Quantum algebras and symplectic reflection algebras for wreath products

Guay, Nicolas

doi:10.1090/S1088-4165-10-00366-3

Quantum algebras and symplectic reflection algebras for wreath products
HTML articles powered by AMS MathViewer

by Nicolas Guay

Represent. Theory 14 (2010), 148-200

DOI: https://doi.org/10.1090/S1088-4165-10-00366-3

Published electronically: February 9, 2010

PDF | Request permission

Abstract:

To a finite subgroup $\Gamma$ of $SL_2(\mathbb {C})$, we associate a new family of quantum algebras which are related to symplectic reflection algebras for wreath products $S_l\wr \Gamma$ via a functor of Schur-Weyl type. We explain that they are deformations of matrix algebras over rank-one symplectic reflection algebras for $\Gamma$ and construct for them a PBW basis. When $\Gamma$ is a cyclic group, we are able to give more information about their structure and to relate them to Yangians.

References

J. Alev, M. A. Farinati, T. Lambre, and A. L. Solotar, Homologie des invariants d’une algèbre de Weyl sous l’action d’un groupe fini, J. Algebra 232 (2000), no. 2, 564–577 (French, with English and French summaries). MR 1792746, DOI 10.1006/jabr.2000.8406
Susumu Ariki, Tomohide Terasoma, and Hirofumi Yamada, Schur-Weyl reciprocity for the Hecke algebra of $(\textbf {Z}/r\textbf {Z})\wr S_n$, J. Algebra 178 (1995), no. 2, 374–390. MR 1359891, DOI 10.1006/jabr.1995.1354
Yuri Berest, Pavel Etingof, and Victor Ginzburg, Cherednik algebras and differential operators on quasi-invariants, Duke Math. J. 118 (2003), no. 2, 279–337. MR 1980996, DOI 10.1215/S0012-7094-03-11824-4
Mitya Boyarchenko, Quantization of minimal resolutions of Kleinian singularities, Adv. Math. 211 (2007), no. 1, 244–265. MR 2313534, DOI 10.1016/j.aim.2006.08.003
William Crawley-Boevey and Martin P. Holland, Noncommutative deformations of Kleinian singularities, Duke Math. J. 92 (1998), no. 3, 605–635. MR 1620538, DOI 10.1215/S0012-7094-98-09218-3
Ivan Cherednik, Double affine Hecke algebras and Macdonald’s conjectures, Ann. of Math. (2) 141 (1995), no. 1, 191–216. MR 1314036, DOI 10.2307/2118632
Ivan Cherednik, Double affine Hecke algebras, London Mathematical Society Lecture Note Series, vol. 319, Cambridge University Press, Cambridge, 2005. MR 2133033, DOI 10.1017/CBO9780511546501
Vyjayanthi Chari and Andrew Pressley, Quantum affine algebras and affine Hecke algebras, Pacific J. Math. 174 (1996), no. 2, 295–326. MR 1405590, DOI 10.2140/pjm.1996.174.295
Vyjayanthi Chari and Andrew Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1994. MR 1300632
Charlotte Dezélée, Generalized graded Hecke algebras of types B and D, Comm. Algebra 34 (2006), no. 6, 2105–2128. MR 2235082, DOI 10.1080/00927870600549618
C. Dezelée, Generalized graded Hecke algebra for complex reflection group of type $G(r,1,n)$, arXiv:math.RT/0605410.
V. G. Drinfel′d, Degenerate affine Hecke algebras and Yangians, Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 69–70 (Russian). MR 831053
V. G. Drinfel′d, A new realization of Yangians and of quantum affine algebras, Dokl. Akad. Nauk SSSR 296 (1987), no. 1, 13–17 (Russian); English transl., Soviet Math. Dokl. 36 (1988), no. 2, 212–216. MR 914215
C. F. Dunkl and E. M. Opdam, Dunkl operators for complex reflection groups, Proc. London Math. Soc. (3) 86 (2003), no. 1, 70–108. MR 1971464, DOI 10.1112/S0024611502013825
Pavel Etingof, Wee Liang Gan, Victor Ginzburg, and Alexei Oblomkov, Harish-Chandra homomorphisms and symplectic reflection algebras for wreath-products, Publ. Math. Inst. Hautes Études Sci. 105 (2007), 91–155. MR 2354206, DOI 10.1007/s10240-007-0005-9
Benjamin Enriquez, PBW and duality theorems for quantum groups and quantum current algebras, J. Lie Theory 13 (2003), no. 1, 21–64. MR 1958574
Pavel Etingof and Victor Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), no. 2, 243–348. MR 1881922, DOI 10.1007/s002220100171
