In short, how do we get the formula for the NC torus? I find the equations in many places (including here) but I still have no idea for how this comes from the torus. If my understanding is correct, then you take the torus and look at its algebra of (smooth or just continuous?) functions. Then make this noncommutative and that is your noncommutative torus. There are different ways to do this which lead to different anti-symmetric matrices $\theta$ which yield different tori. But how do I do this in practice? And is there any way to go backwards: Is there a way to take the information $UV = e^{2\pi i \theta} VU$ and do some math and say, "hey this is a torus in some sense!"?
When looking at noncommutative $\mathbb{R}^4$ for example, it is easy to explain what it has to do with $\mathbb{R}^4$ and how to get it. Is there no other way to simply explain this for the torus? Perhaps because the algebra of functions on the torus is much more complicated than that of $\mathbb{R}^4$?
I am also curious if there is some way to realize $\mathbb{T}^n_{\theta}$ as a quotient of $\mathbb{R}^4_{\theta}$ by some NC lattice (if such a thing exists). That might provide a nice construction of the NC torus.
A reference would be great. I find a lot of references to papers or books that talk about the NC torus a good way into the text and then it often invokes a lot of background. Maybe it really does take that much set up? Any help would be greatly appreciated. Thank you.
*Note: I put this on MO by mistake earlier, but I feel it's better suited here. I think I deleted the MO one...
Recall that any $f \in C^\infty(\mathbb{T}^n)$ can be uniquely written as a convergent Fourier series $$ f = \sum_{\mathbb{k} \in \mathbb{Z}^n} f_{\mathbb{k}} U_{\mathbb{k}}, \quad f_{\mathbb{k}} := \int_{\mathbb{T}^n} e^{-2\pi i \langle \mathbb{k},t \rangle}f(t)\,dt, $$ where for each $\mathbb{k} \in \mathbb{Z}^n$, $$ \forall t \in \mathbb{T}^n, \quad U_{\mathbb{k}}(t) := e^{2\pi i \langle \mathbb{k},t\rangle}. $$ Recall, moreover, that the usual pointwise multiplication of functions corresponds to the convolution product of Fourier series: $$ \forall f, \; g \in C^\infty(\mathbb{T}^n), \quad fg = \sum_{\mathbb{k} \in \mathbb{Z}} \left(\sum_{\mathbb{k}^\prime \in \mathbb{Z}^n} f_{\mathbb{k}-\mathbb{k}^\prime}g_{\mathbb{k}^\prime} \right)U_{\mathbb{k}}. $$ In other words, $C^\infty(\mathbb{T}^n)$ is generated by unitaries $\{U_{\mathbb{k}}\}_{\mathbb{k} \in \mathbb{Z}^n}$ satisfying the relations $$ \forall \mathbb{k}, \; \mathbb{k}^\prime \in \mathbb{Z}^n, \quad U_{\mathbb{k}}U_{\mathbb{k}^\prime} = U_{\mathbb{k}+\mathbb{k}^\prime}, $$ which implies, in turn, that $$ \forall \mathbb{k}, \; \mathbb{k}^\prime \in \mathbb{Z}^n, \quad U_{\mathbb{k}^\prime}U_{\mathbb{k}} = U_{\mathbb{k}}U_{\mathbb{k}^\prime}. $$
Now, suppose that $\Theta : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{T} := \mathbb{R}/\mathbb{Z}$ is a (normalised) $2$-cocycle, i.e., that it satisfies
Then you can replace the usual convolution of Fourier series by a deformed convolution of Fourier series to obtain the noncommutative $n$-torus $C^\infty(\mathbb{T}^n_\Theta) := (C^\infty(\mathbb{T}^n),\star_\Theta)$: $$ \forall f,g \in C^\infty(\mathbb{T}^n), \quad f \star_\Theta g := \sum_{\mathbb{k} \in \mathbb{Z}} \left(\sum_{\mathbb{k}^\prime \in \mathbb{Z}^n} e^{-2\pi i \Theta(\mathbb{k}-\mathbb{k}^\prime,\mathbb{k}^\prime)}f_{\mathbb{k}-\mathbb{k}^\prime}g_{\mathbb{k}^\prime} \right)U_{\mathbb{k}}. $$ In terms of our unitary generators $\{U_{\mathbb{k}}\}_{\mathbb{k} \in \mathbb{Z}^n}$, this reduces to defining $$ \forall \mathbb{k}, \; \mathbb{k}^\prime \in \mathbb{Z}^n, \quad U_{\mathbb{k}} \star_\Theta U_{\mathbb{k}^\prime} := e^{-2\pi i \Theta(\mathbb{k},\mathbb{k}^\prime)}U_{\mathbb{k}+\mathbb{k}^\prime}, $$ which, in turn, implies the commutation relations $$ \forall \mathbb{k}, \; \mathbb{k}^\prime \in \mathbb{Z}^n, \quad U_{\mathbb{k}^\prime} \star_\Theta U_{\mathbb{k}} = e^{2\pi i \theta(\mathbb{k},\mathbb{k}^\prime)}U_{\mathbb{k}} \star_\Theta U_{\mathbb{k}^\prime}, $$ where $\theta : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{T}$ is defined by $$ \forall \mathbb{k}, \; \mathbb{k}^\prime \in \mathbb{Z}^n, \quad \theta(\mathbb{k},\mathbb{k}^\prime) := \Theta(\mathbb{k},\mathbb{k}^\prime) - \Theta(\mathbb{k}^\prime,\mathbb{k}). $$ As it turns out, $\theta$ is an alternating bicharacter, i.e.,
Conversely, every alternating bicharacter $\theta : \mathbb{Z}^n \to \mathbb{Z}^n$ can be induced in this way from a (normalised) $2$-cocycle $\Theta : \mathbb{Z}^n \to \mathbb{Z}^n$.
