Polyadic sigma matrices

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 Σ^het-matrices of order $4qn−14$⁠. Some examples of the lowest arities are presented.