The universal Askey–Wilson algebra AW(3) can be obtained as the commutant of in . We analyze the commutant of in q-oscillator representations of and show that it also realizes AW(3). These two pictures of AW(3) are shown to be dual in the sense of Howe; this is made clear by highlighting the role of the intermediate Casimir elements of each member of the dual pair . We also generalize these results. A higher rank extension of the Askey–Wilson algebra denoted AW(n) can be defined as the commutant of in , and a dual description of AW(n) as the commutant of in q-oscillator representations of is offered by calling upon the dual pair .
REFERENCES
1.
A. S.
Zhedanov
, ““Hidden symmetry” of Askey-Wilson polynomials
,” Theor. Math. Phys.
89
, 1146
–1157
(1991
).2.
P.
Baseilhac
, “Deformed Dolan-Grady relations in quantum integrable models
,” Nucl. Phys. B
709
, 491
–521
(2005
); e-print arXiv:0404149v3 [hep-th].3.
B.
Aneva
, “Tridiagonal symmetries of models of nonequilibrium physics
,” Symmetry, Integrability Geom.: Methods Appl.
4
, 056
(2008
); e-print arXiv:0807.4391.4.
L.
Vinet
and A.
Zhedanov
, “Quasi-linear algebras and integrability (the Heisenberg picture)
,” Symmetry, Integrability Geom.: Methods Appl.
4
, 015
(2008
); e-print arXiv:0802.0744.5.
P.
Baseilhac
, “An integrable structure related with tridiagonal algebras
,” Nucl. Phys. B
705
, 605
–619
(2005
); e-print arXiv:0408025 [math-ph].6.
P.
Baseilhac
and K.
Koizumi
, “A new (in)finite-dimensional algebra for quantum integrable models
,” Nucl. Phys. B
720
, 325
–347
(2005
); e-print arXiv:0503036 [math-ph].7.
P.
Terwilliger
and R.
Vidunas
, “Leonard pairs and the Askey-Wilson relations
,” J. Algebra Appl.
03
, 411
–426
(2005
); e-print arXiv:0305356 [math].8.
P.
Terwilliger
, “The universal Askey-Wilson algebra
,” Symmetry, Integrability Geom.: Methods Appl.
7
, 069
(2011
); e-print arXiv:1104.2813.9.
P.
Terwilliger
, “The universal Askey-Wilson algebra and DAHA of type
,” Symmetry, Integrability Geom.: Methods Appl.
9
, 047
(2013
); e-print arXiv:1202.4673.10.
P.
Terwilliger
, “The q-Onsager algebra and the universal Askey–Wilson algebra
,” Symmetry, Integrability Geom.: Methods Appl.
14
, 044
(2018
); e-print arXiv:1801.06083.11.
D.
Bullock
and J. H.
Przytycki
, “Multiplicative structure of Kauffman bracket skein module quantizations
,” Proc. Am. Math. Soc.
128
, 923
–931
(1999
); e-print arXiv:9902117 [math].12.
T. H.
Koornwinder
, “The relationship between Zhedanov’s algebra AW(3) and the double affine Hecke algebra in the rank one case
,” Symmetry, Integrability Geom.: Methods Appl.
3
, 063
(2006
); e-print arXiv:0612730 [math].13.
T. H.
Koornwinder
, “Zhedanov’s algebra AW(3) and the double affine Hecke algebra in the rank one case. II. The spherical subalgebra
,” Symmetry, Integrability Geom.: Methods Appl.
4
, 052
(2008
); e-print arXiv:0711.2320v3.14.
M.
Mazzocco
, “Confluences of the Painlevé equations, Cherednik algebras and q-Askey scheme
,” Nonlinearity
29
, 2565
–2608
(2016
); e-print arXiv:1307.6140.15.
Y. A.
Granovskii
and A.
Zhedanov
, “Hidden symmetry of the Racah and Clebsch-Gordan problems for the quantum algebra
,” J. Group Theory Phys.
1
, 161
–171
(1993
); e-print arXiv:9304138 [hep-th].16.
H. W.
Huang
, “An embedding of the universal Askey-Wilson algebra into Uq(sl2) ⊗ Uq(sl2) ⊗ Uq(sl2)
,” Nucl. Phys. B
922
, 401
–434
(2017
); e-print arXiv:1611.02130.17.
R.
Howe
, “Remarks on classical invariant theory
,” Trans. Am. Math. Soc.
313
, 539
–570
(1989
).18.
R.
Howe
, “Transcending classical invariant theory
,” J. Am. Math. Soc.
2
, 535
–552
(1989
).19.
R.
Howe
, “Dual pairs in physics: Harmonic oscillators, photons, electrons, singletons
,” in Proceedings of the Applications of Group Theory in Physics and Mathematical Physics
, edited by M.
Flato
, P.
Sally
, and G.
Zuckerman
(AMS
, Chicago
, 1987
), Chap. 6, pp. 179
–206
.20.
M.
