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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 $U_q(sl(2))$ 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]).