Abstract
The question of quantization of two-dimensional mappings representing the discrete Painleve equations and their autonomous limits is examined. The authors show that for all these mappings it is possible to find a consistent quantization scheme, inspired from the commutation relations encountered in quantum groups. In the autonomous case they show that the classical invariant survives after the quantization, provided one introduces adequate quantum corrections in both the mapping and the invariant. For the discrete Painleve equations themselves the integrability constraints are so stringent that they suffice even for the quantized case. In all the known cases the classical Lax pairs can be transcribed as quantal ones requiring only a (straightforward) choice of ordering for some of them.
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