Projective representation of $\mathbb{Z}_N\times\mathbb{Z}_N$ by $su(N)$ generators

Question

[The same question has been posted in MathStackExchange.]

The projective representation (rep.) of $\mathbb{Z}_N\times\mathbb{Z}_N$ is $\mathbb{Z}_N$-classified. It can be understood by embedding into a $SU(N)$ representation.

For example, taking $N=2$, the nontrivial projective rep. of $\mathbb{Z}_2\times\mathbb{Z}_2$ can be written by: \begin{eqnarray} V=\exp(i\pi S^x);\,\,\,W=\exp(i\pi S^z), \end{eqnarray} where $S^x=\sigma_1/2$ and $S^z=\sigma_3/2$ with Pauli matrices $\vec{\sigma}$ in the fundamental representation (namely spin-$1/2$ rep.). The projective nature is characterized by their commutator \begin{eqnarray} VWV^{-1}W^{-1}=-1. \end{eqnarray} For general spin-$s$ rep., we have $VWV^{-1}W^{-1}=(-1)^{2s}$, which is $\mathbb{Z}_2$-classified by $2s$ mod $2$.

My question is whether there is any explicit $\mathbb{Z}_N\times\mathbb{Z}_N$ treatment like above. In other words, can we explicitly write down \begin{eqnarray} V_N=\exp(iT_V);\,\,W_N=\exp(iT_W), \end{eqnarray} with two certain $T_V$ and $T_W$ in Lie algebra $su(N)$, so that for the fundamental rep., we have \begin{eqnarray} V_NW_NV_N^{-1}W_N^{-1}=\exp\left(i\frac{2\pi}{N}\right). \end{eqnarray} For instance, can we write down explicitly $T_V$ and $T_W$ in a certain $su(N)$ basis?

Answer 1

Yes, this is the exponential Weyl form of the Heisenberg group commutation relations, through bounded, unitary operators, of utility in the Stone-von Neumann theorem, $\exp\left[\frac{i}{\hbar} Q \hat{p}~\right] \exp\left[\frac{i}{\hbar}P\hat{q}~\right]$ $= e^{iPQ/\hbar} \exp\left[\frac{i}{\hbar}P\hat{q}~\right]\exp\left[\frac{i}{\hbar} Q \hat{p}~\right]$.

Weyl, in his celebrated book, illustrates this structure in "quantum mechanics around the clock" (of N hours), utilizing Sylvester's clock and shift matrices limiting to elements of the Heisenberg group at large N.

Specifically, as Sylvester defined them in 1882 and Weyl used them in 1927, the clock N×N matrix $V\sim \Sigma_3$ and the discrete shift matrix $W\sim \Sigma_1$ braid to $$ \Sigma_3 \Sigma _1= e^{2\pi i /N} \Sigma_1 \Sigma _3, $$ in a clock of N hours. They generalize the Pauli matrices $\sigma_3, \sigma_1$, for $\omega= \exp(2\pi i/N)$, $$ \Sigma _1 = \begin{bmatrix} 0 & 0 & 0 & \cdots &0 & 1\\ 1 & 0 & 0 & \cdots & 0 & 0\\ 0 & 1 & 0 & \cdots & 0 & 0\\ 0 & 0 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots &\vdots &\vdots \\ 0 & 0 &0 & \cdots & 1 & 0\\ \end{bmatrix} \\\Sigma _3 = \begin{bmatrix} 1 & 0 & 0 & \cdots & 0\\ 0 & \omega & 0 & \cdots & 0\\ 0 & 0 &\omega ^2 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \cdots & \omega ^{N-1} \end{bmatrix}. $$

These and their products and powers comprise a set of $N^2$ independent U(N) group matrices, which, coincidentally, may further serve as a basis of the su(N) algebra, your concern here. (The perfect "analytic" basis for su(N) with trigonometric structure constants, reviewed here. Yes, this Sylvester basis preceded the Gell-Mann basis by 80 years).

If you wished to take their logarithms, this has also been done, cleverly, by use of the discrete Fourier transform, by Santhanam & Tekumalla, but the "momentum" exponent is a bit of a mess; do you understand why? It still does the job, and illustrates a mystifying aspect of the tracelessness of the commutator of the exponents in a finite space.

Answer 2

I think what you're looking for is closely related to the matrices described in

Patera, J. and Zassenhaus, H., 1988. The Pauli matrices in n dimensions and finest gradings of simple Lie algebras of type $A_{n− 1}$. Journal of mathematical physics, 29(3), pp.665-673.

For $\mathfrak{su}(2)$, these are the regular Pauli matrices (up to a phase) and they are labelled by elements in $\mathbb{Z}_2\times \mathbb{Z}_2$. To be explicit: \begin{align} \sigma_1&=\left(\begin{array}{cc} 0&1\\ -1 &0\end{array}\right) \to \sigma_{(1,0)}\, , \\ \sigma_2&=\left(\begin{array}{cc} 0&i\\ -i &0\end{array}\right)\to \sigma_{(1,1)}\, , \\ \sigma_3&=\left(\begin{array}{cc} i&0\\ 0 &-i\end{array}\right) \to \sigma_{(0,1)}\, ,\\ \sigma_0&=\left(\begin{array}{cc} 1&0\\ 0 &1\end{array}\right)\to \sigma_{(0,0)}\ \end{align} These matrices are proportional to the Pauli matrices and are also unitary.

The commutation relations are of the form \begin{align} [\sigma_{(p,q)},\sigma_{(p',q')}]= k \sigma_{(p+p',q+q')} \end{align} where $k=\pm$ depending on $(p,q)$ and $(p',q')$ and addition is taken mod 2.

The $8$ matrices $\pm \sigma_k$ form a subgroup of $SU(2)$ (under multiplication of course) and thus, if you just use the set of $\sigma_k$, you get what you want.

The nice thing about the P&Z paper is they generalize this to any $\mathfrak{su}(n)$. For $n=3$, the generalized Pauli matrices are generate by products of \begin{align} A=\left( \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \\ \end{array} \right)\to\sigma_{(0,-1)}\, ,\qquad D= \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \omega & 0 \\ 0 & 0 & \omega ^2 \\ \end{array} \right)\to \sigma_{(1,0)} \end{align} where $\omega^3=1$. A total of 27 matrices can be obtained by multiplication of these generating elements. They do satisfy in general \begin{align} X_kY_{k'}X_k^{-1} Y_{k}^{-1}= \omega^j \end{align} for some power of $\omega$. Moreover, under commutation they also satisfy $[X_{k},X_{k'}]=\omega^{j'} X_{k+k'}$ where $k+k'$ is taken modulo $3$ in each entry. A subset of $8$ of these, plus the identity, spans the $\mathfrak{su}(3)$ algebra. Note that the matrices are (of course) not hermitian.

Unfortunately, they are not given the form $V_N=\exp(i T_V)$ although presumably this can be worked out by expanding $T_V$ in terms of the $8$ elements chosen as basis set for $\mathfrak{su}(3)$, plus $\mathbb{1}$.