M Theory Lesson 109

In the easy book Special matrices of Mathematical Physics the chapter on circulant matrices begins with the statement: making a Fourier transform is equivalent to diagonalising a circulant matrix, and an inverse Fourier transform takes a diagonal matrix into a circulant matrix. In fact, for a circulant built from the numbers $X_1 , X_2 , \cdots , X_n$ the eigenvalues $\lambda_k$ are related by the following pair of $\mathbb{Z}_n$ (Abelian) transforms

$\lambda_k = \sum_{j} e^{\frac{2 \pi i}{n} jk} X_j$
$X_j = \frac{1}{n} \sum_{k} e^{- \frac{2 \pi i}{n} jk} \lambda_k$

Perhaps non-commutative Fourier transforms should be about circulants of circulants! In fact, examples of quantum groups (Hopf algebras) in the book are built from matrices of matrices. The simple quantum analogue of phase space is viewed as the space underlying the circulant/diagonal matrix space, such as the discrete torus $\mathbb{Z}_n \otimes \mathbb{Z}_n$, where the cyclic group $\mathbb{Z}_n$ belongs to the self-dual schizophrenic object $U(1)$ of Stone (Pontrjagin) duality.

A circulant $C$ is diagonalised via $M^{-1} C M$ with $M_{ij} = \frac{\omega^{- ij}}{\sqrt{n}}$ for $\omega$ the primitive $n$-th root of unity. Given that the eigenspace projectors are circulant, a circulant matrix can be written in the basis of projectors. It can also be written as a degree $n$ polynomial in a nice circulant $S$ representing a basic shift operator. In the $3 \times 3$ case this is the familiar

001
100
010

corresponding to the permutation $(312)$, and a circulant takes the form $X_1 + X_2 (231) + X_3 (312)$. A spectral function $f$ on a circulant is defined to be the circulant

$f(C) = \frac{1}{n} \sum_j (\sum_k f(\lambda_k) \omega^{-jk}) S^j$

The Fourier transform pair is particularly simple in terms of the matrices $S^j$ and the projectors. Thus it is useful to define a matrix product via convolution, $G.H = \sum_j (G*H)_j S^j$ where

$(G*H)_j = \sum_k G_k H_{j - k}$

Now let the diagonal matrix with entries $\delta_{ij} \omega^{i}$ be denoted $D$. Then one has $DS = \omega SD$, a simple non-commutativity condition, which extends to the Weyl rule $D^m S^n = \omega^{mn} S^n D^m$ for a quantum torus. This setup has the feature that the $n \rightarrow \infty$ process is associated with a continuum limit, which is just what we want to do with n-categories. To quote Weyl: The problems of mathematics are not problems in a vacuum ...