Let $G\rightrightarrows M$ be a Lie groupoid with structural functions $\alpha$ , $\beta$ , $m$ and $\epsilon$ , $(\Omega,\omega)$ be a l.c.s. structure on $G$ , $\sigma:G\to\hbox{\ddpp R}$ be a multiplicative function and $\theta$ be the 1-form on $G$ defined by

$$\theta=e^{\sigma}(\delta\sigma-\omega).$$

(13)

Then, $(G\rightrightarrows M,\Omega,\omega,\sigma)$ is a l.c.s. groupoid if the following properties hold:

$$m^{\ast}\Omega=\pi_{1}^{\ast}\Omega+e^{(\sigma\circ\pi_{1})}\pi_{2}^{\ast}\Omega;$$

(14)

$$\tilde{\alpha}\circ\omega=0,\quad\tilde{\beta}\circ\theta=0;$$

(15)

$$m^{\ast}\omega=\pi_{1}^{\ast}\omega,\quad m^{\ast}\theta=e^{(\sigma\circ\pi_{1})}\pi_{2}^{\ast}\theta;$$

(16)

$$\Lambda(\omega,\theta)=0,\quad(\theta+\omega-\tilde{\epsilon}\circ\tilde{\beta}\circ\omega)\circ\epsilon=0;$$

(17)

where $\pi_{i}:G^{(2)}\to G$ , $i=1,2$ , are the canonical projections, $(\Lambda,E)$ is the Jacobi structure associated with the l.c.s. structure $(\Omega,\omega)$ and $\tilde{\alpha}$ , $\tilde{\beta}$ , $\oplus_{T^{\ast}G}$ and $\tilde{\epsilon}$ are the structural functions of the cotangent groupoid $T^{\ast}G\rightrightarrows A^{\ast}G$ .

Let $(\mathfrak{g},[,])$ be a $\mathbb{Q}$ -Lie algebra and let $c$ be a $2$ -cocycle of $\mathfrak{g}$ with values in $\mathbb{Q}$ i.e., $c$ satisfies the following equation:

$$c([x,y],z)+c([y,z],x)+c([z,x],y)=0$$

for all $x,y,z\in\mathfrak{g}$ . The Sridharan algebra $U_{c}(\mathfrak{g})$ associated to $c$ is $T(\mathfrak{g})/I_{c}(\mathfrak{g})$ where $T(\mathfrak{g})$ is the tensorial algebra of $\mathfrak{g}$ and $I_{c}(\mathfrak{g})$ is the ideal generated by elements $x\cdot y-y\cdot x-[x,y]-c(x,y)\cdot 1$ with $x,y\in\mathfrak{g}$ .