Moshinsky
and C.
Quesne
, “Linear canonical transformations and their unitary representations
,” J. Math. Phys.
12
, 1772
–1780
(1971
).21.
H. P.
Jakobsen
and M.
Vergne
, “Wave and Dirac operators, and representations of the conformal group
,” J. Funct. Anal.
24
, 52
–106
(1977
).22.
D. J.
Rowe
, J.
Repka
, and M. J.
Carvalho
, “Simple unified proofs of four duality theorems
,” J. Math. Phys.
52
, 013507
(2011
).23.
D. J.
Rowe
, M. J.
Carvalho
, and J.
Repka
, “Dual pairing of symmetry and dynamical groups in physics
,” Rev. Mod. Phys.
84
, 711
–757
(2012
); e-print arXiv:1207.0148.24.
J.
Gaboriaud
, L.
Vinet
, S.
Vinet
, and A.
Zhedanov
, “The Racah algebra as a commutant and Howe duality
,” J. Phys. A: Math. Theor.
51
, 50LT01
(2018
); e-print arXiv:1808.05261.25.
J.
Gaboriaud
, L.
Vinet
, S.
Vinet
, and A.
Zhedanov
, “The generalized Racah algebra as a commutant
,” J. Phys.: Conf. Ser.
1194
, 012034
(2019
); e-print arXiv:1808.09518v1.26.
M.
Noumi
, T.
Umeda
, and M.
Wakayama
, “Dual pairs, spherical harmonics and a Capelli identity in quantum group theory
,” Compos. Math.
104
, 227
–277
(1996
).27.
A.
Klimyk
and K.
Schmüdgen
, Quantum Groups and Their Representations
(Springer-Verlag Berlin Heidelberg
, 1997
), p. 552
.28.
A. M.
Gavrilik
and N. Z.
Iorgov
, “On Casimir elements of q-algebras (son) and their eigenvalues in representations
,” Proc. Inst. Math. NAS Ukraine
30
, 310
–314
(2000
); e-print arXiv:9911201 [math].29.
A. U.
Klimyk
, “The nonstandard q-deformation of enveloping algebra U(son): Results and problems
,” Czech. J. Phys.
51
, 331
–340
(2001
).30.
A. U.
Klimyk
, “On classification of irreducible representations of q-deformed algebra (son) related to quantum gravity
,” Proc. Inst. Math. NAS Ukraine
43
, 407
–418
(2002
) [math].31.
N. Z.
Iorgov
and A. U.
Klimyk
, “Classification theorem on irreducible representations of the q-deformed algebra (son)
,” Int. J. Math. Math. Sci.
2005
(2
), 225
–262
; e-print arXiv:0702482 [math].32.
M.
Noumi
, “Macdonald’s symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces
,” Adv. Math.
123
, 16
–77
(1996
); e-print arXiv:9503224 [math].33.
G.
Letzter
, “Coideal subalgebras and quantum symmetric pairs
,” in New Directions in Hopf Algebras
, edited by S.
Montgomery
and H.-J.
Schneider
(MSRI Publications
, Cambridge
, 2002
), pp. 117
–165
; e-print arXiv:0103228 [math].34.
C.
Itzykson
, “Remarks on boson commutation rules
,” Commun. Math. Phys.
4
, 92
–122
(1967
).35.
M.
Feigin
and T.
Hakobyan
, “On Dunkl angular momenta algebra
,” J. High Energy Phys.
2015
, 107
; e-print arXiv:1409.2480.36.
H.
De Bie
, V. X.
Genest
, W.
van de Vijver
, and L.
Vinet
, “A higher rank Racah algebra and the Laplace-Dunkl operator
,” J. Phys. A: Math. Theor.
51
, 025203
(2017
); e-print arXiv:1610.02638.37.
S.
Post
and A.
Walter
, “A higher rank extension of the Askey-Wilson algebra
,” e-print arXiv:1705.01860v2 (2017
).38.
H.
De Bie
, H.
De Clercq
, and W.
Van De Vijver
, “The higher rank q-deformed Bannai-Ito and Askey-Wilson algebra
,” Commun. Math. Phys.
374
, 277
–316
(2020
).39.
E. G.
Kalnins
, W.
Miller
, and S.
Post
, “Wilson polynomials and the generic superintegrable system on the 2-sphere
,” J. Phys. A: Math. Theor.
40
, 11525
–11538
(2007
).40.
P.
Iliev
, “The generic quantum superintegrable system on the sphere and Racah operators
,” Lett. Math. Phys.
107
, 2029
–2045
(2017
); e-print arXiv:1608.04590.41.
J.
Gaboriaud
, L.
Vinet
, S.
Vinet
, and A.
Zhedanov
, “The dual pair , the Dirac equation and the Bannai–Ito algebra
,” Nucl. Phys. B
937
, 226
–239
(2018
); e-print arXiv:1810.00130.© 2019 Author(s).
2019
Author(s)
You do not currently have access to this content.
