an equality in the space ${\mathcal{V}}_{\lambda,\mu}$.

\[20130513:def\] A *non-local Poisson vertex algebra* (PVA) is a differential algebra ${\mathcal{V}}$ endowed with an admissible non-local skew-symmetric $\lambda$-bracket $\{\cdot\,_\lambda\,\cdot\}:\,{\mathcal{V}}\otimes {\mathcal{V}}\to {\mathcal{V}}((\lambda^{-1}))$ satisfying the Jacobi identity.

Dirac reduction {#sub:non_local_2}
---------------

Let ${\mathcal{V}}$ be a (non-local) Poisson vertex algebra with $\lambda$-bracket $\{\cdot\,_{\lambda}\,\cdot\}$. Let $\theta_1,\dots,\theta_m$ be elements of ${\mathcal{V}}$, and let ${\mathcal{I}}=\langle\theta_1,\dots,\theta_m\rangle_{{\mathcal{V}}}$ be the differential ideal generated by them. Consider the matrix pseudodifferential operator $C(\partial)=(C_{\alpha\beta}(\partial))_{\alpha,\beta=1}^m
\in\operatorname{Mat}_{m\times m}{\mathcal{V}}((\partial^{-1}))$, whose symbol is $$\label{eq:C}
C_{\alpha\beta}(\lambda)=\{\theta_{\beta}{}_{\lambda}\theta_{\alpha}\}\,.$$ By the skew-symmetry condition, the pseudodifferential operator $C(\partial)$ is skewadjoint. We shall assume that the matrix pseudodifferential operator $C(\partial)$ is invertible, and we denote its inverse by $C^{-1}(\partial)=\big((C^{-1})_{\alpha\beta}(\partial)\big)_{\alpha,\beta=1}^m$.

\[20130514:def\] The *Dirac modification* of the PVA $\lambda$-bracket $\{\cdot\,_{\lambda}\,\cdot\}$, associated to the elements $\theta_1,\dots,\theta_m$, is the map $\{\cdot\,_{\lambda}\,\cdot\}^D:\,{\mathcal{V}}\times{\mathcal{V}}\to{\mathcal{V}}((\lambda^{-1}))$ given by ($a,b\in{\mathcal{V}}$): $$\label{eq:dirac}
\{a_{\lambda}b\}^D
=\{a_{\lambda}b\}
-\sum_{\alpha,\beta=1}^m
\{{\theta_{\beta}}_{\lambda+\partial}b\}_{\to}
(C^{-1})_{\beta\alpha}(\lambda+\partial)
\{a_{\lambda}\theta_{\alpha}\}\,.$$

\[prop:dirac\]

(a) The Dirac modification $\{\cdot\,_\lambda\,\cdot\}^D$ is a PVA $\lambda$-bracket on ${\mathcal{V}}$.

(b) All the elements $\theta_i,\,i=1,\dots,m$, are central with respect to the Dirac modified $\lambda$-bracket: $\{a_\lambda\theta_i\}^D=\{{\theta_i}_\lambda a\}^D=0$ for all $i=1,\dots,m$ and $a\in{\mathcal{V}}$.

(c) The differential ideal ${\mathcal{I}}=\langle\theta_1,\dots,\theta_m\rangle_{{\mathcal{V}}}\subset{\mathcal{V}}$, generated by $\theta_1,\dots,\theta_m$, is such that $\{{\mathcal{I}}\,_\lambda\,{\mathcal{V}}\}^D$, $\{{\mathcal{V}}\,_\lambda\,{\mathcal{I}}\}^D\,
    \subset{\mathcal{I}}((\lambda^{-1}))$.

The quotient space ${\ensuremath{\raisebox{.40ex}{\ensuremath{{\mathcal{V}}}}
\! \big / \! \raisebox{-.40ex}{\ensuremath{{\mathcal{I}}}}}}$ is a (non-local) PVA, with $\lambda$-bracket induced by $\{\cdot\,_\lambda\,\cdot\}^D$, which we call 