On *-representations of U q ( s l ( 2 ) ) : more real forms

Eduard Vaysleb

Banach Center Publications (1997)

  • Volume: 40, Issue: 1, page 59-65
  • ISSN: 0137-6934

Abstract

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The main goal of this paper is to do the representation-theoretic groundwork for two new candidates for locally compact (nondiscrete) quantum groups. These objects are real forms of the quantized universal enveloping algebra U q ( s l ( 2 ) ) and do not have real Lie algebras as classical limits. Surprisingly, their representations are naturally described using only bounded (in one case only two-dimensional) operators. That removes the problem of describing their Hopf structure ’on the Hilbert space level’([W]).

How to cite

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Vaysleb, Eduard. "On *-representations of $U_{q}(sl(2))$: more real forms." Banach Center Publications 40.1 (1997): 59-65. <http://eudml.org/doc/252183>.

@article{Vaysleb1997,
abstract = {The main goal of this paper is to do the representation-theoretic groundwork for two new candidates for locally compact (nondiscrete) quantum groups. These objects are real forms of the quantized universal enveloping algebra $U_q(sl(2))$ and do not have real Lie algebras as classical limits. Surprisingly, their representations are naturally described using only bounded (in one case only two-dimensional) operators. That removes the problem of describing their Hopf structure ’on the Hilbert space level’([W]).},
author = {Vaysleb, Eduard},
journal = {Banach Center Publications},
keywords = {locally compact quantum groups; real forms on the quantized universal enveloping algebra},
language = {eng},
number = {1},
pages = {59-65},
title = {On *-representations of $U_\{q\}(sl(2))$: more real forms},
url = {http://eudml.org/doc/252183},
volume = {40},
year = {1997},
}

TY - JOUR
AU - Vaysleb, Eduard
TI - On *-representations of $U_{q}(sl(2))$: more real forms
JO - Banach Center Publications
PY - 1997
VL - 40
IS - 1
SP - 59
EP - 65
AB - The main goal of this paper is to do the representation-theoretic groundwork for two new candidates for locally compact (nondiscrete) quantum groups. These objects are real forms of the quantized universal enveloping algebra $U_q(sl(2))$ and do not have real Lie algebras as classical limits. Surprisingly, their representations are naturally described using only bounded (in one case only two-dimensional) operators. That removes the problem of describing their Hopf structure ’on the Hilbert space level’([W]).
LA - eng
KW - locally compact quantum groups; real forms on the quantized universal enveloping algebra
UR - http://eudml.org/doc/252183
ER -

References

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  1. [CP] A. Pressley, V. Chari, A guide to quantum groups, Cambridge Univ. Press, Cambridge, 1994. Zbl0839.17009
  2. [J] M. Jimbo, A q-difference analog of U(𝐺) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63-69. Zbl0587.17004
  3. [L] G. Lusztig, Modular representations and quantum groups, Contemp. Math. 82. Classical groups and related topics, Amer. Math. Soc., Providence, 1990, pp. 59-77. 
  4. [MM] T. Masuda, K. Mimachi, Y. Nakagami, M. Noumi, Y. Saburi, K. Ueno, Unitary representations of the quantum group S U q ( 1 , 1 ) , Lett. Math. Phys. 19 (1990), 187-204. Zbl0704.17007
  5. [T] E. Twietmeyer, Real forms of U q ( ) , Lett. Math. Phys. 24 (1992), 49-58. Zbl0759.17008
  6. [V1] E. Vaysleb, Infinite-dimensional *-representations of the Sklyanin algebra and of the quantum algebra U q ( s l ( 2 ) ) , Selecta Mathematica formerly Sovietica 12 (1993), 57-73. Zbl0816.17008
  7. [V2] E. Vaysleb, Collections of commuting selfadjoint operators satisfying some relations with a non-selfadjoint one, Ukrain. Matem. Zh. 42 (1990), 1258-1262; Engish transl. in Ukrain. Math. J. 42 (1990), 1119-1123. 
  8. [W] S. L. Woronowicz, Unbounded elements affiliated with C*-algebras and non-compact quantum groups, Commun. Math. Phys. 136 (1991), 399-432. Zbl0743.46080

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