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 Representation Theory
Representation Theory
ISSN 1088-4165

     

Quantum algebras and symplectic reflection algebras for wreath products

Author(s): Nicolas Guay
Journal: Represent. Theory 14 (2010), 148-200.
MSC (2010): Primary 17B37; Secondary 20C08
Posted: February 9, 2010
MathSciNet review: 2593918
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Abstract | References | Similar articles | Additional information

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:

[AFLS]
J. Alev, M.A. Farinati, T. Lambre, 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. MR 1792746 (2002c:16047)

[ATY]
S. Ariki, T. Terasoma, H. Yamada, Schur-Weyl reciprocity for the Hecke algebra of $ (\mathbb{Z}/r\mathbb{Z})\wr S\sb n$, J. Algebra 178 (1995), no. 2, 374-390. MR 1359891 (96k:20010)

[BEG]
Y. Berest, P. Etingof, V. Ginzburg, Cherednik algebras and differential operators on quasi-invariants, Duke Math. J. 118 (2003), no. 2, 279-337. MR 1980996 (2004f:16039)

[Bo]
D. Boyachenko, Quantization of minimal resolutions of Kleinian singularities, Adv. Math. 211 (2007), no. 1, 244-265. MR 2313534 (2008f:14001)

[CBHo]
W. Crawley-Boevey, M.P. Holland, Noncommutative deformations of Kleinian singularities, Duke Math. J. 92 (1998), no. 3, 605-635. MR 1620538 (99f:14003)

[Ch1]
I. Cherednik, Double affine Hecke algebras and Macdonald's conjectures, Ann. of Math. (2) 141 (1995), no. 1, 191-216. MR 1314036 (96m:33010)

[Ch2]
I. Cherednik, Double affine Hecke algebras, London Mathematical Society Lecture Note Series, 319, Cambridge University Press, Cambridge, 2005. xii+434 pp. MR 2133033 (2007e:32012)

[ChPr1]
V. Chari, A. Pressley, Quantum affine algebras and affine Hecke algebras, Pacific J. Math. 174 (1996), no. 2, 295-326. MR 1405590 (97i:17011)

[ChPr2]
V. Chari, A. Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1994. xvi+651 pp. MR 1300632 (95j:17010)

[De1]
C. Dezelée, Generalized graded Hecke algebras of types B and D, Comm. Algebra 34 (2006), no. 6, 2105-2128. MR 2235082 (2008b:20009)

[De2]
C. Dezelée, Generalized graded Hecke algebra for complex reflection group of type $ G(r,1,n)$, arXiv:math.RT/0605410.

[Dr1]
V. Drinfeld, Degenerate affine Hecke algebras and Yangians, (Russian) Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 69-70. MR 831053 (87m:22044)

[Dr2]
V. Drinfeld, A new realization of Yangians and of quantum affine algebras, Soviet Math. Dokl. 36 (1988), no. 2, 212-216. MR 914215 (88j:17020)

[DuOp]
C. Dunkl, E. Opdam, Dunkl operators for complex reflection groups, Proc. London Math. Soc. (3) 86 (2003), no. 1, 70-108. MR 1971464 (2004d:20040)

[EGGO]
P. Etingof, W.L. Gan, V. Ginzburg, A. Oblomkov, Harish-Chandra homomorphisms and symplectic reflection algebras for wreath-products, Publ. Math. Inst. Hautes Études Sci. No. 105 (2007), 91-155. MR 2354206 (2009j:16016)

[En]
B. Enriquez, PBW and duality theorems for quantum groups and quantum current algebras, J. Lie Theory 13 (2003), no. 1, 21-64. MR 1958574 (2004a:17016)

[EtGi]
P. Etingof, V. Ginzburg, Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002), no. 2, 243-348. MR 1881922 (2003b:16021)

[Fa]
M. Farinati, Hochschild duality, localization, and smash products, J. Algebra 284 (2005), no. 1, 415-434. MR 2115022 (2005j:16009)

[FFNR]
B. Feigin, M. Finkelberg, A. Negut, L. Rybnikov, Yangians and cohomology rings of Laumon spaces, arXiv:0812.465 [math.AG].

