A homological model for Uq𝔰𝔩2 Verma modules and their braid representations

Martel, Jules¹

¹ Max Planck Institute for Mathematics, Bonn, Germany

Geometry & topology, Tome 26 (2022) no. 3, pp. 1225-1289.

Résumé

We extend Lawrence’s representations of the braid groups to relative homology modules and we show that they are free modules over a ring of Laurent polynomials. We define homological operators and we show that they actually provide a representation for an integral version for $U_q\mathfrak{𝔰}\mathfrak{𝔩}(2)$. We suggest an isomorphism between a given basis of homological modules and the standard basis of tensor products of Verma modules and we show it preserves the integral ring of coefficients, the action of $U_q\mathfrak{𝔰}\mathfrak{𝔩}(2)$, the braid group representation and its grading. This recovers an integral version for Kohno’s theorem relating absolute Lawrence representations with the quantum braid representation on highest-weight vectors. This is an extension of the latter theorem as we get rid of generic conditions on parameters, and as we recover the entire product of Verma modules as a braid group and a $U_q\mathfrak{𝔰}\mathfrak{𝔩}(2)$–module.

Détail

DOI : 10.2140/gt.2022.26.1225

Classification : 17B37, 20F36, 57M27, 57R56, 55N25, 55R80, 57M10
Keywords: quantum groups, braid representations, Verma modules, configuration spaces, twisted homology

Bibliographie

[1] S Baseilhac, Quantum coadjoint action and the 6j–symbols of Uqsl2, from: "Interactions between hyperbolic geometry, quantum topology and number theory" (editors A Champanerkar, O Dasbach, E Kalfagianni, I Kofman, W Neumann, N Stoltzfus), Contemp. Math. 541, Amer. Math. Soc. (2011) 103 | DOI

[2] S J Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14 (2001) 471 | DOI

[3] S Bigelow, A homological definition of the Jones polynomial, from: "Invariants of knots and –manifolds" (editors T Ohtsuki, T Kohno, T Le, J Murakami, J Roberts, V Turaev), Geom. Topol. Monogr. 4, Geom. Topol. Publ. (2002) 29 | DOI

[4] S Bigelow, Homological representations of the Iwahori–Hecke algebra, from: "Proceedings of the Casson Fest" (editors C Gordon, Y Rieck), Geom. Topol. Monogr. 7, Geom. Topol. Publ. (2004) 493 | DOI

[5] V Chari, A Pressley, A guide to quantum groups, Cambridge Univ. Press (1994)

[6] C De Concini, C Procesi, Quantum groups, from: "–modules, representation theory, and quantum groups" (editors G Zampieri, A D’Agnolo), Lecture Notes in Math. 1565, Springer (1993) 31 | DOI

[7] V G Drinfeld, Quasi-Hopf algebras, Algebra i Analiz 1 (1989) 114

[8] J M Droz, E Wagner, Grid diagrams and Khovanov homology, Algebr. Geom. Topol. 9 (2009) 1275 | DOI

[9] G Felder, C Wieczerkowski, Topological representations of the quantum group Uq(sl2), Comm. Math. Phys. 138 (1991) 583 | DOI

[10] K Habiro, An integral form of the quantized enveloping algebra of sl2 and its completions, J. Pure Appl. Algebra 211 (2007) 265 | DOI

[11] T Ito, Reading the dual Garside length of braids from homological and quantum representations, Comm. Math. Phys. 335 (2015) 345 | DOI

[12] C Jackson, T Kerler, The Lawrence–Krammer–Bigelow representations of the braid groups via Uq(sl2), Adv. Math. 228 (2011) 1689 | DOI

[13] C Kassel, Quantum groups, 155, Springer (1995) | DOI

[14] T Kohno, Monodromy representations of braid groups and Yang–Baxter equations, Ann. Inst. Fourier (Grenoble) 37 (1987) 139

[15] T Kohno, Hyperplane arrangements, local system homology and iterated integrals, from: "Arrangements of hyperplanes" (editors H Terao, S Yuzvinsky), Adv. Stud. Pure Math. 62, Math. Soc. Japan (2012) 157 | DOI

[16] T Kohno, Quantum and homological representations of braid groups, from: "Configuration spaces" (editors A Bjorner, F Cohen, C De Concini, C Procesi, M Salvetti), CRM Series 14, Ed. Norm. (2012) 355 | DOI

[17] D Krammer, Braid groups are linear, Ann. of Math. 155 (2002) 131 | DOI

[18] R J Lawrence, Homological representations of the Hecke algebra, Comm. Math. Phys. 135 (1990) 141 | DOI

[19] G Lusztig, Finite-dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc. 3 (1990) 257 | DOI

[20] C Manolescu, Nilpotent slices, Hilbert schemes, and the Jones polynomial, Duke Math. J. 132 (2006) 311 | DOI

[21] L Paoluzzi, L Paris, A note on the Lawrence–Krammer–Bigelow representation, Algebr. Geom. Topol. 2 (2002) 499 | DOI

[22] M Salvetti, Topology of the complement of real hyperplanes in CN, Invent. Math. 88 (1987) 603 | DOI

[23] V V Schechtman, A N Varchenko, Quantum groups and homology of local systems, from: "Algebraic geometry and analytic geometry" (editors A Fujiki, K Kato, T Katsura, Y Kawamata, Y Miyaoka), Springer (1991) 182 | DOI

[24] H Zheng, A reflexive representation of braid groups, J. Knot Theory Ramifications 14 (2005) 467 | DOI