PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
1997 | 40 | 1 | 21-40
Tytuł artykułu

Semilinear relations and *-representations of deformations of so(3)

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We study a family of commuting selfadjoint operators $𝔸=(A_k)_{k=1}^n$, which satisfy, together with the operators of the family $𝔹=(B_j)_{j=1}^{n}$, semilinear relations $⅀ _{i} f_{ij}(𝔸) B_j g_{ij}(𝔸) = h(𝔸)$, ($f_{ij}$, $g_{ij}$, $h_j: ℝ^n → ℂ$ are fixed Borel functions). The developed technique is used to investigate representations of deformations of the universal enveloping algebra U(so(3)), in particular, of some real forms of the Fairlie algebra $U_q'(so(3))$.
Słowa kluczowe
Rocznik
Tom
40
Numer
1
Strony
21-40
Opis fizyczny
Daty
wydano
1997
Twórcy
  • Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivska Str., Kiev, 252601 Ukraine
  • Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivska Str., Kiev, 252601 Ukraine
Bibliografia
  • [1] Yu. N. Bespalov, Yu. S. Samoĭlenko, and V. S. Shul'man, On families of operators connected by semilinear relations, in: Application of Methods of Functional Analysis in Mathematical Physics, Math. Inst. Akad. Nauk Ukrain. SSR, Kiev, (1991), 28-51, (Russian).
  • [2] C. Daskaloyannis, Generalized deformed oscillator and nonlinear algebras, J. Phys. A 24 (1991), L789-L794.
  • [3] V. D. Drinfeld, Quantum groups, Zap. nauch. sem. LOMI. 115 (1980), 19-49.
  • [4] D. B. Fairlie, Quantum deformations of SU(2), J. Phys. A: Math. Gen. 23 (1990), L183-L187.
  • [5] O. M. Gavrilik, A. U. Klimyk, Rpersentations of the q-deformed algebras $U_q(so_2, 1)$, $U_q(so_3, 1)$, J. Math. Phys. 35 (1994), no. 2, 100-121. %preprint ITP-93-29E.
  • [6] M. F. Gorodniy, G. B. Podkolzin, Irreducible representations of graduated Lie algebra, Spectral theory of operators and infinite-dimensional analisys, Math. Inst. Akad. Nauk Ukrain. SSR, Kiev, (1984), 66-77, (Russian).
  • [7] P. Halmos, A Hilbert space problem book, Van Nostrand, Princeton, 1967.
  • [8] M. Jimbo, q-difference analogue of U(n) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63-69.
  • [9] M. Jimbo, Quantum R matrix related to the generalizated Toda system: an algebraic approach, Lect. Notes Phys. 246 (1986), 334-361.
  • [10] P. E. T. Jοrgensen, L. M. Schmitt, and R. F Werner, Positive representation of general commutation relations allowing Wick ordering, Preprint Osnabrück, (1993).
  • [11] A. A. Kirillov, Elements of the theory of representations, Springer, Berlin, (1970).
  • [12] S. Klimek and A. Lesniewski, Quantum Riemann surfaces. I. The unit disc, Commun. Math. Phys. 146 (1992), 103 - 122.
  • [13] M. Havliček, A. U. Klimyk, E. Pelantová, Fairlie algebra $U_q'(so_3)$: oscillator realizations, root of unity, reduction $U_q(sl_3) ⊃ U_q'(so_3)$, J. Phys. A. (to appear)
  • [14] S. A. Kruglyak and Yu. S. Samoĭlenko, On unitary equivalence of collections of self-adjoint operators, Funct. Anal. i Prilozhen. 14 (1980), no. 1, 60 - 62, (Russian).
  • [15] V. L. Ostrovskiĭ and Yu. S. Samoĭlenko, Unbounded operators satisfying non-Lie commutation relations, Repts. Math. Phys. 28 (1989), no. 1, 91-103.
  • [16] V. L. Ostrovskyĭ and Yu. S. Samoĭlenko, On pairs of self-adjoint operators, Seminar Sophus Lie 3 (1993), no. 2, 185-218.
  • [17] V. L. Ostrovskyĭ and Yu. S. Samoĭlenko, Representations of *-algebras and dynamical system, Non-linear Math. Phys. 2 (1995), no. 2, 133-150.
  • [18] V. L. Ostrovskyĭ and Yu. S. Samoĭlenko, On representations of the Heisenberg relations for the quantum e(2) group, Ukr. Mat. Zhurn. 47 (1995), no. 5, 700-710.
  • [19] V. L. Ostrovskyĭ and L. B. Turovskaya, Representations of *-algebras and multidimensional dynamical systems, Ukr. Mat. Zhurn. 47 (1995), no. 4, 488-497.
  • [20] A. Yu. Piryatinskaya and Yu. S. Samoĭlenko, Wild problems in representation theory of *-algebras with generators and relations, Ukr. Mat. Zhurn. 47 (1995), no. 1, 70-78.
  • [21] W. Pusz and S. L. Woronowicz, Twisted second quantization, Reports Math. Phys. 27 (1989), 231-257.
  • [22] N. Yu. Reshetikhin, L. A. Takhtajan, and L. D. Faddeev, Quantization of Lie groups and Lie algebras, Algebra i Analiz 1 (1989), no. 1, 178-206.
  • [23] W. Rudin, Functional Analysis, McGraw-Hill, New York (1973).
  • [24] Yu. S. Samoĭlenko, Spectral theory of families of selfadjoint operators, 'Naukova Dumka', Kiev (1984) (Russian).
  • [25] Yu. S. Samoĭlenko, V. S. Shul'man and L. B. Turovskaya, Semilinear relations and their *-representations, Preprint Augsburg, 1995.
  • [26] Yu. S. Samoĭlenko and L. B. Turovskaya, On *-representations of semilinear relations, in: Met hods of Functional Analysis in problems of Mathematical Physics, Inst. Math Acad. Sci Ukraine, Kiev, (1992), 97-108.
  • [27] Yu. S. Samoĭlenko and L. B. Turowska, Representations of cubic semilinear relations and real forms of the Fairlie algebra, Repts. Math. Phys. (to appear).
  • [28] K. Schmüdgen, Unbounded operator algebras and representation theory, Akademie-Verlag, Berlin, (1990).
  • [29] V. S. Shul'man, Multiplication operators and spectral synthesis, Dokl. Akad. Nauk SSSR 313 (1990), no. 5, (1047-1051); English transl. in Soviet Math. 42 (1991), no. 1.
  • [30] Ya. S. Soibelman and L. L. Vaksman, The algebra of functions on the quantum group SU(n+1) and odd-dimensional quantum spheres, Algebra i Analiz. 2 (1990), no. 5, 101-120.
  • [31] L. Turovskaya, Representations of some real forms of $U_q(sl(3))$, Algebras, groups and geometries, 12 (1995), 321-338.
  • [32] E. Ye. Vaisleb, Representations of relations which connect a family of commuting operators with non-sefadjoint one, Ukrain. Math. Zh. 42 (1990), 1258 - 1262, (Russian).
  • [33] E. Ye. Vaisleb and Yu. S. Samoĭlenko, Representations of operator relations by unbounded operators and multi-dimensional dynamical systems, Ukrain. Math. Zh. 42 (1990), no. 9, 1011 - 1019, (Russian).
  • [34] S. L. Woronowicz, Quantum E(2) group and its Pontryagin dual, Lett. Math. Phys. 23 (1991), 251 - 263.
  • [35] D. P. Zhelobenko, Compact Lie groups and their representations, 'Nauka', Moscow, (1970) (Russian).
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-bcpv40z1p21bwm
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.