Orthosymplectic Z 2 × Z 2 Z 2 × Z 2 -graded Lie superalgebras and parastatistics

Source abstract

Abstract A Z 2 × Z 2 -graded Lie superalgebra g is a Z 2 × Z 2 -graded algebra with a bracket [ [ ⋅ , ⋅ ] ] that satisfies certain graded versions of the symmetry and Jacobi identity. In particular, despite the common terminology, g is not a Lie superalgebra. We construct the most general orthosymplectic Z 2 × Z 2 -graded Lie superalgebra o s p ( 2 m 1 + 1 , 2 m 2 | 2 n 1 , 2 n 2 ) in terms of defining matrices. A special case of this algebra appeared already in work of Tolstoy in 2014. Our construction is based on the notion of graded supertranspose for a Z 2 × Z 2 -graded matrix. Since the orthosymplectic Lie superalgebra o s p ( 2 m + 1 | 2 n ) is closely related to the definition of parabosons, parafermions and mixed parastatistics, we investigate here the new parastatistics relations following from o s p ( 2 m 1 + 1 , 2 m 2 | 2 n 1 , 2 n 2 ) . Some special cases are of particular interest, even when one is dealing with parabosons only.