非交换导出预栈上的移位双辛结构与双泊松结构
Shifted bisymplectic and double Poisson structures on non-commutative derived prestacks
AI总结：
本文在非交换导出模函子上引入移位双辛与双泊松结构，证明其与移位辛结构的关联及典范等价，并利用形式积分性质揭示Calabi--Yau结构与商预栈上对应结构的联系。
摘要：
我们引入了微分分次结合代数上的移位双辛结构与移位双泊松结构的概念，更一般地，引入了具有良好性质余切复形的非交换导出模函子上的这些结构。对于集中在0次的平滑代数，这些结构恢复了双辛与双泊松结构的经典概念，但通常它们包含无穷的高阶同伦数据层级，以确保在拟同构下保持不变。这些结构在底层的交换导出模函子及表示函子上诱导出移位辛结构与移位泊松结构。我们证明了移位双辛结构空间与非退化n-移位双泊松结构空间之间存在典范等价。我们还给出了Calabi--Yau dg范畴上模的各种导出非交换模函子的典范移位双辛与双拉格朗日结构。与交换对应物不同，这些结构具有形式积分性质，我们利用它证明了dg代数上的Calabi--Yau与pre-Calabi--Yau结构分别对应于其伴随G_m-作用商预栈上的双辛与双泊松结构。
英文摘要：
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.
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