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

SIGMA 11 (2015), 024, 30 pages      arXiv:1401.4338      https://doi.org/10.3842/SIGMA.2015.024
Contribution to the Special Issue on New Directions in Lie Theory

### The Feigin Tetrahedron

Dylan Rupel
Department of Mathematics, Northeastern University, Boston, MA 02115, USA

Received September 11, 2014, in final form March 03, 2015; Published online March 19, 2015

Abstract
The first goal of this note is to extend the well-known Feigin homomorphisms taking quantum groups to quantum polynomial algebras. More precisely, we define generalized Feigin homomorphisms from a quantum shuffle algebra to quantum polynomial algebras which extend the classical Feigin homomorphisms along the embedding of the quantum group into said quantum shuffle algebra. In a recent work of Berenstein and the author, analogous extensions of Feigin homomorphisms from the dual Hall-Ringel algebra of a valued quiver to quantum polynomial algebras were defined. To relate these constructions, we establish a homomorphism, dubbed the quantum shuffle character, from the dual Hall-Ringel algebra to the quantum shuffle algebra which relates the generalized Feigin homomorphisms. These constructions can be compactly described by a commuting tetrahedron of maps beginning with the quantum group and terminating in a quantum polynomial algebra. The second goal in this project is to better understand the dual canonical basis conjecture for skew-symmetrizable quantum cluster algebras. In the symmetrizable types it is known that dual canonical basis elements need not have positive multiplicative structure constants, while this is still suspected to hold for skew-symmetrizable quantum cluster algebras. We propose an alternate conjecture for the symmetrizable types: the cluster monomials should correspond to irreducible characters of a KLR algebra. Indeed, the main conjecture of this note would establish this ''KLR conjecture'' for acyclic skew-symmetrizable quantum cluster algebras: that is, we conjecture that the images of rigid representations under the quantum shuffle character give irreducible characters for KLR algebras. We sketch a proof in the symmetric case giving an alternative to the proof of Kimura-Qin that all non-initial cluster variables in an acyclic skew-symmetric quantum cluster algebra are contained in the dual canonical basis. With these results in mind we interpret the cluster mutations directly in terms of the representation theory of the KLR algebra.

Key words: cluster algebra; Hall algebra; quantum group; quiver Hecke algebra; KLR algebra; dual canonical basis; Feigin homomorphism; categorification.

pdf (534 kb)   tex (46 kb)

