Asymptotic representation thoery of quantum groups and related topics

[1] P. Biane, Quantum random walk on the dual of SU (n), Probab. Th. Rel. Fields 89 (1991) 117–129.

[2] B. Blackadar, Operator Algebras : Theory of C∗-Algebras and von Neumann Algebras, Encyclopedia of Math- ematical Sciences 122, Springer-Verlag Berlin Heidelberg, 2006.

[3] A. Borodin, I. Corwin, Macdonald processes, Probab. Theory and Related Fields 158 (2014), 225–400.

[4] A. Borodin, P. L. Ferrari, Anisotropic growth of random surfaces in 2+1 dimensions, Commun. Math. Phys. 325 (2014), 603–684.

[5] A. Borodin, V. Gorin, Markov processes of infinitely many nonintersecting random walk, Probab. Theory Relat. Fields. 155 (2013), 935–997.

[6] A. Borodin, G. Olshanski, Markov processes o the path space of the Gelfand–Tsetlin graph and on its boundary, J. Funct. Anal. 263 (2012), 248–303.

[7] A. Borodin, L. Petrov, Integrable probability: From representation theory to Macdonald processes, Probab. Surv. 11, 1–58.

[8] O. Bratteli, D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1. Equilibrium states. Models in quantum statistical mechanics. Second edition, Texts and Monographs in Physics, Springer-Verlag, Berlin, Heidelberg, 1997.

[9] O. Bratteli, D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics 2. Equilibrium states. Models in quantum statistical mechanics. Second edition, Texts and Monographs in Physics, Springer-Verlag, Berlin, Heidelberg, 1997.

[10] A. Borodin, G. Olshanski, Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes, Ann. of Math. (2) 161 (2005), no. 3, 1319–1422

[11] A. Borodin, G. Olshanski, Representations of the Infinite Symmetric Group, Cambridge University Press, 2016.

[12] R. P. Boyer, Infinite traces of AF-algebras and characters of U (∞), J. Operator Theory 9 (1983), 205–236.

[13] R. P. Boyer, Characters and factor representations of the infinite dimensional classical groups, J. Operator Theory 28(1992), no. 2, pp. 281–307.

[14] G. Brown, A. H. Dooley, J. Lake, On the Krieger–Araki–Woods ratio set, Tˆohoku Math. J. 47 (1995), 1–13.

[15] F. Cipriani, U. Franz, A. Kula, Symmetries of L´evy processes on compact quantum groups, their Markov semigroups and potential theory, Journal of Funct. Anal., 266 (2014), no. 5, 2789–2844.

[16] C. Cuenca, Asymptotic Formulas for Macdonald Polynomials and the Boundary of the (q, t)-Gelfand–Tsetlin graph, SIGMA 14 (2018), No. 001, 66pp.

[17] C. Cuenca, V. Gorin, q-Deformed character theory for infinite-dimensional symplectic and orthogonal groups, Selecta Mathematica 26 (2020), pp. 1–55.

[18] V. Chari, A. Presseley, A Guide to Quantum Groups, Cambridge University Press, 1994.

[19] J. Dixmier, C∗-algebras, North-Holland Mathematical Library, North-Holland Publishing Co. Amsterdam-New York-Oxford, 1997.

[20] R. Durrett, Probability: theory and examples, Fourth edition, Cambridge Series in Statistical and Probabilistic Mathematics, 31. Cambridge University Press, 2010.

[21] M. Enock, J.-M. Schwartz, Kac Algebras and Duality of Locally Compact Groups, Springer-Verlag Berlin Hei- delberg, 1992.

[22] T. Enomoto, M. Izumi, Indecomposable characters of infinite dimensional groups associated with operator algebras, J. Math. Soc. Japan 68 (2016), no. 3, 1231–1270

[23] S. N. Ethier, T. G. Kurtz, Markov processes – Characterization and Convergence, Wiley–Interscience, New York, 1986.

[24] U. Franz, L´evy processes on quantum group and dual groups, In: Quantum independent increment processes II., Springer, Berlin, Heidelberg (2006), 161–257.

[25] W. Fulton, J. Harris, Representation Theory: A First Course, Vol. 129. Springer Science & Business Media, 2013.

[26] V. Gorin, The q-Gelfand–Tsetlin graph, Gibbs measures and q-Toeplitz matrices, Adv. Math. 229 (2012), 201– 266.

[27] T. Hirai, E. Hirai, Positive definite class functions on a topological group and characters of factor representations, J. Math. Kyoto Univ. 45 (2005), no. 2, 355–376

[28] T. Hirai, H. Shimomura, N. Tatsuma, E. Hirai, Inductive limits of topologies, their direct products, and problems related to algebraic structures, J. Math. Kyoto Univ. 41 (2001), 475–505.

[29] M. Izumi, Non-coummutative Poisson boundaries and compact quantum group actions, Adv. in Math. 169(2002), no.1, 1–57.

[30] M. Izumi, S. Neshveyev, L. Tuset, Poisson boundary of the dual of SUq (n), Commun. Math. Phys. 262 (2006), 505–531.

[31] S. V. Kerov, Asymptotic Representation Theory of the Symmetric Group and its Applications in Analysis, Trans. Math. Mono. 219, Amer. Math. Soc., 2003.

