Polyadic sigma matrices

Abstract

We generalize $\sigma$-matrices to higher arities using the polyadization procedure proposed by the author. We build the nonderived $n$-ary version of $SU\left( 2\right) $ using cyclic shift block matrices. We define a new function, the polyadic trace, which has an additivity property analogous to the ordinary trace for block diagonal matrices and which can be used to build the corresponding invariants. The elementary $\Sigma$-matrices introduced here play a role similar to ordinary matrix units, and their sums are full $\Sigma$-matrices which can be treated as a polyadic analog of $\sigma$-matrices. The presentation of $n$-ary $SU\left( 2\right) $ in terms of full $\Sigma$-matrices is done using the Hadamard product. We then generalize the Pauli group in two ways: for the binary case we introduce the extended phase shifted $\sigma$-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 $\Sigma$-matrices of order $4q\left( n-1\right) +1$, and the finite $n$-ary group of phase-shifted full $\Sigma$-matrices of order $4q$. Finally, we introduce the finite $n$-ary group of heterogeneous full $\mathit{\Sigma}^{het}$-matrices of order $\left( 4q\left( n-1\right) \right) ^{4}$. Some examples of the lowest arities are presented.