References

1. Benkart G., Kang S., Oh S., Park E., Construction of irreducible representations over Khovanov-Lauda-Rouquier algebras of finite classical type, arXiv:1108.1048.
2. Berenstein A., Group-like elements in quantum groups and Feigin's conjecture, q-alg/9605016.
3. Berenstein A., Rupel D., Quantum cluster characters of Hall algebras, Selecta Math., to appear, arXiv:1308.2992.
4. Berenstein A., Zelevinsky A., Quantum cluster algebras, Adv. Math. 195 (2005), 405-455, math.QA/0404446.
5. Chen X., Xiao J., Exceptional sequences in Hall algebras and quantum groups, Compositio Math. 117 (1999), 161-187.
6. Chriss N., Ginzburg V., Representation theory and complex geometry, Birkhäuser Boston, Inc., Boston, MA, 1997.
7. Green J.A., Hall algebras, hereditary algebras and quantum groups, Invent. Math. 120 (1995), 361-377.
8. Hausel T., Rodriguez-Villegas F., Mixed Hodge polynomials of character varieties, Invent. Math. 174 (2008), 555-624, math.AG/0612668.
9. Hernandez D., Leclerc B., Quantum Grothendieck rings and derived Hall algebras, J. Reine Angew. Math., to appear, arXiv:1109.0862.
10. Iohara K., Malikov F., Rings of skew polynomials and Gel'fand-Kirillov conjecture for quantum groups, Comm. Math. Phys. 164 (1994), 217-237, hep-th/9306138.
11. Jiang Y., Sheng J., Xiao J., The elements in crystal bases corresponding to exceptional modules, Chin. Ann. Math. Ser. B 31 (2010), 1-20, arXiv:0902.1216.
12. Joseph A., Sur une conjecture de Feigin, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 1441-1444.
13. Kang S., Kashiwara M., Kim M., Symmetric quiver Hecke algebras and $R$-matrices of quantum affine algebras II, arXiv:1308.0651.
14. Khovanov M., Lauda A.D., A diagrammatic approach to categorification of quantum groups. I, Represent. Theory 13 (2009), 309-347, arXiv:0803.4121.
15. 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.
16. Kimura Y., Qin F., Graded quiver varieties, quantum cluster algebras and dual canonical basis, Adv. Math. 262 (2014), 261-312, arXiv:1205.2066.
17. Kleshchev A., Cuspidal systems for affine Khovanov-Lauda-Rouquier algebras, Math. Z. 276 (2014), 691-726, arXiv:1210.6556.
18. Kleshchev A., Ram A., Homogeneous representations of Khovanov-Lauda algebras, J. Eur. Math. Soc. 12 (2010), 1293-1306, arXiv:0809.0557.
19. Kleshchev A., Ram A., Representations of Khovanov-Lauda-Rouquier algebras and combinatorics of Lyndon words, Math. Ann. 349 (2011), 943-975, arXiv:0909.1984.
20. Lampe P., A quantum cluster algebra of Kronecker type and the dual canonical basis, Int. Math. Res. Not. 2011 (2011), 2970-3005, arXiv:1002.2762.
21. Lampe P., Quantum cluster algebras of type $A$ and the dual canonical basis, Proc. Lond. Math. Soc. 108 (2014), 1-43, arXiv:1101.0580.
22. Lauda A.D., Vazirani M., Crystals from categorified quantum groups, Adv. Math. 228 (2011), 803-861, arXiv:0909.1810.
23. Leclerc B., Dual canonical bases, quantum shuffles and $q$-characters, Math. Z. 246 (2004), 691-732, math.QA/0209133.
24. Lusztig G., Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), 365-421.
25. Lusztig G., Introduction to quantum groups, Progress in Mathematics, Vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993.
26. Lusztig G., Problems on canonical bases, in Algebraic Groups and their Generalizations: Quantum and Infinite-Dimensional Methods (University Park, PA, 1991), Proc. Sympos. Pure Math., Vol. 56, Amer. Math. Soc., Providence, RI, 1994, 169-176.
27. McNamara P.J., Finite dimensional representations of Khovanov-Lauda-Rouquier algebras I: Finite type, arXiv:1207.5860.
28. Qin F., Quantum cluster variables via Serre polynomials, J. Reine Angew. Math. 668 (2012), 149-190, arXiv:1004.4171.
29. Ringel C.M., Hall algebras and quantum groups, Invent. Math. 101 (1990), 583-591.
30. Ringel C.M., Hall algebras revisited, in Quantum Deformations of Algebras and their Representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992), Israel Math. Conf. Proc., Vol. 7, Bar-Ilan Univ., Ramat Gan, 1993, 171-176.
31. Rosso M., Quantum groups and quantum shuffles, Invent. Math. 133 (1998), 399-416.
32. Rouquier R., 2-Kac-Moody algebras, arXiv:0812.5023.
33. Rouquier R., Quiver Hecke algebras and 2-Lie algebras, Algebra Colloq. 19 (2012), 359-410, arXiv:1112.3619.
34. Rupel D., On a quantum analog of the Caldero-Chapoton formula, Int. Math. Res. Not. 2011 (2011), 3207-3236.
35. Rupel D., Quantum cluster characters of valued quivers, arXiv:1109.6694.
36. Toën B., Derived Hall algebras, Duke Math. J. 135 (2006), 587-615, math.QA/0501343.
37. Varagnolo M., Vasserot E., Canonical bases and KLR-algebras, J. Reine Angew. Math. 659 (2011), 67-100, math.RT/0107177.
38. Webster B., Weighted Khovanov-Lauda-Rouquier algebras, arXiv:1209.2463.