Symplectic Groupoid

Multiplicative forms

Let $\mathcal{G}\rightrightarrows M$ be a Lie groupoid. Let $\mathrm{m}$ be the groupoid multiplication map $$ \mathrm{m}:\mathcal{G}^{(2)}\to \mathcal{G}: (g,h)\mapsto g\circ h. $$ Consider a $0$-form, namely, a function $f\in C^\infty(\mathcal{G}$. In order for the groupoid structure to be compatible with the $0$-form, $f\colon\mathcal{G} \to \mathbb{R}$ should be a homomorphism of groupoids: $$ f(g\cdot h) = f(g) + f(h), \forall (g,h)\in \mathcal{G}^{(2)}, $$ or equivalently $$ \mathrm{m}^*(f) = \operatorname{pr}^\ast_1 f+ \operatorname{pr}^\ast_2 f. $$

Definition. A multiplicative form on a Lie groupoid $\mathcal{G}$ is a differential form $\omega \in \Omega^k(\mathcal{G})$ s.t. $$ \mathrm{m}^*(\omega) = \operatorname{pr}^\ast_1 \omega+ \operatorname{pr}^\ast_2\omega. $$

Remark(Lie group) . For $\mathcal{G}=G$ a Lie group, multiplicative $1$-forms are bi-invariant, and multiplicative $k$-forms $(k\geq 2)$ all vanish. This follows immediately after writing down the pullbacks explicitly. For $1$-form $\omega$, $ (\mathrm{m}^\ast\omega)(V,0) = \operatorname{pr}_1^\ast \omega(V,0) + \operatorname{pr}_2^\ast\omega(V,0) $ over $(g,h)$ and note that $\mathrm{m}_{\ast}(V_g,0_h)=R_{h\ast}V_g$, this gives $\omega(V)|_{gh} = \omega(V)|_g$. Now use $(0,W)$ instead of $(V,0)$, the bi-invariance follows. To arrive from bi-invariance at multiplicativity, pullback along $\mathrm{m}_\ast: (V,0)_{g,h}\mapsto V_{gh}$, $\mathrm{pr}_{1\ast}: (V,0)_{g,h} \mapsto V_g$ and $\mathrm{pr}_{2\ast}:(V,0)_{g,h}\mapsto 0_h$. For $k$-form $\omega$, it is sufficient to consider the case $k=2$, which is immediate from the computation of $\mathrm{m}^\ast\omega((0,V),(W,0))$. ◈

Multiplicativity condition

Two properties that might illustrate the character of multiplicativity condition.

Multiplicative forms are right/left-invariant on $\mathsf{s}$-/$\mathsf{t}$-fibers. Note that here right actions are precomposition of morphisms (we use the right-to-left convention for morphisms).
Multiplicativity is a cohomological condition.

The first property: In exactly the same way as the above bi-invariance of multiplicative $1$-forms on Lie groups is proved, one can see that

Lemma. A multiplicative form $\omega\in \Omega^k(\mathcal{G})$ for $k\geq 1$ is right(resp. left)-invariant when restricted to the $\mathsf{s}$(resp. $\mathsf{t}$)-fibers.

The second property. Define maps from base to morphisms and then to the fiber product: $$ \delta:\Omega^k(\mathcal{G})\to \Omega^k(\mathcal{G}^{[2]}): \omega \mapsto (\operatorname{pr}^*_1 -\mathrm{m}^\ast + \operatorname{pr}^\ast_2)\omega $$ $$ \delta:\Omega^k(M)\to \Omega^k(\mathcal{G}): \alpha \mapsto (\mathsf{s}^\ast-\mathsf{t}^\ast)\alpha $$ then $\delta\circ\delta =0$. This can be checked by evaluating $\delta\circ\delta \alpha$ at $(g_2,g_1)\in \mathcal{G}^{[2]}$: $$ p_2 \xleftarrow{g_2} p_1 \xleftarrow{g_1} p_0. $$

The condition that $\omega\in\Omega^k(\mathcal{G})$ is multiplicative means that $\omega$ is a cocycle, $$ \delta \omega = 0. $$ The boundaries, or multiplicatively exact forms, are those $\omega \in \Omega^{k-1}(\mathcal{G})$ of the form $$ \omega = \delta(\alpha) = (\mathsf{s}^\ast -\mathsf{t}^\ast)\alpha,\quad (\alpha \in \Omega^{k-1}(M)). $$

Definition. A symplectic groupoid is a Lie groupoid $\Sigma\rightrightarrows M$ together with a multiplicative symplectic form $\omega \in \Omega^2(\Sigma)$.

A good example will be the action groupoid associated with the coadjoint action. Let $G$ be a Lie group and let $\Sigma = G\rtimes \mathfrak{g}^\ast \rightrightarrows \mathfrak{g}^\ast$ be the action groupoid. Spelled out, this is the groupoid with $\mathsf{s}(g,\xi)=\xi,\mathsf{t}(g,\xi)=g\cdot\xi = \operatorname{Ad}_g^\ast \xi$.

Symplectic groupoid actions

Definition. Let $(\Sigma,\Omega)\rightrightarrows M$ be a symplectic groupoid and $(S,\omega)$ a symplectic manifold. A left groupoid action $A:\sigma\times_M S\to S$ along amap $\mu:S\to M$ is called a sympelctic groupoid action if $$ A^\ast \omega = \operatorname{pr}^\ast_1\Omega + \operatorname{pr}^\ast_2\omega $$ where $(\operatorname{pr}_1,\operatorname{pr}_2)$ are projections onto $(\Sigma,S)$ from $\Sigma\times_M S$.

The space $(S,\omega)$ is called a Hamiltonian $(\Sigma,\Omega)$-space with the moment map $\mu:S\to M$.

Remark(Hamiltonian spaces) . Symplectic groupoid actions of the action groupoid $G\rtimes \mathfrak{g}^\ast$ are the same as $G$-Hamiltonian spaces $(S,\omega,\mu)$. ◈