35. Nakajima, H.: Quiver varieties and cluster algebras. Kyoto J. Math. 51(1), 71–126 (2011). https://doi.

org/10.1215/0023608X-2010-021

36. Nakajima, H., Weekes, A.: Coulomb branches of quiver gauge theories with symmetrizers. J. Eur. Math.

Soc. (JEMS) 25(1), 203–230 (2023). https://doi.org/10.4171/JEMS/1176

37. Ringel, C.M.: Representations of K -species and bimodules. J. Algebra 41(2), 269–302 (1976). https://

doi.org/10.1016/0021-8693(76)90184-8

38. Söderberg, C.: Preprojective algebras of d-representation finite species with relations, preprint.

arXiv:2109.15187 v3 (2021)

39. Speyer, D.E.: Powers of Coxeter elements in infinite groups are reduced. Proc. Amer. Math. Soc. 137(4),

1295–1302 (2009). https://doi.org/10.1090/S0002-9939-08-09638-X

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