# C Heisenberg group

## C.1 Heisenberg group action on the contact space

We consider the unitary shift operators $$T_\tau,M_\omega\in\mathcal{U}(L^2(\mathbb{R}))$$. The commutation relation of these unitary operators is given by $\begin{equation} T_\tau M_\omega T_\tau^{-1} M_\omega^{-1} = e^{-2\pi i\omega\tau} \mathrm{Id} \end{equation}$

Indeed, $$\forall s\in L^2(\mathbb{R})$$ \begin{align*} T_\tau M_\omega T_\tau^{-1} M_\omega^{-1} s(t) &= T_\tau M_\omega T_\tau^{-1} e^{-2\pi i\omega t} s(t)\\ &= T_\tau M_\omega e^{-2\pi i\omega(t+\tau)} s(t+\tau)\\ &= T_\tau e^{-2\pi i\omega\tau} s(t+\tau)\\ &= e^{-2\pi i\omega\tau} s(t) \end{align*}

We also consider the representation $$U$$ from the Heisenberg group $$\mathbb{H}^1$$ defined $$U:\mathbb{H}^1\rightarrow\mathcal{U}(L^2(\mathbb{R}^2))$$ by $\begin{equation} U(\tau,\omega,\lambda) = e^{-2\pi i\lambda}T_\tau M_\omega \end{equation}$

Therefore, the operator algebra generated by the unitary time and frequency shift operators coincides with the Heisenberg group $$\mathbb{H}^1$$ .

## C.2 Introducing the chirpiness to the Heisenberg group

An automorphism of a group $$G$$ is a group isomorphism from $$G$$ onto $$G$$.

An isomorphism from $$(G_1,*)$$ to $$(G_2,\cdot)$$ is a bijective function $$f:G_1\rightarrow G_2$$ such as $$\forall x,y\in G_1$$

$\begin{equation} f(x*y) = f(x)\cdot f(y) \end{equation}$

It follows that the automorphism of a group $$(G,\cdot)$$ is a bijective function $$g:G\rightarrow G$$ such as $$\forall x,y\in G$$

$\begin{equation}\label{eq:aut_g} g(x\cdot y) = g(x)\cdot g(y) \end{equation}$

Let’s consider the affine group group $$H=\mathbb{R}_+\ltimes\mathbb{R}$$ equipped with the square matrix multiplication.

$\begin{equation} H = \left\{\left.\begin{pmatrix}a & t\\0 & 1\end{pmatrix}\right\vert a\in\mathbb{R}_+, t\in\mathbb{R}\right\} \end{equation}$

We denote $$(a,t)\equiv \begin{pmatrix}a & t\\0 & 1\end{pmatrix}\in H$$

We hence have $$(a,t)(b,s)\equiv\begin{pmatrix}a & t\\0 & 1\end{pmatrix}\begin{pmatrix}b & s\\0 & 1\end{pmatrix}=\begin{pmatrix}ab & as+t\\0 & 1\end{pmatrix}\equiv(ab,as+t)$$

In order to find $$\mathrm{Aut}(H)$$ we need to define the group action $$(H,\psi)$$ that verifies () $\begin{equation}\label{eq:aut_h} \psi((a,t)(b,s)) = \psi(a,t)\psi(b,s) \end{equation}$ We notice that the right conjugation by $$h$$ defined as $$\psi_h(x)=h^{-1}xh$$ is a group action verifying (). For all $$x,y\in H$$, $\begin{equation} \psi_h(xy)=h^{-1}xy h=(h^{-1}xh)(h^{-1}yh)=\psi_h(x)\psi_h(y) \end{equation}$ $$\psi_h$$ is therefore an inner automorphism of $$H$$.

Let $$h=(b,s)\in H$$ therefore $$\psi_h$$ is defined $$\forall(a,t)\in H$$ $\begin{equation} \psi_h(a,t)=(b,s)^{-1}(a,t)(b,s)=\left(a, \frac{t}{b}+\frac{(a-1)s}{b}\right) \end{equation}$ Let $$h=(b,0)\in H$$, the inner automorphism is defined as $$\psi_h(a,t)=(a,t/b)$$.

This corresponds to multiplying the chirpiness by $$b\in\mathbb{R}_+$$. Hence allowing to redefine the augmented space of $$(\nu,a,t)$$ as the semidirect product $$G=\mathbb{R}\ltimes H$$ with respect to $$\psi\in\mathrm{Aut}(H)$$. Nevertheless, further work is needed in order to construct such space from the Wavelet time-frequency representation.

### References


Ugo Boscain, Dario Prandi, Ludovic Sacchelli, and Giuseppina Turco. 2021. A bio-inspired geometric model for sound reconstruction. The Journal of Mathematical Neuroscience 11, 1 (January 2021), 2. DOI:https://doi.org/10.1186/s13408-020-00099-4