论文标题
在Goldman-Millson定理上，以$ a_ \ intgy $ - 代数为intunary特征
On the Goldman-Millson theorem for $A_\infty$-algebras in arbitrary characteristic
论文作者
论文摘要
完整的过滤$ A_ \ infty $ -Algebras模型在非交通设置中某些变形问题。组表示的形式变形理论是一个经典的例子。考虑到此类应用，我们为$ l_ \ infty $ -Algebras提供了来自Maurer-Cartan理论的几个关键定理的$ A_ \ infty $类似物。与$ l_ \ infty $案例相反，我们的结果在任意特征的领域中保持不变。我们首先利用一些抽象的同源代数来简明证明$ a_ \ infty $ -goldman-millson定理：神经函数，该函数分配了一个简单的$ \ nathcal {n} _ {n} _ {\ bullet}（\ bullet}（a）（a）$等价。然后，我们将$ \ MATHCAL {N} _ \ bullet（a）$的同型组中的同类组中的组成组成，以同时的代数$ h（a）$及其一组准简化元素。最后，我们返回到特征性的零案例，并表明$ a $的神经是同质的，相当于其换向器$ l_ \ infty $ -Algebra的简单毛拉 - 卡丹集合。这回答了N. de Kleijn和F. Wierstra在Arxiv中提出的问题：1809.07743。
Complete filtered $A_\infty$-algebras model certain deformation problems in the noncommutative setting. The formal deformation theory of a group representation is a classical example. With such applications in mind, we provide the $A_\infty$ analogs of several key theorems from the Maurer-Cartan theory for $L_\infty$-algebras. In contrast with the $L_\infty$ case, our results hold over a field of arbitrary characteristic. We first leverage some abstract homotopical algebra to give a concise proof of the $A_\infty$-Goldman-Millson theorem: The nerve functor, which assigns a simplicial set $\mathcal{N}_{\bullet}(A)$ to an $A_\infty$-algebra $A$, sends filtered quasi-isomorphisms to homotopy equivalences. We then characterize the homotopy groups of $\mathcal{N}_\bullet(A)$ in terms of the cohomology algebra $H(A)$, and its group of quasi-invertible elements. Finally, we return to the characteristic zero case and show that the nerve of $A$ is homotopy equivalent to the simplicial Maurer-Cartan set of its commutator $L_\infty$-algebra. This answers a question posed by N. de Kleijn and F. Wierstra in arXiv:1809.07743.