[GaGi]
W.L. Gan, V. Ginzburg, Deformed preprojective algebras and symplectic reflection algebras for wreath products, J. Algebra 283 (2005), no. 1, 350-363. MR 2102087 (2005h:16049)

[GGOR]
V. Ginzburg, N. Guay, E. Opdam, R. Rouquier, On the category $ \mathcal{O}$ for rational Cherednik algebras, Invent. Math. 154 (2003), no. 3, 617-651. MR 2018786 (2005f:20010)

[GHL]
N. Guay, D. Hernandez, S. Loktev, Double affine Lie algebras and finite groups, to appear in the Pacific Journal of Mathematics.

[GKV]
V. Ginzburg, M. Kapranov, E. Vasserot, Langlands reciprocity for algebraic surfaces, Math. Res. Lett. 2 (1995), no. 2, 147-160. MR 1324698 (96f:11086)

[GoSm]
I. Gordon, P. Smith, Representations of symplectic reflection algebras and resolutions of deformations of symplectic quotient singularities, Math. Ann. 330 (2004), no. 1, 185-200. MR 2091684 (2006f:14013)

[GoSt]
I. Gordon, J.T. Stafford, Rational Cherednik algebras and Hilbert schemes, Adv. Math. 198 (2005), no. 1, 222-274. MR 2183255 (2008i:14006)

[Gu1]
N. Guay, Cherednik algebras and Yangians, Int. Math. Res. Not. 2005, no.57, 3551-3593. MR 2199856 (2006m:16040)

[Gu2]
N. Guay, Affine Yangians and deformed double current algebras in type $ A$, Adv. Math. 211 (2007), no. 2, 436-484. MR 2323534 (2008d:17020)

[Gu3]
N. Guay, Quantum algebras and quivers, Selecta Math. (N.S.) 14 (2009), no. 3-4, 667-700. MR 2511195

[He1]
D. Hernandez, Representations of quantum affinizations and fusion product, Transformation Groups 10 (2005), no. 2, 163-200. MR 2195598 (2006k:17025)

[He2]
D. Hernandez, Drinfeld coproduct, quantum fusion tensor category and applications, Proc. Lond. Math. Soc. (3) 95 (2007), no. 3, 567-608. MR 2368277 (2008k:17017)

[Ka]
C. 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), J. Pure Appl. Algebra 34 (1984), no. 2-3, 265-275. MR 772062 (86h:17013)

[KaLo]
C. Kassel, J.L. Loday, Extensions centrales d'algèbres de Lie, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 4, 119-142 (1983). MR 694130 (85g:17004)

[Le1]
S.Z. Levendorski, On PBW bases for Yangians, Lett. Math. Phys. 27 (1993), no. 1, 37-42. MR 1212024 (94b:17027)

[Le2]
S.Z. Levendorski, On generators and defining relations of Yangians, J. Geom. Phys. 12 (1993), no. 1, 1-11. MR 1226802 (94d:17017)

[Lu]
G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), no. 3, 599-635. MR 991016 (90e:16049)

[MRY]
R.V. Moody, S. E. Rao, T. Yokonuma, Toroidal Lie algebras and vertex representations, Geom. Dedicata 35 (1990), no. 1-3, 283-307. MR 1066569 (91i:17032)

[RaSh]
A. Ram, A. Shepler, Classification of graded Hecke algebras for complex reflection groups, Comment. Math. Helv. 78 (2003), no. 2, 308-334. MR 1988199 (2004d:20007)

[VaVa1]
E. Vasserot, M. Varagnolo, Schur duality in the toroidal setting, Comm. Math. Phys. 182 (1996), no. 2, 469-483. MR 1447301 (98a:17024)

[VaVa2]
M. Varagnolo, E. Vasserot, Double-loop algebras and the Fock space, Invent. Math. 133 (1998), no. 1, 133-159. MR 1626481 (99g:17035)

[VaVa3]
M. Varagnolo, E. Vasserot, On the $ K$-theory of the cyclic quiver variety, Internat. Math. Res. Notices 1999, no. 18, 1005-1028. MR 1722361 (2000m:14011)

[We]
C. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, Cambridge, 1994. xiv+450 pp. MR 1269324 (95f:18001)

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Additional Information:

Nicolas Guay
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, CAB 632, Edmonton, Alberta T6G 2G1, Canada
Email: nguay@math.ualberta.ca

DOI: 10.1090/S1088-4165-10-00366-3
PII: S 1088-4165(10)00366-3
Received by editor(s): October 19, 2007
Received by editor(s) in revised form: September 29, 2009
Posted: February 9, 2010
Copyright of article: Copyright 2010, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




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