Now, what happens when $\Theta$ and $\Theta^\prime$ both induce the same alternating bicharacter $\theta$? It's an old theorem of Kleppner's that this happens if and only if $\Theta$ and $\Theta^\prime$ are cohomologous, i.e., there exists some $T : \mathbb{Z}^n \to \mathbb{T}$ such that $$ \forall \mathbb{k}, \; \mathbb{k}^\prime \in \mathbb{Z}^n, \quad \Theta^\prime(\mathbb{k},\mathbb{k}^\prime) = \Theta(\mathbb{k},\mathbb{k}^\prime) + T(\mathbb{k}) + T(\mathbb{k}^\prime) - T(\mathbb{k}+\mathbb{k}^\prime) ; $$ in other words, $\Theta^\prime - \Theta = dT$. In that case, we can define an explicit $\mathbb{T}^n$-equivariant isomorphism $\Psi_T : C^\infty(\mathbb{T}^n_\Theta) \to C^\infty(\mathbb{T}^n_{\Theta^\prime})$ by $$ \forall \mathbb{k} \in \mathbb{Z}^n, \quad \Psi_T(U_{\mathbb{k}}) = e^{2\pi i T(\mathbb{k})} U_{\mathbb{k}}. $$ Hence, up to equivariant isomorphism, for any alternating bicharacter $\theta$ (or equivalently, by Kleppner, for any class $\theta \in H^2(\mathbb{Z}^n,\mathbb{T})$, where $H^2(\mathbb{Z}^n,\mathbb{T})$ is the second group cohomology of $\mathbb{Z}^n$ with coefficients in $\mathbb{T}$), we can define the noncommutative $n$-torus $C^\infty(\mathbb{T}^n_\theta)$ as the algebra generated by unitaries $\{U_{\mathbb{k}}\}_{\mathbb{k} \in \mathbb{Z}^n}$ satisfying the commutation relations $$ \forall \mathbb{k}, \; \mathbb{k}^\prime \in \mathbb{Z}^n, \quad U_{\mathbb{k}^\prime} U_{\mathbb{k}} = e^{2\pi i \theta(\mathbb{k},\mathbb{k}^\prime)}U_{\mathbb{k}} U_{\mathbb{k}^\prime}; $$ to construct it, we just take $C^\infty(\mathbb{T}^n_\theta) := C^\infty(\mathbb{T}^n_\Theta)$ for any $2$-cocycle $\Theta$ inducing $\theta$.
Let me be more explicit about what appears in the literature. We have an isomorphism $$ \{\text{alternating bicharacters $\theta : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{T}$}\} \cong \mathbb{T}^{n(n-1)/2} $$ given by $$ \theta \mapsto (\theta(e_i,e_j))_{1 \leq i < j \leq n}, $$ where $\{e_i\}_{i=1}^n$ is the standard ordered basis for $\mathbb{R}^n$; conversely, the alteranting bicharacter $\theta$ corresponding to $(\theta_{ij})_{1\leq i < j \leq n} \in \mathbb{T}^{n(n-1)/2}$ is given by $$ \forall \mathbb{k}, \; \mathbb{k}^\prime \in \mathbb{Z}^n, \quad \theta(\mathbb{k},\mathbb{k}^\prime) = \sum_{1 \leq i < j \leq n} \theta_{ij}(k_i k^\prime_j - k_j k^\prime_i). $$ Then there are three conventions that tend to appear for constructing $2$-cocycles $\Theta : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{T}$ inducing $\theta$:
The first two conventions give you non-alternating bicharacters, which complicates a fair bit of the algebra, but turn out to be far more robust from a group-cohomological perspective; in particular, they give honest-to-goodness splittings of the short exact sequence $$ 0 \to \ker(\Theta \mapsto \theta) \to \{\text{$2$-cocycles}\} \xrightarrow{\Theta \mapsto \theta} \{\text{alternating bicharacters}\} \to 0. $$ The third convention gives you alternating bicharacters, which simplifies a fair bit of the algebra, but all the non-uniqueness can give you headaches when dealing with certain technicalities.