Deformed Heisenberg algebra with reflection

A universality of the deformed Heisenberg algebra involving the reflection operator is revealed. It is shown that in addition to the well-known infinite-dimensional representations related to parabosons, the algebra has also finite-dimensional representations of parafermionic nature. We demonstrate that finite-dimensional representations are representations of the deformed parafermionic algebra with internal Z(2)-grading structure. On the other hand, any finite- or infinite-dimensional representation of the algebra supplies us with irreducible representations of the osp(1\2) superalgebra. We show that the normalized form of the deformed Heisenberg algebra with reflection has the structure of a guon algebra related to the generalized statistics. (C) 1997 Elsevier Science B.V.