[32] A. A. Kirillov, Introduction to the theory of representations and noncommutative harmonic analysis, Represen- tation Theory and Noncommutative Harmonic Analysis I, Springer, Berlin, Heidelberg, 1994.

[33] A. Klimyk, K. Schmu¨dgen, Quantum Groups and Their Representations, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997.

[34] Z. Kosloff, On a type III1 type Bernoulli shift, Ergod. Th. & Dynam. Sys. 31 (2011), no. 6, 1727–1747

[35] J. Kuan, Two construction of Markov chains on the dual of U (n), Journal of Th. Prob. 31 (2018), 1411–1428.

[36] J. Kustermans, S. Vaes, Locally compact quantum groups, Ann. Sci. E´cole Norm. Sup., (4)33(6) (2000), 837– 934.

[37] T. Masuda, Y. Nakagami, A von Neumann algebra framework for the duality of the quantum groups, Publ. Res. Inst. Math. Sci. 30(1994), no. 5, 799–850

[38] S. Neshveyev, L. Tuset, Compact Quantum Groups and Their Representation Categories, Soc. Math. France., 2013.

[39] M. Noumi, H. Yamada, K. Mimachi, Finite dimensional representations of the quantum group GLq (n; C) and the zonal spherical functions on Uq (n − 1)\Uq (n), Japan J. Math. 19 (1993), no. 1, 31–80

[40] A. Okounkov, G. Olshanski, Limits of BC-type orthogonal polynomials as the number of variables goes to infinity, Contemporary Mathathematics 417 (2006), 281–318.

[41] G. Olshanski, Unitary representations of infinite-dimensional pairs (G, K) and the formalism of R. Howe, In Rep- resentation of Lie groups and related topics (A. M. Vershik and D. P. Zhelobenko, eds.), Adv. Stud. Comtemp. Math. 7, Gordon and Breach, New York, 1990, 269–468

[42] G. Olshanski, The problem of harmonic analysis on the infinite-dimensional unitary group, Journal of Funct. Anal., 205 (2003), 464–524.

[43] G. Olshanski, The representation ring of the unitary groups and Markov processes of algebraic origin, Adv. Math. 300 (2016), 544–615.

[44] L. Petrov, Asymptotics of random lozenge tilings via Gelfand–Tsetlin schemes, Probab. Theory Rel. Fields 160 (2013), 429–487.

[45] R. R. Phelps, Lectures on Choquet’s Theorem, Lecture Note in Mathematics 1757, Springer-Verlag, Berlin-New York, 2001.

[46] R. Sato, Quantized Vershik–Kerov theory and quantized central probability measures on branching graphs, Journal of Funct. Anal., 277 (2019), 2522–2557

[47] R. Sato, Type classification of extreme quantized characters, Ergodic Theory Dynam. Systems, 41(2021), 593– 605.

[48] R. Sato, Inductive limits of compact quantum groups and their unitary representations, Lett Math Phys (2021), published online.

[49] R. Sato, Markov semigroups on unitary duals generated by quantized characters, arXiv:2102.09082.

[50] R. Sato, Characters of infinite-dimensional quantum classical groups: BCD cases, arXiv:2106.12771.

[51] S. Stratila, D. Voiculescu, Representations of AF-Algebras and of the Group U(∞), Lecture Notes in Mathe- matics 486, Springer-Verlag, Berlin-New York, 1975.

[52] Z. Takeda, On the representations of operator algebra, Proc. Japan Acad. 30 (1954), 299–304.

[53] Z. Takeda, On the representations of operator algebra II, Tˆohoku Math. J. (2)6 (1954), 212–219.

[54] Z. Takeda, Inductive limit and infinite direct product of operator algebras, Tˆohoku Math. J. (2)7 (1955), 67–86.

[55] M. Takesaki, Algebraic equivalence of locally normal representations, Pacific J. Math. 34 (1970), 807–816.

[56] M. Takesaki, Theory of Operator Algebras II, Encyclopedia of Mathematical Sciences 127, Springer, 2003.

[57] E. Thoma, Die unzerlegbaren, positive-definiten Klassenfunktionen der abz¨ahlbar unendlichen, symmetrischen Gruppe, Math. Zeitschr. 85(1964), pp. 40–61.

[58] R. Tomatsu, A characterization of right coideals of quotient type and its application to classification of Poisson boundaries, Commun. Math. Phys. 275 (2007), 271–296

[59] Y. Ueda, Spherical representations of C∗-flows, preprint, arXiv:2010.15324

[60] A. M. Vershik, S. V. Kerov, Characters and factor representations of the infinite symmetric group, Dokl. Akad. Nauk. SSSR 257 (1981), no. 5, 1037–1040

[61] A. M. Vershik, S. V. Kerov, Characters and factor representations of the infinite unitary group, Sov. Math. Dokl. 26 (1982), 570–574.

[62] D. Voiculescu, Repr´esentations factorielles de type II de U (∞), J. Math. Pures Appl. 55 (1976), 1–20.

[63] S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613–665.

[64] S. Yamagami, On unitary representation theories of compact quantum groups, Commun. Math. Phys., 167 (1995), 509–529

[65] M. Yoshida, Odometer action on Risesz product, J. Austral. Math. Soc. (Series A) 61 (1996), 141–149.

[66] D. P. Z˘elobenko, Compact Lie groups and their representations, Translations of Mathematical Monographs 40, Amer. Math. Soc., 1973.