Canonical derivations of the noncommutative torus

Core idea

In $A_\theta$ , with $vu=e^{2\pi i\theta}uv$ , let $\mathcal P_\theta$ be the algebra of finite sums of $v^mu^n$ . The canonical derivations are the complex-linear maps

\delta_1(v^mu^n)=2\pi im\,v^mu^n,\qquad \delta_2(v^mu^n)=2\pi in\,v^mu^n.

They obey the Leibniz rule and commute on $\mathcal P_\theta$ .

Product-rule verification

Multiplying two monomials adds their respective exponents and introduces only a constant phase from the commutation relation. Applying $\delta_j$ multiplies by the sum of the corresponding frequencies, which equals the two Leibniz terms.

Smooth algebra

The smooth noncommutative torus is a dense algebra of rapidly decreasing Fourier series, preserved by both derivations. At $\theta=0$ , these derivations become the two usual partial derivatives on the smooth ordinary torus.

References

Alain Connes, C-algebras and differential geometry*, English translation of the 1980 note, pp. 4–5 (torus action, derivations, and rapid-decay series).