论文标题
泊松类乘形式
Multiplicative forms on Poisson groupoids
论文作者
论文摘要
给定一个lie groupoid $ \ mathcal {g} $上的$ m $，$ a $ $ \ mathcal {g} $和$ρ：a \ rightarrow tm $ adncar tm $ adncal tm $，我们提供了一个分解了一个任意乘以$ k $ -form-k $ -form-form-unific $ quy $ qu \ g}的公式。第一部分是$ e $，$ 1 $ -COCYCL的$ \ Mathfrak {J} \ Mathcal {g} $中的$ \ wedge^k t^*m $，第二部分是γ\inγ（a^*\ otimes inγ（\ wedge^wedge^{k-1} t^*m）$ - $〜$ u \ in $ u \ in $ in $ u \。我们称这对数据$（e，θ）$ $（0，k）$ - 特征对$θ$。接下来，我们证明，如果$ \ mathcal {g} $是一个poisson lie caltoid，那么$ω^{\ bullet} _ {\ mathrm {mathrm {mult}}（\ mathcal {g}）$ yathcal {g} $ nignal {g} $ nifected lie alge alge alge alge a al al al al a al al al a al al al a al al al al al a al al al al al a al al al a al al al al a al al albra（dg）（dg）此外，当与$ω^\ bullet（m）$结合使用时，这是基本歧管$ m $，$ω^{\ bullet} _ {\ mathrm { + mathrm { + mathcal {g}）$ canonical dgla crossed crossered模块的$ω^\ bullet（m）$。这补充了一个以前已知的事实，即$ \ Mathcal {g} $上的乘数多夜间字段与Schouten代数$γ（\ wedge^\ bullet a）$ crossed Module形成了DGLA交叉模块。
Given a Lie groupoid $\mathcal{G}$ over $M$, $A$ the tangent Lie algebroid of $\mathcal{G}$, and $ρ: A\rightarrow TM$ the anchor map, we provide a formula that decomposes an arbitrary multiplicative $k$-form $Θ$ on $\mathcal{G}$ into two parts. The first part is $e$, a $1$-cocycle of $\mathfrak{J}\mathcal{G}$ valued in $\wedge^k T^*M$, and the second part is $θ\in Γ(A^*\otimes (\wedge^{k-1} T^*M))$ which is $ρ$-compatible, meaning that $ι_{ρ(u)}θ(u)=0$ for all $u\in A$. We call this pair of data $(e,θ)$ the $(0,k)$-characteristic pair of $Θ$. Next, we prove that if $\mathcal{G}$ is a Poisson Lie groupoid, then the space $Ω^{\bullet}_{\mathrm{mult}}(\mathcal{G})$ of multiplicative forms on $\mathcal{G}$ has a differential graded Lie algebra (DGLA) structure. Furthermore, when combined with $Ω^\bullet(M)$, which is the space of forms on the base manifold $M$, $Ω^{\bullet}_{\mathrm{mult}}(\mathcal{G})$ forms a canonical DGLA crossed module. This supplements a previously known fact that multiplicative multivector fields on $\mathcal{G}$ form a DGLA crossed module with the Schouten algebra $Γ(\wedge^\bullet A)$ stemming from the tangent Lie algebroid $A$.