Marco Farinati, Hochschild duality, localization, and smash products, J. Algebra 284 (2005), no. 1, 415–434. MR 2115022, DOI 10.1016/j.jalgebra.2004.09.009
B. Feigin, M. Finkelberg, A. Negut, L. Rybnikov, Yangians and cohomology rings of Laumon spaces, arXiv:0812.465 [math.AG].
Wee Liang Gan and Victor Ginzburg, Deformed preprojective algebras and symplectic reflection algebras for wreath products, J. Algebra 283 (2005), no. 1, 350–363. MR 2102087, DOI 10.1016/j.jalgebra.2004.08.007
Victor Ginzburg, Nicolas Guay, Eric Opdam, and Raphaël Rouquier, On the category $\scr O$ for rational Cherednik algebras, Invent. Math. 154 (2003), no. 3, 617–651. MR 2018786, DOI 10.1007/s00222-003-0313-8
N. Guay, D. Hernandez, S. Loktev, Double affine Lie algebras and finite groups, to appear in the Pacific Journal of Mathematics.
Victor Ginzburg, Mikhail Kapranov, and Éric Vasserot, Langlands reciprocity for algebraic surfaces, Math. Res. Lett. 2 (1995), no. 2, 147–160. MR 1324698, DOI 10.4310/MRL.1995.v2.n2.a4
Iain Gordon and S. Paul Smith, Representations of symplectic reflection algebras and resolutions of deformations of symplectic quotient singularities, Math. Ann. 330 (2004), no. 1, 185–200. MR 2091684, DOI 10.1007/s00208-004-0545-y
I. Gordon and J. T. Stafford, Rational Cherednik algebras and Hilbert schemes, Adv. Math. 198 (2005), no. 1, 222–274. MR 2183255, DOI 10.1016/j.aim.2004.12.005
Nicolas Guay, Cherednik algebras and Yangians, Int. Math. Res. Not. 57 (2005), 3551–3593. MR 2199856, DOI 10.1155/IMRN.2005.3551
Nicolas Guay, Affine Yangians and deformed double current algebras in type A, Adv. Math. 211 (2007), no. 2, 436–484. MR 2323534, DOI 10.1016/j.aim.2006.08.007
Nicolas Guay, Quantum algebras and quivers, Selecta Math. (N.S.) 14 (2009), no. 3-4, 667–700. MR 2511195, DOI 10.1007/s00029-009-0496-y
David Hernandez, Representations of quantum affinizations and fusion product, Transform. Groups 10 (2005), no. 2, 163–200. MR 2195598, DOI 10.1007/s00031-005-1005-9
David Hernandez, Drinfeld coproduct, quantum fusion tensor category and applications, Proc. Lond. Math. Soc. (3) 95 (2007), no. 3, 567–608. MR 2368277, DOI 10.1112/plms/pdm017
Christian Kassel, Kähler differentials and coverings of complex simple Lie algebras extended over a commutative algebra, Proceedings of the Luminy conference on algebraic $K$-theory (Luminy, 1983), 1984, pp. 265–275. MR 772062, DOI 10.1016/0022-4049(84)90040-9
C. Kassel and J.-L. Loday, Extensions centrales d’algèbres de Lie, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 4, 119–142 (1983) (French, with English summary). MR 694130, DOI 10.5802/aif.896
Serge Z. Levendorskiĭ, On PBW bases for Yangians, Lett. Math. Phys. 27 (1993), no. 1, 37–42. MR 1212024, DOI 10.1007/BF00739587
S. Z. Levendorskiĭ, On generators and defining relations of Yangians, J. Geom. Phys. 12 (1993), no. 1, 1–11. MR 1226802, DOI 10.1016/0393-0440(93)90084-R
George Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), no. 3, 599–635. MR 991016, DOI 10.1090/S0894-0347-1989-0991016-9
Robert V. Moody, Senapathi Eswara Rao, and Takeo Yokonuma, Toroidal Lie algebras and vertex representations, Geom. Dedicata 35 (1990), no. 1-3, 283–307. MR 1066569, DOI 10.1007/BF00147350
Arun Ram and Anne V. Shepler, Classification of graded Hecke algebras for complex reflection groups, Comment. Math. Helv. 78 (2003), no. 2, 308–334. MR 1988199, DOI 10.1007/s000140300013
M. Varagnolo and E. Vasserot, Schur duality in the toroidal setting, Comm. Math. Phys. 182 (1996), no. 2, 469–483. MR 1447301, DOI 10.1007/BF02517898
M. Varagnolo and E. Vasserot, Double-loop algebras and the Fock space, Invent. Math. 133 (1998), no. 1, 133–159. MR 1626481, DOI 10.1007/s002220050242
M. Varagnolo and E. Vasserot, On the $K$-theory of the cyclic quiver variety, Internat. Math. Res. Notices 18 (1999), 1005–1028. MR 1722361, DOI 10.1155/S1073792899000525
Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324, DOI 10.1017/CBO9781139644136