the *Dirac reduction* of ${\mathcal{V}}$ by the constraints $\theta_1,\dots,\theta_m$.

\[gfz\_reduced\] Let $N\geq1$ and let us consider the generalized GFZ PVA $R_N$ defined in Example \[gfzN\] (associated to the symmetric matrix $S=\big(s_{ij}\big)_{i,j=1}^N$). Let $\theta=v_1+\dots+v_N\in R_N$. We have $\{\theta_\lambda\theta\}=\lambda s$, where $s=\sum_{h,k=1}^N s_{hk}\in{\mathbb{F}}$. Provided that $s\neq0$, we can consider the Dirac reduction of $R_N$ by $\theta$, and we can identify it with $R_{N-1}={\mathbb{F}}[v_i^{(n)}\mid i\in\{1,\dots,N-1\},n\in{\mathbb{Z}}_+]$. The corresponding Dirac modified $\lambda$-bracket on $R_{N-1}$ is given by $$\label{formula}
\{{v_i}_\lambda v_j\}^D=\left(s_{ij}-\frac{s_is_j}{s}\right)\lambda\,,$$ for all $i,j=1,\ldots,N-1$, where $s_i=\sum_{k=1}^Ns_{ik}\in{\mathbb{F}}$.

\[20131002:lem1\] Let ${\mathcal{V}}_1$ and ${\mathcal{V}}_2$ be PVAs and let $\varphi:{\mathcal{V}}_1\rightarrow{\mathcal{V}}_2$ be a PVA homomorphism. Let $\theta_1,\dots\theta_m\in{\mathcal{V}}_1$ be such that the matrix $C(\partial)\in\operatorname{Mat}_{m\times m}{\mathcal{V}}_1((\partial^{-1}))$ defined in is invertible. Then, $\varphi$ induces a PVA homomorphism of the corresponding Dirac reduced PVAs $$\varphi:\,{\ensuremath{\raisebox{.40ex}{\ensuremath{{\mathcal{V}}_1}}
\! \big / \! \raisebox{-.40ex}{\ensuremath{\langle\theta_1,\dots,\theta_m\rangle_{{\mathcal{V}}_1}}}}}
\to{\ensuremath{\raisebox{.40ex}{\ensuremath{{\mathcal{V}}_2}}
\! \big / \! \raisebox{-.40ex}{\ensuremath{\langle\varphi(\theta_1),\dots,\varphi(\theta_m)\rangle_{{\mathcal{V}}_2}}}}}
\,.$$

Let $D_{\alpha\beta}(\lambda)
=\{\varphi(\theta_\beta)_{\lambda}\varphi(\theta_\alpha)\}_2=\varphi(C_{\alpha\beta}(\lambda))$, $\alpha,\beta=1,\dots,m$. Since $\varphi$ is a differential algebra homomorphism, the matrix $D(\partial)\in\operatorname{Mat}_{m\times m}{\mathcal{V}}_2((\partial^{-1}))$ is invertible, and $D^{-1}(\partial)=\varphi\big(C^{-1}(\partial)\big)$. Hence, by we have $$\varphi\left(\{a_\lambda b\}_1^D\right)
=\{\varphi(a)_{\lambda}\varphi(b)\}_2^D\,,$$ for all $a,b\in{\mathcal{V}}_1$, as required.

In general, if we have two compatible PVA $\lambda$-brackets $\{\cdot\,_\lambda\,\cdot\}_0$ and $\{\cdot\,_\lambda\,\cdot\}_1$ on ${\mathcal{V}}$ (recall Definition \[def:compatible\]), and we take their Dirac reductions by a finite number of constraints $\theta_1,\dots,\theta_m$, we do NOT get compatible PVA $\lambda$-brackets 