Twisted Yangians, twisted quantum loop algebras and affine Hecke algebras of type $BC$
HTML articles powered by AMS MathViewer
- by Hongjia Chen, Nicolas Guay and Xiaoguang Ma PDF
- Trans. Amer. Math. Soc. 366 (2014), 2517-2574 Request permission
Abstract:
We study twisted Yangians of type AIII which have appeared in the literature under the name of reflection algebras. They admit $q$-versions which are new twisted quantum loop algebras. We explain how these can be defined equivalently either via the reflection equation or as coideal subalgebras of Yangians of $\mathfrak {gl}_n$ (resp. of quantum loop algebras of $\mathfrak {gl}_n$). The connection with affine Hecke algebras of type $BC$ comes from a functor of Schur-Weyl type between their module categories.References
- Tomoyuki Arakawa, Drinfeld functor and finite-dimensional representations of Yangian, Comm. Math. Phys. 205 (1999), no. 1, 1–18. MR 1706920, DOI 10.1007/s002200050664
- Tomoyuki Arakawa and Takeshi Suzuki, Duality between $\mathfrak {s}\mathfrak {l}_n(\textbf {C})$ and the degenerate affine Hecke algebra, J. Algebra 209 (1998), no. 1, 288–304. MR 1652134, DOI 10.1006/jabr.1998.7530
- 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
- Pascal Baseilhac and Samuel Belliard, Generalized $q$-Onsager algebras and boundary affine Toda field theories, Lett. Math. Phys. 93 (2010), no. 3, 213–228. MR 2679971, DOI 10.1007/s11005-010-0412-6
- S. Belliard and N. Crampé, Coideal algebras from twisted Manin triples, J. Geom. Phys. 62 (2012), no. 10, 2009–2023. MR 2944789, DOI 10.1016/j.geomphys.2012.05.008
- S. Belliard and V. Fomin, Generalized $q$-Onsager algebras and dynamical $K$-matrices, J. Phys. A 45 (2012), no. 2, 025201, 17. MR 2871402, DOI 10.1088/1751-8113/45/2/025201
- I. V. Cherednik, A new interpretation of Gel′fand-Tzetlin bases, Duke Math. J. 54 (1987), no. 2, 563–577. MR 899405, DOI 10.1215/S0012-7094-87-05423-8
- Ivan Cherednik, Lectures on Knizhnik-Zamolodchikov equations and Hecke algebras, Quantum many-body problems and representation theory, MSJ Mem., vol. 1, Math. Soc. Japan, Tokyo, 1998, pp. 1–96. MR 1724948
- Hongjia Chen and Nicolas Guay, Twisted affine Lie superalgebra of type $Q$ and quantization of its enveloping superalgebra, Math. Z. 272 (2012), no. 1-2, 317–347. MR 2968227, DOI 10.1007/s00209-011-0935-2
- H. Chen, N. Guay, Central extensions of matrix Lie superalgebras over $\mathbb {Z}/2\mathbb {Z}$-graded algebras, to appear in Algebras and Representation Theory.
- Vyjayanthi Chari and Andrew Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1994. MR 1300632
- Vyjayanthi Chari and Andrew Pressley, Quantum affine algebras and affine Hecke algebras, Pacific J. Math. 174 (1996), no. 2, 295–326. MR 1405590
- Mathijs S. Dijkhuizen and Jasper V. Stokman, Some limit transitions between $BC$ type orthogonal polynomials interpreted on quantum complex Grassmannians, Publ. Res. Inst. Math. Sci. 35 (1999), no. 3, 451–500. MR 1710751, DOI 10.2977/prims/1195143610
- G. W. Delius, N. J. MacKay, and B. J. Short, Boundary remnant of Yangian symmetry and the structure of rational reflection matrices, Phys. Lett. B 522 (2001), no. 3-4, 335–344. MR 1879394, DOI 10.1016/S0370-2693(01)01275-8
- V. G. Drinfel′d, Degenerate affine Hecke algebras and Yangians, Funktsional. Anal. i Prilozhen. 20 (1986), no. 1, 69–70 (Russian). MR 831053
- Pavel Etingof, Rebecca Freund, and Xiaoguang Ma, A Lie-theoretic construction of some representations of the degenerate affine and double affine Hecke algebras of type $BC_n$, Represent. Theory 13 (2009), 33–49. MR 2480387, DOI 10.1090/S1088-4165-09-00345-8
- Edward Frenkel and Evgeny Mukhin, The Hopf algebra $\textrm {Rep}\,U_q\widehat {\mathfrak {g}\mathfrak {l}}_\infty$, Selecta Math. (N.S.) 8 (2002), no. 4, 537–635. MR 1951206, DOI 10.1007/PL00012603
- 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 symplectic reflection algebras for wreath products, Represent. Theory 14 (2010), 148–200. MR 2593918, DOI 10.1090/S1088-4165-10-00366-3
- Nicolas Guay, David Hernandez, and Sergey Loktev, Double affine Lie algebras and finite groups, Pacific J. Math. 243 (2009), no. 1, 1–41. MR 2550136, DOI 10.2140/pjm.2009.243.1
- David Jordan and Xiaoguang Ma, Quantum symmetric pairs and representations of double affine Hecke algebras of type $C^\vee C_n$, Selecta Math. (N.S.) 17 (2011), no. 1, 139–181. MR 2765001, DOI 10.1007/s00029-010-0037-8
- Shin-ichi Kato, Irreducibility of principal series representations for Hecke algebras of affine type, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 929–943 (1982). MR 656065
- 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
- David Kazhdan and George Lusztig, Equivariant $K$-theory and representations of Hecke algebras. II, Invent. Math. 80 (1985), no. 2, 209–231. MR 788408, DOI 10.1007/BF01388604
- Sergey Khoroshkin and Maxim Nazarov, Yangians and Mickelsson algebras. I, Transform. Groups 11 (2006), no. 4, 625–658. MR 2278142, DOI 10.1007/s00031-005-1125-2
- Sergey Khoroshkin and Maxim Nazarov, Yangians and Mickelsson algebras. II, Mosc. Math. J. 6 (2006), no. 3, 477–504, 587 (English, with English and Russian summaries). MR 2274862, DOI 10.17323/1609-4514-2006-6-3-477-504
- Sergey Khoroshkin and Maxim Nazarov, Twisted Yangians and Mickelsson algebras. I, Selecta Math. (N.S.) 13 (2007), no. 1, 69–136. MR 2330588, DOI 10.1007/s00029-007-0036-6
- M. Nazarov and S. Khoroshkin, Twisted Yangians and Mickelsson algebras. II, Algebra i Analiz 21 (2009), no. 1, 153–228 (Russian); English transl., St. Petersburg Math. J. 21 (2010), no. 1, 111–161. MR 2553055, DOI 10.1090/S1061-0022-09-01088-7
- S. Kolb, Quantum symmetric Kac-Moody pairs, in preparation.
