BMW algebra, quantized coordinate algebra and type 𝐶 Schur–Weyl duality

Hu, Jun

doi:10.1090/S1088-4165-2011-00369-1

BMW algebra, quantized coordinate algebra and type $C$ Schur–Weyl duality
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by Jun Hu

Represent. Theory 15 (2011), 1-62

DOI: https://doi.org/10.1090/S1088-4165-2011-00369-1

Published electronically: January 10, 2011

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Abstract:

We prove an integral version of the Schur–Weyl duality between the specialized Birman–Murakami–Wenzl algebra $\mathfrak {B}_n(-q^{2m+1},q)$ and the quantum algebra associated to the symplectic Lie algebra $\mathfrak {sp}_{2m}$. In particular, we deduce that this Schur–Weyl duality holds over arbitrary (commutative) ground rings, which answers a question of Lehrer and Zhang in the symplectic case. As a byproduct, we show that, as a $\mathbb {Z}[q,q^{-1}]$-algebra, the quantized coordinate algebra defined by Kashiwara (which he denoted by $A_q^{\mathbb {Z}}(g)$) is isomorphic to the quantized coordinate algebra arising from a generalized Faddeev–Reshetikhin–Takhtajan construction.

References

Henning Haahr Andersen, Patrick Polo, and Ke Xin Wen, Representations of quantum algebras, Invent. Math. 104 (1991), no. 1, 1–59. MR 1094046, DOI 10.1007/BF01245066
A. A. Beilinson, G. Lusztig, and R. MacPherson, A geometric setting for the quantum deformation of $\textrm {GL}_n$, Duke Math. J. 61 (1990), no. 2, 655–677. MR 1074310, DOI 10.1215/S0012-7094-90-06124-1
Richard Brauer, On algebras which are connected with the semisimple continuous groups, Ann. of Math. (2) 38 (1937), no. 4, 857–872. MR 1503378, DOI 10.2307/1968843
Wm. P. Brown, An algebra related to the orthogonal group, Michigan Math. J. 3 (1955), 1–22. MR 72122
William P. Brown, The semisimplicity of $\omega _f^n$, Ann. of Math. (2) 63 (1956), 324–335. MR 75931, DOI 10.2307/1969613
Joan S. Birman and Hans Wenzl, Braids, link polynomials and a new algebra, Trans. Amer. Math. Soc. 313 (1989), no. 1, 249–273. MR 992598, DOI 10.1090/S0002-9947-1989-0992598-X
Roger W. Carter and George Lusztig, On the modular representations of the general linear and symmetric groups, Math. Z. 136 (1974), 193–242. MR 354887, DOI 10.1007/BF01214125
Vyjayanthi Chari and Andrew Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1994. MR 1300632
C. de Concini and C. Procesi, A characteristic free approach to invariant theory, Advances in Math. 21 (1976), no. 3, 330–354. MR 422314, DOI 10.1016/S0001-8708(76)80003-5
Richard Dipper, Stephen Doty, and Jun Hu, Brauer algebras, symplectic Schur algebras and Schur-Weyl duality, Trans. Amer. Math. Soc. 360 (2008), no. 1, 189–213. MR 2342000, DOI 10.1090/S0002-9947-07-04179-7
Richard Dipper and Gordon James, Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. (3) 52 (1986), no. 1, 20–52. MR 812444, DOI 10.1112/plms/s3-52.1.20
Richard Dipper and Gordon James, The $q$-Schur algebra, Proc. London Math. Soc. (3) 59 (1989), no. 1, 23–50. MR 997250, DOI 10.1112/plms/s3-59.1.23
S. Donkin, Good filtrations of rational modules for reductive groups, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 69–80. MR 933351, DOI 10.1090/pspum/047.1/933351
Stephen Donkin, Invariants of several matrices, Invent. Math. 110 (1992), no. 2, 389–401. MR 1185589, DOI 10.1007/BF01231338
Stephen Doty, Polynomial representations, algebraic monoids, and Schur algebras of classical type, J. Pure Appl. Algebra 123 (1998), no. 1-3, 165–199. MR 1492900, DOI 10.1016/S0022-4049(96)00082-5
Stephen Doty, Presenting generalized $q$-Schur algebras, Represent. Theory 7 (2003), 196–213. MR 1990659, DOI 10.1090/S1088-4165-03-00176-6
Stephen Doty, Anthony Giaquinto, and John Sullivan, Presenting generalized Schur algebras in types $B$, $C$, $D$, Adv. Math. 206 (2006), no. 2, 434–454. MR 2263710, DOI 10.1016/j.aim.2005.09.006
Stephen Doty and Jun Hu, Schur-Weyl duality for orthogonal groups, Proc. Lond. Math. Soc. (3) 98 (2009), no. 3, 679–713. MR 2500869, DOI 10.1112/plms/pdn044
V. G. Drinfel′d, Hopf algebras and the quantum Yang-Baxter equation, Dokl. Akad. Nauk SSSR 283 (1985), no. 5, 1060–1064 (Russian). MR 802128
Jie Du, A note on quantized Weyl reciprocity at roots of unity, Algebra Colloq. 2 (1995), no. 4, 363–372. MR 1358684
Jie Du, Brian Parshall, and Leonard Scott, Quantum Weyl reciprocity and tilting modules, Comm. Math. Phys. 195 (1998), no. 2, 321–352. MR 1637785, DOI 10.1007/s002200050392
