No Access

FOCK REPRESENTATIONS AND QUANTUM MATRICES

D. SHKLYAROV

Institute for Low Temperature Physics & Engineering, National Academy of Sciences of Ukraine, Ukraine

Search for more papers by this author

S. SINEL'SHCHIKOV

Institute for Low Temperature Physics & Engineering, National Academy of Sciences of Ukraine, Ukraine

Search for more papers by this author

, and

L. VAKSMAN

Institute for Low Temperature Physics & Engineering, National Academy of Sciences of Ukraine, Ukraine

Search for more papers by this author

https://doi.org/10.1142/S0129167X04002600Cited by:7 (Source: Crossref)

Abstract

In this paper we study the Fock representation of a certain *-algebra which appears naturally in the framework of quantum group theory. It is also a generalization of the twisted CCR-algebra introduced by Pusz and Woronowicz. We prove that the Fock representation is a faithful irreducible representation of the algebra by bounded operators in a Hilbert space, and, moreover, it is the only (up to unitary equivalence) representation possessing these properties.

Keywords:

AMSC: 20G42, 46L52

References

W. B. Arveson, Acta Math. 123, 122 (1969). Google Scholar
G. M. Bergman, Adv. Math. 29, 178 (1978). Crossref, Web of Science, Google Scholar
V. Chari and A. Pressley , A Guide to Quantum Groups ( Cambridge University Press , 1995 ) . Google Scholar
I. Damiani and C. De Concini, Representations of Lie Groups and Quantum Groups, Pitman Research Notes in Mathematics 311, eds. V. Baldoniet al. (Longman Scientific & Technical, 1994) pp. 1–45. Google Scholar
R. Dipper and S. Donkin, Proc. London Math. Soc. 63, 165 (1991). Web of Science, Google Scholar
V. G. Drinfeld, Quantum groups, Proceedings of the International Congress of Mathematicians, ed. A. M. Gleason (American Mathematical Society, Providence, R. I., 1987) pp. 798–820. Google Scholar
L. D. Faddeev, N. Yu. Reshetikhin and L. A. Takhtajan, Quantization of Lie groups and Lie algebras, in Algebraic Analysis; 1, eds. M. Kashiwara and T. Kawai (Academic Press, Inc., Boston, New York, 1988); LOMI preprint E-14-87, 1987 . Google Scholar
J. C. Jantzen, Lectures in Quantum Groups, Graduate Studies in Mathematics (American Mathematical Society) . Google Scholar
A. Joseph, Quantum Groups and their Primitive Ideals (Springer, Berlin, 1995) p. 384. Crossref, Google Scholar
A. Joyal and R. Street, Adv. Math. 102, 20 (1993). Crossref, Web of Science, Google Scholar
V. G. Kac , Infinite Dimensional Lie Algebras ( Cambridge University Press , 1985 ) . Google Scholar
A. N. Kirillov and N. Yu. Reshetikhin, Commun. Math. Phys. 34, 421 (1973). Web of Science, Google Scholar
A. Klimyk and K. Schmüdgen , Quantum Groups and Their Representations ( Springer-Verlag , 1997 ) . Crossref, Google Scholar
L. Korogodski and Ya. Soibelman , Algebras of Functions on Quantum Groups ( American Mathematical Society , Providence RI , 1998 ) . Crossref, Google Scholar
G. Lusztig, Geom. Dedicata 35, 89 (1990). Web of Science, Google Scholar
V. A. Lunts and A. L. Rosenberg, Selecta Math. (N.S.) 5, 123 (1999). Crossref, Web of Science, Google Scholar
S. Z. Levendorskii and Ya. S. Soibelman, Commun. Math. Phys. 139, 141 (1991). Crossref, Web of Science, Google Scholar
S. Z. Levendorskii and Ya. S. Soibelman, J. Geom. Phys. 7, 241 (1990). Crossref, Google Scholar
M. Noumi, CWI Quarterly 5, 293 (1992). Google Scholar
M. Noumi, H. Yamada and K. Mimachi, Japan J. Math. 19, 31 (1993). Crossref, Google Scholar
B. Parshall and J. Wang , Quantum Linear Groups , Memoirs of the American Mathematical Society 89 ( American Mathematical Society , Providence RI , 1991 ) . Google Scholar
D. Proskurin and L. Turowska, Meth. Funct. Anal. Topology 7, 89 (2001). Google Scholar
W. Pusz and S. Woronowicz, Rep. Math. Phys. 27, 231 (1989). Crossref, Google Scholar
M. Rosso, Seminaire Bourbaki 43(744), 443 (1991). Google Scholar
S. Sinel'shchikov and L. Vaksman, On q-analogues of bounded symmetric domains and Dolbeault complexes, Math. Phys. Anal. Geom. 1 (1998) 75–100; e-print 1997, q-alg/9703005 . Google Scholar
Y. Soibelman, The algebra of functions on a compact quantum group, and its representations, Leningrad Math. J. 2 (1991) 161–178; correction ibid. 2 (1991), p. 126 . Google Scholar
Y. Soibelman and L. Vaksman, On some problems in the theory of quantum groups, in Representation Theory and Dynamical Systems, ed. A. M. Vershik, Advances in Soviet Mathematics, Vol. 9 (American Mathematical Society, Providence, RI), pp. 3–55 . Google Scholar
L. Turowska, J. Phys. A 34, 2063 (2001). Crossref, Google Scholar
L. Vaksman, Mat. Fiz. Anal. Geom. 8(4), 366 (2001). Google Scholar
L. Vaksman and Y. Soibelman, Funct. Anal. Appl. 22, 170 (1988). Crossref, Web of Science, Google Scholar
L. Vaksman and Y. Soibelman, Leningrad Math. J. 2(5), 1023 (1991). Google Scholar