### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 050, 43 pages      arXiv:2201.10931      https://doi.org/10.3842/SIGMA.2022.050

### Spherical Representations of $C^*$-Flows II: Representation System and Quantum Group Setup

Yoshimichi Ueda
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan

Received February 07, 2022, in final form June 26, 2022; Published online July 05, 2022

Abstract
This paper is a sequel to our previous study of spherical representations in the operator algebra setup. We first introduce possible analogs of dimension groups in the present context by utilizing the notion of operator systems and their relatives. We then apply our study to inductive limits of compact quantum groups, and establish an analogue of Olshanski's notion of spherical unitary representations of infinite-dimensional Gelfand pairs of the form $G$ < $G\times G$ (via the diagonal embedding) in the quantum group setup. This, in particular, justifies Ryosuke Sato's approach to asymptotic representation theory for quantum groups.

Key words: spherical representation; KMS state; ordered $*$-vector space; operator system; inductive limit; quantum group; $\sigma$-$C^*$-algebra.

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References

1. Borodin A., Bufetov A., Plancherel representations of $U(\infty)$ and correlated Gaussian free fields, Duke Math. J. 163 (2014), 2109-2158, arXiv:1301.0511.
2. Borodin A., Olshanski G., Representations of the infinite symmetric group, Cambridge Studies in Advanced Mathematics, Vol. 160, Cambridge University Press, Cambridge, 2017.
3. Boyer R.P., Characters of the infinite symplectic group - a Riesz ring approach, J. Funct. Anal. 70 (1987), 357-387.
4. Bratteli O., Robinson D.W., Operator algebras and quantum statistical mechanics. 1. $C^*$- and $W^*$-algebras. Symmetry groups. Decomposition of states, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987.
5. Bratteli O., Robinson D.W., Operator algebras and quantum statistical mechanics. 2. Equilibrium states. Models in quantum statistical mechanics, 2nd ed., Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997.
6. Brown N.P., Ozawa N., $C^*$-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, Vol. 88, Amer. Math. Soc., Providence, RI, 2008.
7. Dijkhuizen M.S., Some remarks on the construction of quantum symmetric spaces, Acta Appl. Math. 44 (1996), 59-80, arXiv:math.QA/9512225.
8. Dijkhuizen M.S., Stokman J.V., Some limit transitions between $BC$ type orthogonal polynomials interpreted on quantum complex Grassmannians, Publ. Res. Inst. Math. Sci. 35 (1999), 451-500, arXiv:math.QA/9806123.
9. Effros E.G., Dimensions and $C^{\ast}$-algebras, CBMS Regional Conference Series in Mathematics, Vol. 46, Conference Board of the Mathematical Sciences, Washington, D.C., 1981.
10. Geissinger L., Hopf algebras of symmetric functions and class functions, in Combinatoire et Représentation du Groupe Symétrique (Actes Table Ronde C.N.R.S., Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976), Lecture Notes in Math., Vol. 579, Springer, Berlin - Heidelberg, 1977, 168-181.
11. Gorin V., The $q$-Gelfand-Tsetlin graph, Gibbs measures and $q$-Toeplitz matrices, Adv. Math. 229 (2012), 201-266, arXiv:1011.1769.
12. Kerov S.V., Asymptotic representation theory of the symmetric group and its applications in analysis, Translations of Mathematical Monographs, Vol. 219, Amer. Math. Soc., Providence, RI, 2003.
13. Kirchberg E., On subalgebras of the CAR-algebra, J. Funct. Anal. 129 (1995), 35-63.
14. Kolb S., Stokman J.V., Reflection equation algebras, coideal subalgebras, and their centres, Selecta Math. (N.S.) 15 (2009), 621-664, arXiv:0812.4459.
15. Mahanta S., Mathai V., Operator algebra quantum homogeneous spaces of universal gauge groups, Lett. Math. Phys. 97 (2011), 263-277, arXiv:1012.5893.
16. Masuda T., Nakagami Y., A von Neumann algebra framework for the duality of the quantum groups, Publ. Res. Inst. Math. Sci. 30 (1994), 799-850.
