# On *-representations of ${U}_{q}\left(sl\left(2\right)\right)$: more real forms

Banach Center Publications (1997)

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

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topVaysleb, 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

top- [CP] A. Pressley, V. Chari, A guide to quantum groups, Cambridge Univ. Press, Cambridge, 1994. Zbl0839.17009
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- [L] G. Lusztig, Modular representations and quantum groups, Contemp. Math. 82. Classical groups and related topics, Amer. Math. Soc., Providence, 1990, pp. 59-77.
- [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
- [T] E. Twietmeyer, Real forms of ${U}_{q}\left(\right)$, Lett. Math. Phys. 24 (1992), 49-58. Zbl0759.17008
- [V1] E. Vaysleb, Infinite-dimensional *-representations of the Sklyanin algebra and of the quantum algebra ${U}_{q}\left(sl\left(2\right)\right)$, Selecta Mathematica formerly Sovietica 12 (1993), 57-73. Zbl0816.17008
- [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.
- [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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