Polyadic sigma matrices

Abstract

We generalize σ-matrices to higher arities using the polyadization procedure proposed by the author. We build the nonderived n-ary version of SU2 using cyclic shift block matrices. We introduce the polyadic trace, which has an additivity property analogous to the ordinary trace for block diagonal matrices. The so called elementary Σ-matrices are ordinary matrix units, their sums are full Σ-matrices which can be treated as a polyadic analog of σ-matrices. The expression of n-ary SU2 in terms of full Σ-matrices is given using the Hadamard product. We then generalize the Pauli group in two ways: for the binary case we introduce the extended phase shifted σ-matrices with multipliers in cyclic groups of order 4q (q > 4), and for the polyadic case we construct the correspondent finite n-ary semigroup of phase-shifted elementary Σ-matrices of order 4qn−1+1, and the finite n-ary group of phase-shifted full Σ-matrices of order 4q. Finally, we introduce the finite n-ary group of heterogeneous full Σhet-matrices of order 4qn−14. Some examples of the lowest arities are presented.