17. Mawhinney L., Todorov I.G., Inductive limits in the operator system and related categories, Dissertationes Math. 536 (2018), 1-57, arXiv:1705.04663.
18. Neshveyev S., Tuset L., Compact quantum groups and their representation categories, Cours Spécialisés, Vol. 20, Société Mathématique de France, Paris, 2013.
19. Noumi M., Macdonald's symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces, Adv. Math. 123 (1996), 16-77, arXiv:math.QA/9503224.
20. Noumi M., Yamada H., Mimachi K., Finite-dimensional representations of the quantum group ${\rm GL}_q(n;{\bf C})$ and the zonal spherical functions on ${\rm U}_q(n-1)\backslash{\rm U}_q(n)$, Japan. J. Math. (N.S.) 19 (1993), 31-80.
21. Oblomkov A.A., Stokman J.V., Vector valued spherical functions and Macdonald-Koornwinder polynomials, Compos. Math. 141 (2005), 1310-1350, arXiv:math.QA/0311512.
22. Okada S., Littlewood-Richardson rings, in Proceedings of Symposium on Representation Theory, Publication Society of the Proceedings of the Symposium on Representation Theory, Japan, 1994, 82-96.
23. Olshanski G., Unitary representations of infinite-dimensional pairs $(G,K)$ and the formalism of R. Howe, in Representation of Lie Groups and Related Topics, Adv. Stud. Contemp. Math., Vol. 7, Gordon and Breach, New York, 1990, 269-463.
24. Olshanski G., An introduction to harmonic analysis on the infinite symmetric group, in Asymptotic Combinatorics with Applications to Mathematical Physics (St. Petersburg, 2001), Lecture Notes in Math., Vol. 1815, Editors A.M. Vershik, Y. Yakubovich, Springer, Berlin, 2003, 127-160, arXiv:math.RT/0311369.
25. Olshanski G., The problem of harmonic analysis on the infinite-dimensional unitary group, J. Funct. Anal. 205 (2003), 464-524, arXiv:math.RT/0109193.
26. Olshanski G., The representation ring of the unitary groups and Markov processes of algebraic origin, Adv. Math. 300 (2016), 544-615, arXiv:1504.01646.
27. Onn U., Stokman J.V., Quantum dimensions and their non-Archimedean degenerations, Int. Math. Res. Pap. 2006 (2006), 54701, 53 pages, arXiv:math.QA/0606222.
28. Paulsen V., Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, Vol. 78, Cambridge University Press, Cambridge, 2002.
29. Pietsch A., History of Banach spaces and linear operators, Birkhäuser Boston, Inc., Boston, MA, 2007.
30. Reich H., On the $K$- and $L$-theory of the algebra of operators affiliated to a finite von Neumann algebra, $K$-Theory 24 (2001), 303-326.
31. Sato R., Quantized Vershik-Kerov theory and quantized central measures on branching graphs, J. Funct. Anal. 277 (2019), 2522-2557, arXiv:1804.02644.
32. Sato R., Inductive limits of compact quantum groups and their unitary representations, Lett. Math. Phys. 111 (2021), 122, 20 pages, arXiv:1908.03988.
33. Sato R., Type classification of extreme quantized characters, Ergodic Theory Dynam. Systems 41 (2021), 593-605, arXiv:1903.07454.
34. Tomatsu R., A characterization of right coideals of quotient type and its application to classification of Poisson boundaries, Comm. Math. Phys. 275 (2007), 271-296, arXiv:math.OA/0611327.
35. Ueda Y., Spherical representations of $C^*$-flows I, arXiv:2010.15324.
36. Vershik A.M., Kerov S.V., Asymptotic theory of characters of the symmetric group, Funct. Anal. Appl. 15 (1981), 246-255.
37. Vershik A.M., Kerov S.V., Characters and factor representations of the infinite unitary group, Sov. Math. Dokl. 26 (1982), 570-574.
38. Vershik A.M., Kerov S.V., Locally semisimple algebras, combinatorial theory and the $K_0$-functor, J. Sov. Math. 38 (1987), 1701-1733.
39. Vershik A.M., Kerov S.V., The Grothendieck group of infinite symmetric group and symmetric functions (with the elements of the theory of $K_0$-functor of AF-algebras), in Representation of Lie Groups and Related Topics, Adv. Stud. Contemp. Math., Vol. 7, Gordon and Breach, New York, 1990, 39-117.
40. Yamagami S., On unitary representation theories of compact quantum groups, Comm. Math. Phys. 167 (1995), 509-529.
41. Yamashita M., Elements of quantum groups, Kyoritsu Shuppan, Japan, 2017.