ИСТИНА

Para-Bose and para-Fermi statistics are generalizations of the usual Bose and Fermi statistics. Quantum fields corresponding to these para-statistics satisfy Green's trilinear relations (H.S. Green, Phys. Rev. 90, 270 (1953)). It was proved that the trilinear relations are related with an orthogonal Lie algebra so(2m+1) (in the case of the para-Fermi statistic) and with a Z_2-graded orthosymplectic Lie superalgebra osp(1|2n)) (in the case of the para-Bose statistic). Here m (n) denotes a number of parafermionic (para-bosonic) degrees of freedom. There are also two mixed cases of the para-Bose and para-Fermi systems which are called "relative para-Fermi sets" P_{FB} and "relative para-Bose sets" P_{BF} (O.W. Greenberg and A.M.L. Messian, Phys. Rev. 138, 1155 (1965)). It was established that the relative para-Fermi sets P_{FB} are associated with the Z_2-graded superalgebra osp(2m+1|2n). In the case of the relative para-Bose sets $P_{BF} a similar connection was not know up to now. In this talk we prove a new result that the case of the relative para-Bose sets P_{BF} is associated with the Z_2 xZ_2 -graded superalgebra osp(2m+1|2n).