I have a kind of vague question. 


Two non-degenerate symplectic vector space of same dimension (say $\mathbb{R}^{2n}$) are isomorphic. Then why all noncommutative tori of same dimension aren't isomorphic?



isomorphism of noncommutative tori
If you view a noncommutative $2n$-torus not as a strict deformation quantisation along a $\mathbb{R}^{2n}$-action (that happens to be periodic) but as a strict deformation quantisation along a $\mathbb{T}^{2n}$-action, then what actually matters isn't the non-degenerate symplectic form $\theta : \mathbb{R}^{2n} \times \mathbb{R}^{2n} \to \mathbb{R}$ but rather the induced alternating bicharacter $\lambda\_\theta : \mathbb{Z}^{2n} \times \mathbb{Z}^{2n} \to U(1)$, defined by
$
 \lambda\_\theta(\mathbf{m},\mathbf{n}) := e^{2\pi i \theta(\mathbf{m},\mathbf{n})}
$,
which governs the commutation relations on the corresponding noncommutative $2n$-torus. Indeed, in general, the noncommutative $N$-tori are classified up to $\mathbb{T}^{N}$-equivariant $\ast$-isomorphism by the $U(1)$-valued alternating bicharacters on $\mathbb{Z}^{N}$, the Pontrjagin dual of $\mathbb{T}^{N}$. It's now easy to see that two non-degenerate symplectic forms $\theta$ and $\theta^\prime$ on $\mathbb{R}^{2n}$ needn't give isomorphic $U(1)$-valued alternating bicharacters $\lambda\_\theta$ and $\lambda\_{\theta^\prime}$ on $\mathbb{Z}^{2n}$ (in the sense that $\lambda\_{\theta^\prime} = \Phi^\ast \lambda\_\theta$ for some $\Phi \in GL(2n,\mathbb{Z})$), which suggests that one really shouldn't expect all noncommutative $2n$-tori to be isomorphic.


For example, let's take a closer look at the case when $n=1$. Given $\theta$, $\theta^\prime \in \mathbb{R} \setminus \{0\}$, one can check that the non-degenerate symplectic forms $\theta dx \wedge dy$ and $\theta^\prime dx \wedge dy$ on $\mathbb{R}^2$ give isomorphic $U(1)$-valued alternating bicharacters on $\mathbb{Z}^2$ if and only if $\theta^\prime = \pm \theta \bmod \mathbb{Z}$. As it turns out, when $\theta$ and $\theta^\prime$ are both irrational, a $K$-theoretic argument confirms that the irrational noncommutative $2$-tori $C(\mathbb{T}^2\_\theta)$ and $C(\mathbb{T}^2\_{\theta^\prime})$ are $\ast$-isomorphic if and only if $\theta^\prime = \pm \theta \bmod \mathbb{Z}$. 


In the higher-rank case, $\ast$-isomorphism of $C^\ast$-algebraic noncommutative tori becomes a bit more delicate. Still, at the level of Frechet pre-$C^\ast$-algebras, at least, Cuntz--Elliott--Goodman--Jorgensen [1985] showed that for $\theta$ and $\theta^\prime$ in a certain dense subset of $\wedge^2 \mathbb{R}^N$, the smooth simple noncommutative $N$-tori $C^\infty(\mathbb{T}^N\_\theta)$ and $C^\infty(\mathbb{T}^N\_{\theta^\prime})$ are $\ast$-isomorphic if and only if the induced $U(1)$-valued alternating bicharacters on $\mathbb{Z}^N$ are isomorphic.