John Enyang, Cellular bases for the Brauer and Birman-Murakami-Wenzl algebras, J. Algebra 281 (2004), no. 2, 413–449. MR 2098377, DOI 10.1016/j.jalgebra.2003.03.002
N. Yu. Reshetikhin, L. A. Takhtadzhyan, and L. D. Faddeev, Quantization of Lie groups and Lie algebras, Algebra i Analiz 1 (1989), no. 1, 178–206 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 1, 193–225. MR 1015339
Jochen Gruber and Gerhard Hiss, Decomposition numbers of finite classical groups for linear primes, J. Reine Angew. Math. 485 (1997), 55–91. MR 1442189, DOI 10.1515/crll.1997.485.55
J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math. 123 (1996), no. 1, 1–34. MR 1376244, DOI 10.1007/BF01232365
James A. Green, Polynomial representations of $\textrm {GL}_{n}$, Lecture Notes in Mathematics, vol. 830, Springer-Verlag, Berlin-New York, 1980. MR 606556, DOI 10.1007/BFb0092296
I. Grojnowski and G. Lusztig, A comparison of bases of quantized enveloping algebras, Linear algebraic groups and their representations (Los Angeles, CA, 1992) Contemp. Math., vol. 153, Amer. Math. Soc., Providence, RI, 1993, pp. 11–19. MR 1247495, DOI 10.1090/conm/153/01304
D. Ju. Grigor’ev, An analogue of the Bruhat decomposition for the closure of the cone of a Chevalley group of the classical series, Sov. Math. Doklady 23 (1981), 393–397.
Takahiro Hayashi, Quantum deformation of classical groups, Publ. Res. Inst. Math. Sci. 28 (1992), no. 1, 57–81. MR 1147851, DOI 10.2977/prims/1195168856
Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, vol. 42, American Mathematical Society, Providence, RI, 2002. MR 1881971, DOI 10.1090/gsm/042
Jun Hu, Specht filtrations and tensor spaces for the Brauer algebra, J. Algebraic Combin. 28 (2008), no. 2, 281–312. MR 2430306, DOI 10.1007/s10801-007-0103-2
Michio Jimbo, A $q$-difference analogue of $U({\mathfrak {g}})$ and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), no. 1, 63–69. MR 797001, DOI 10.1007/BF00704588
Michio Jimbo, A $q$-analogue of $U({\mathfrak {g}}{\mathfrak {l}}(N+1))$, Hecke algebra, and the Yang-Baxter equation, Lett. Math. Phys. 11 (1986), no. 3, 247–252. MR 841713, DOI 10.1007/BF00400222
Masaki Kashiwara, Global crystal bases of quantum groups, Duke Math. J. 69 (1993), no. 2, 455–485. MR 1203234, DOI 10.1215/S0012-7094-93-06920-7
Masaki Kashiwara, Crystal bases of modified quantized enveloping algebra, Duke Math. J. 73 (1994), no. 2, 383–413. MR 1262212, DOI 10.1215/S0012-7094-94-07317-1
Louis H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), no. 2, 417–471. MR 958895, DOI 10.1090/S0002-9947-1990-0958895-7
Bertram Kostant, Groups over $Z$, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 90–98. MR 0207713
G. I. Lehrer and R. B. Zhang, Strongly multiplicity free modules for Lie algebras and quantum groups, J. Algebra 306 (2006), no. 1, 138–174. MR 2271576, DOI 10.1016/j.jalgebra.2006.03.043
G. Lusztig, Quantum deformations of certain simple modules over enveloping algebras, Adv. in Math. 70 (1988), no. 2, 237–249. MR 954661, DOI 10.1016/0001-8708(88)90056-4
George Lusztig, Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra, J. Amer. Math. Soc. 3 (1990), no. 1, 257–296. MR 1013053, DOI 10.1090/S0894-0347-1990-1013053-9
George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
G. Lusztig, Canonical bases in tensor products, Proc. Nat. Acad. Sci. U.S.A. 89 (1992), no. 17, 8177–8179. MR 1180036, DOI 10.1073/pnas.89.17.8177
H.R. Morton and A.J. Wassermann, A basis for the Birman–Murakami–Wenzl algebra, unpublished paper, 2000, http://www.liv.ac.uk/ su14/knotprints.html.
Jun Murakami, The Kauffman polynomial of links and representation theory, Osaka J. Math. 24 (1987), no. 4, 745–758. MR 927059
G. E. Murphy, The representations of Hecke algebras of type $A_n$, J. Algebra 173 (1995), no. 1, 97–121. MR 1327362, DOI 10.1006/jabr.1995.1079
Sebastian Oehms, Centralizer coalgebras, FRT-construction, and symplectic monoids, J. Algebra 244 (2001), no. 1, 19–44. MR 1856529, DOI 10.1006/jabr.2001.8909
Sebastian Oehms, Symplectic $q$-Schur algebras, J. Algebra 304 (2006), no. 2, 851–905. MR 2264282, DOI 10.1016/j.jalgebra.2005.07.030
I. Schur, Über die rationalen Darstellungen der allgemeinen linearen Gruppe, (1927). Reprinted in I. Schur, Gesammelte Abhandlungen, Vol. III, pp. 68–85, Springer-Verlag, Berlin, 1973.
Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. MR 1488158