论文标题
在非共同衍生的prestack上移动的双链和双泊松结构
Shifted bisymplectic and double Poisson structures on non-commutative derived prestacks
论文作者
论文摘要
我们介绍了差异分级的联想代数上移动的双胞质和转移双泊松结构的概念，更笼统地介绍了具有表现良好的cotangangent复合物的非交换性衍生模量函数。对于集中度为$ 0 $的平滑代数，这些结构恢复了双链和双泊松结构的经典概念，但通常它们涉及较高同位数据的无限层次结构，以确保它们在准iSomorphism下是不变的。这些结构诱导了基础派生的模量函数以及基础表示函子上移动的符号和转移的泊松结构。 我们表明，转移的双链结构的空间与非降级$ n $缩短的双泊松结构之间存在规范对等。我们还为模块的各种派生的非交通模量函数提供了典范的双链和双ragrangian结构，而YAU DG类别则提供了模块。 这些结构与他们的交换属性不同，我们享有正式的整合属性，我们利用这表明DG代数上的calabi-yau和calabi-yau结构分别对应于双骨和双泊松结构，以其eapectient prostacks的eapeient和双泊松结构，其experaceent Prestacks aepandeent Prestacks aepandeent Prestacks aepandeent $ \ \ m nathbb $ \ mathbb {g g} _m $ $ -saction。
We introduce the notions of shifted bisymplectic and shifted double Poisson structures on differential graded associative algebras, and more generally on non-commutative derived moduli functors with well-behaved cotangent complexes. For smooth algebras concentrated in degree $0$, these structures recover the classical notions of bisymplectic and double Poisson structures, but in general they involve an infinite hierarchy of higher homotopical data, ensuring that they are invariant under quasi-isomorphism. The structures induce shifted symplectic and shifted Poisson structures on the underlying commutative derived moduli functors, and also on underlying representation functors. We show that there are canonical equivalences between the spaces of shifted bisymplectic structures and of non-degenerate $n$-shifted double Poisson structures. We also give canonical shifted bisymplectic and bi-Lagrangian structures on various derived non-commutative moduli functors of modules over Calabi--Yau dg categories. Unlike their commutative counterparts, these structures enjoy a formal integration property, which we exploit to show that Calabi--Yau and pre-Calabi--Yau structures on a dg algebra correspond respectively to bisymplectic and double Poisson structures on its quotient prestacks by the adjoint $\mathbb{G}_m$-action.
