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An explicit description of the pairing between the universal enveloping algebra Uq (sl(2)) and the algebra of functions over SLq(2) is obtained and the regular representation of the quantum double is explicitly constructed and studied. The structure of the root subspaces of the Casimir operator is revealed and described in terms of the elements of SLq(2). Bibliography:8 titles.

The regular representation of the quantum Heisenberg double {U q sl(2)),Fun q (SL(2))} (q is a root of unity)