Abstract
In this paper we study *-representations for polynomial algebras on quantum matrix spaces. We deal with two special cases of the polynomial algebras, namely the algebra of polynomials on quantum complex matrices Mat2 and on quantum complex symmetric matrices \(\mathrm{Mat_2^{sym}}\). For the second algebra we classify all irreducible *-representations by bounded operators in a Hilbert space (up to a unitary equivalence). Moreover, we present a construction of *-representations of the above algebras which enables to obtain the full list of *-representations (sometimes by passing to subrepresentations).
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References
Bershtein, O.: Regular functions on the Shilov boundary. J. Algebra Appl. 4(6), 613–629 (2005)
Korogodsky, L., Soibelman, Ya.: Algebra of Functions on Quantum Groups: Part 1. Amer. Math. Soc., Providence RI (1998)
Ostrovsky, V., Samoilenko, Yu.: Introduction to the Theory Representation of Finitely Presented *-Algebras. 1. Representations by bounded operators. The Gordon and Breach Publ. Group, London (1999)
Ostrovskyi, V., Turowskaya, L.: Representations and dynamical systems. Ukranian Math. J. 47(4), 488–497 (1995)
Sinel’shchikov, S., Vaksman, L.: On q-analogs of bounded symmetric domains and Dolbeaut complexes. Math. Phys. Anal. Geom. 1(1), 75–100 (1998)
Shklyarov, D., Sinel’shchikov, S., Vaksman, L.: Geometric realizations for some series of representations of the quantum group SU 2,2. Math Phys. Anal. Geom. 8(1), 90–110 (2001) (in Russian), Engl. vers.: In: Vaksman, L. (ed.) Lectures on q-Analogues of Cartan Domains and Associated Harish-Chandra Modules, pp. 73–90 (2001). arXiv:math.QA/0109198
Shklyarov, D., Sinel’shchikov, S., Vaksman, L.: Fock representations and quantum matrices. Int. J. Math. 15(9), 1–40 (2004)
Turowska, L.: On representations of a q-analogue of the *-algebra Pol(Mat2). J. Phys. A: Math. Gen. 34, 2063–2070 (2001)
Vaksman, L.: Quantum Bounded Symmetric Domains. Amer. Math.Soc., Providence RI (2010)
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Bershtein, O. On *-Representations of Polynomial Algebras in Quantum Matrix Spaces of Rank 2. Algebr Represent Theor 17, 1083–1093 (2014). https://doi.org/10.1007/s10468-013-9433-z
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DOI: https://doi.org/10.1007/s10468-013-9433-z