ИСТИНА

Algebraic structure of all supersymmetries known today in models of relativistic and nuclear physics is associated with Z_2-graded Lie superalgebras (or supergroups). In this talk alternative supersymmetries based on Z_2×Z_2-graded Lie superalgebras are discussed. Algebraic structure of parastatistics is also considered and it will be show that one of mixed parastatistics of Greenberg and Messian is related with Z_2xZ_2-graded orthosymplectic Lie superalgebras osp(1,2m|2n). It turns out that there exits a more general case of parastatictics which is associated with Z_2×Z_2-graded orthosymplectic Lie superalgebras osp(2m_1,2m_2|2n_1,2n_2) and which includes (as degenerate cases) all known parastatistics of Green and Greenberg-Messian. Potential applications of Z_2×Z_2-graded Lie superalgebras in SUSY models of nuclei are briefly discussed.