## 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**

## Bibliographic Information

**Jun Hu**- Affiliation: School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia
- Email: junhu303@yahoo.com.cn
- Received by editor(s): March 8, 2009
- Received by editor(s) in revised form: October 14, 2009
- Published electronically: January 10, 2011
- Additional Notes: This research was supported by National Natural Science Foundation of China (Project 10771014), the Program NCET and partly by an Australian Research Council discovery grant. The author also acknowledges the support of the Chern Institute of Mathematics during his visit in March of 2007.
- © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Represent. Theory
**15**(2011), 1-62 - MSC (2000): Primary 17B37, 20C20; Secondary 20C08
- DOI: https://doi.org/10.1090/S1088-4165-2011-00369-1
- MathSciNet review: 2754334