论文标题
在减少乘法多发射器的空间上
On the reduced space of multiplicative multivectors
论文作者
论文摘要
严格的谎言$ 2 $ -Algebra $γ（\ wedge^\ bullet a）\ stackrel {t} {\ rightArow} \ Mathfrak {x} _ {\ Mathrm {mathrm {mathrm {mathrm {mathrm {mathrm {mathcal {g}在这里，$γ（\ wedge^\ bullet a）$是切线的schouten代数，lie algebroid $ a $ a $ \ nathcal {g} $和$ \ m athfrak {x} _ {\ mathrm {mathrm {mult}}}}}}}}}}^\ mathcal（g}） $ \ MATHCAL {G} $。商$ {r} _ {\ mathrm { + mathrm { + bult：= \ mathfrak {x} _ {\ mathrm {\ mathrm {mathrm {mult}}^\ bulter（\ mathcal {g}）乘法多送子。我们证明了$ \ mathfrak中的元素的规范分解公式，{x} _ {\ Mathrm {Mathrm {Mult}}}}^\ bullet（\ Mathcal {g}）$并建立$ {r} _ {r} _ {\ Mathrm {Mathrm {mult}}^k $和co { ^1（\ Mathfrak {J} \ Mathcal {g}，\ wedge ^k a）$其中$ \ Mathfrak {J} \ Mathcal {G} $是$ \ Mathcal {g} $的JET glotoid，而$ 1 \ leqslant k \ leqslant k \ leqslant \ Mathrm}我们还研究$ {r} _ {\ mathrm {diff}}}^\ bullet $，$ a $上的lie algebroid差异空间。通过服用InfInitesimals，$ \barδ：$ $ {r} _ {\ MATHRM {MULT}}}}^\ bullet $ $ $ \ to $ $ {r} _ {\ Mathrm {diff}}}^\ bullet，两个还原的空间是相关的。我们发现，$ \barδ$的内核与范斯特映射$ \ mathrm {h}^1的内核是同构的（\ mathcal {g}，\ wedge^k \kerρ）\ to \ to \ mathrm {h}
A strict Lie $2$-algebra $Γ(\wedge^\bullet A) \stackrel{T}{\rightarrow} \mathfrak{X}_{\mathrm{mult}}^\bullet(\mathcal{G})$ is associated with any Lie groupoid $\mathcal{G}$. Here, $Γ(\wedge^\bullet A)$ is the Schouten algebra of the tangent Lie algebroid $A$ of $\mathcal{G}$ and $\mathfrak{X}_{\mathrm{mult}}^\bullet(\mathcal{G})$ is the space of multiplicative multivectors on $\mathcal{G}$. The quotient ${R}_{\mathrm{mult}}^\bullet:=\mathfrak{X}_{\mathrm{mult}}^\bullet(\mathcal{G})/\mathrm{Img} T$, a Morita invariant of $\mathcal{G}$, is called the reduced space of multiplicative multivectors. We prove a canonical decomposition formula of elements in $\mathfrak{X}_{\mathrm{mult}}^\bullet(\mathcal{G})$ and establish a key relation between ${R}_{\mathrm{mult}}^k$ and the cohomology $\mathrm{H} ^1(\mathfrak{J} \mathcal{G},\wedge^k A)$ where $\mathfrak{J} \mathcal{G}$ is the jet groupoid of $\mathcal{G}$ and $1\leqslant k\leqslant \mathrm{rank} A$. We also study ${R}_{\mathrm{diff}}^\bullet $, the reduced space of Lie algebroid differentials on $A$. By taking infinitesimals, $\barδ: $ ${R}_{\mathrm{mult}}^\bullet $ $\to $ ${R}_{\mathrm{diff}}^\bullet $, the two reduced spaces are related. We find that the kernel of $\barδ$ is isomorphic to the kernel of the Van Est map $\mathrm{H}^1(\mathcal{G},\wedge^k \kerρ)\to \mathrm{H}^1(A,\wedge^k \kerρ)$, where $ρ$ is the anchor of $A$.