- Cathy Kriloff and Arun Ram, Representations of graded Hecke algebras, Represent. Theory 6 (2002), 31–69. MR 1915086, DOI 10.1090/S1088-4165-02-00160-7
- 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
- N. J. MacKay, Rational $K$-matrices and representations of twisted Yangians, J. Phys. A 35 (2002), no. 37, 7865–7876. MR 1945798, DOI 10.1088/0305-4470/35/37/302
- N. J. MacKay, Twisted Yangians and symmetric pairs, arXiv:0305285v1 [math.QA].
- A. Molev, M. Nazarov, and G. Ol′shanskiĭ, Yangians and classical Lie algebras, Uspekhi Mat. Nauk 51 (1996), no. 2(308), 27–104 (Russian); English transl., Russian Math. Surveys 51 (1996), no. 2, 205–282. MR 1401535, DOI 10.1070/RM1996v051n02ABEH002772
- A. I. Molev and E. Ragoucy, Representations of reflection algebras, Rev. Math. Phys. 14 (2002), no. 3, 317–342. MR 1894013, DOI 10.1142/S0129055X02001156
- A. I. Molev, E. Ragoucy, and P. Sorba, Coideal subalgebras in quantum affine algebras, Rev. Math. Phys. 15 (2003), no. 8, 789–822. MR 2027560, DOI 10.1142/S0129055X03001813
- M. Nazarov, unpublished note on reflection algebras and degenerated affine Hecke algebras of type $BC$, 2001.
- Maxim Nazarov, Yangian of the queer Lie superalgebra, Comm. Math. Phys. 208 (1999), no. 1, 195–223. MR 1729884, DOI 10.1007/s002200050754
- Masatoshi Noumi and Tetsuya Sugitani, Quantum symmetric spaces and related $q$-orthogonal polynomials, Group theoretical methods in physics (Toyonaka, 1994) World Sci. Publ., River Edge, NJ, 1995, pp. 28–40. MR 1413733
- Erhard Neher, Alistair Savage, and Prasad Senesi, Irreducible finite-dimensional representations of equivariant map algebras, Trans. Amer. Math. Soc. 364 (2012), no. 5, 2619–2646. MR 2888222, DOI 10.1090/S0002-9947-2011-05420-6
- G. I. Ol′shanskiĭ, Twisted Yangians and infinite-dimensional classical Lie algebras, Quantum groups (Leningrad, 1990) Lecture Notes in Math., vol. 1510, Springer, Berlin, 1992, pp. 104–119. MR 1183482, DOI 10.1007/BFb0101183
- E. K. Sklyanin, Boundary conditions for integrable quantum systems, J. Phys. A 21 (1988), no. 10, 2375–2389. MR 953215
- D. Uglov, Yangian actions on higher level irreducible integrable modules of $\widehat {\mathfrak {gl}}_N$, arXiv:9802048v2 [math.QA].
- M. Varagnolo and E. Vasserot, Schur duality in the toroidal setting, Comm. Math. Phys. 182 (1996), no. 2, 469–483. MR 1447301
Additional Information
- Hongjia Chen
- Affiliation: School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of China
- Email: hjchenmath@gmail.com
- Nicolas Guay
- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada
- Email: nguay@ualberta.ca
- Xiaoguang Ma
- Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China
- Email: xgma@math.tsinghua.edu.cn
- Received by editor(s): January 1, 2012
- Received by editor(s) in revised form: July 6, 2012
- Published electronically: January 31, 2014
- Additional Notes: The second author was supported by an NSERC Discovery Grant. Special thanks to Alexander Molev for answering some of our questions and to M. Nazarov for a copy of his unpublished note and for allowing us to reproduce the proof of one of his results, namely Theorem 4.7
The third author was supported by a Postdoctoral Fellowship of the Pacific Institute for the Mathematical Sciences. - © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 2517-2574
- MSC (2010): Primary 17B37, 20C08
- DOI: https://doi.org/10.1090/S0002-9947-2014-05994-1
- MathSciNet review: 3165646