## Temperley–Lieb, Brauer and Racah algebras and other centralizers of $\mathfrak {su}(2)$

HTML articles powered by AMS MathViewer

- by Nicolas Crampé, Loïc Poulain d’Andecy and Luc Vinet PDF
- Trans. Amer. Math. Soc.
**373**(2020), 4907-4932 Request permission

## Abstract:

In the spirit of the Schur–Weyl duality, we study the connections between the Racah algebra and the centralizers of tensor products of three (possibly different) irreducible representations of $\mathfrak {su}(2)$. As a first step we show that the Racah algebra always surjects onto the centralizer. We then offer a conjecture regarding the description of the kernel of the map, which depends on the irreducible representations. If true, this conjecture would provide a presentation of the centralizer as a quotient of the Racah algebra. We prove this conjecture in several cases. In particular, while doing so, we explicitly obtain the Temperley–Lieb algebra, the Brauer algebra and the one-boundary Temperley–Lieb algebra as quotients of the Racah algebra.## References

- 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 - 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 - N. Crampe, L. Frappat, and L. Vinet,
*Bannai–Ito and Brauer algebras,*J. Phys. A**52**(2019), 424001, and arXiv:1906.03936. - Hendrik De Bie, Vincent X. Genest, Wouter van de Vijver, and Luc Vinet,
*A higher rank Racah algebra and the $\Bbb Z^n_2$ Laplace-Dunkl operator*, J. Phys. A**51**(2018), no. 2, 025203, 20. MR**3741969**, DOI 10.1088/1751-8121/aa9756 - Hendrik De Bie, Vincent X. Genest, and Luc Vinet,
*The $\Bbb {Z}_2^n$ Dirac-Dunkl operator and a higher rank Bannai-Ito algebra*, Adv. Math.**303**(2016), 390–414. MR**3552530**, DOI 10.1016/j.aim.2016.08.007 - J. J. Duistermaat and F. A. Grünbaum,
*Differential equations in the spectral parameter*, Comm. Math. Phys.**103**(1986), no. 2, 177–240. MR**826863**, DOI 10.1007/BF01206937 - Ya. A. Granovskiĭ and A. S. Zhedanov,
*Nature of the symmetry group of the $6j$-symbol*, Zh. Èksper. Teoret. Fiz.**94**(1988), no. 10, 49–54 (Russian); English transl., Soviet Phys. JETP**67**(1988), no. 10, 1982–1985 (1989). MR**997934** - Plamen Iliev,
*Bispectral extensions of the Askey-Wilson polynomials*, J. Funct. Anal.**266**(2014), no. 4, 2294–2318. MR**3150161**, DOI 10.1016/j.jfa.2013.06.018 - 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 - Gustav Isaac Lehrer and Ruibin Zhang,
*On endomorphisms of quantum tensor space*, Lett. Math. Phys.**86**(2008), no. 2-3, 209–227. MR**2465755**, DOI 10.1007/s11005-008-0284-1 - G. I. Lehrer, and R. B. Zhang,
*A Temperley-Lieb analogue for the BMW algebra*, arXiv:0806.0687. - Paul Martin and Hubert Saleur,
*On an algebraic approach to higher-dimensional statistical mechanics*, Comm. Math. Phys.**158**(1993), no. 1, 155–190. MR**1243720**, DOI 10.1007/BF02097236 - Paul P. Martin and David Woodcock,
*On the structure of the blob algebra*, J. Algebra**225**(2000), no. 2, 957–988. MR**1741573**, DOI 10.1006/jabr.1999.7948 - A. Nichols, V. Rittenberg, and J. de Gier,
*One-boundary Temperley-Lieb algebras in the $XXZ$ and loop models*, J. Stat. Mech. Theory Exp.**3**(2005), P03003, 30. MR**2140125** - S. Post and A. Walter,
*A higher rank extension of the Askey-Wilson Algebra*, arXiv:1705.01860. - G. Racah,
*Theory of complex spectra II*, Phys. Rev.**62**(1942), 438–462. - H. N. V. Temperley and E. H. Lieb,
*Relations between the “percolation” and “colouring” problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the “percolation” problem*, Proc. Roy. Soc. London Ser. A**322**(1971), no. 1549, 251–280. MR**498284**, DOI 10.1098/rspa.1971.0067 - A. S. Zhedanov,
*“Hidden symmetry” of Askey-Wilson polynomials*, Teoret. Mat. Fiz.**89**(1991), no. 2, 190–204 (Russian, with English summary); English transl., Theoret. and Math. Phys.**89**(1991), no. 2, 1146–1157 (1992). MR**1151381**, DOI 10.1007/BF01015906

## Additional Information

**Nicolas Crampé**- Affiliation: Institut Denis-Poisson CNRS/UMR 7013, Université de Tours, Université d’Orléans, Parc de Grandmont, 37200 Tours, France; and Centre de Recherches Mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal, Québec, H3C 3J7, Canada
- Email: crampe1977@gmail.com
**Loïc Poulain d’Andecy**- Affiliation: Laboratoire de mathématiques de Reims UMR 9008, Université de Reims Champagne-Ardenne, UFR Sciences exactes et naturelles, Moulin de la Housse BP 1039, 51100 Reims, France
- Email: loic.poulain-dandecy@univ-reims.fr
**Luc Vinet**- Affiliation: Centre de Recherches Mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal, Québec, H3C 3J7, Canada
- MR Author ID: 178665
- ORCID: 0000-0001-6211-7907
- Email: vinet@CRM.UMontreal.ca
- Received by editor(s): August 2, 2019
- Received by editor(s) in revised form: November 4, 2019
- Published electronically: March 27, 2020
- Additional Notes: The first and second authors were partially supported by Agence National de la Recherche Projet AHA ANR-18-CE40-0001.

The research of the third author was supported in part by a Discovery Grant from the Natural Science and Engineering Research Council (NSERC) of Canada. - © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**373**(2020), 4907-4932 - MSC (2010): Primary 16S20, 17B35
- DOI: https://doi.org/10.1090/tran/8055
- MathSciNet review: 4127866