发帖
雷达卡
摘要翻译：
本文介绍了一般结合代数a的微分算子的一种新的形式，它用双导子的双模DDer(a)代替了交换代数上Grothendieck的微分算子的概念，从而代替了交换代数上的Grothendieck的微分算子的概念。我们的微分算子不作用于代数A本身，而是作用于F(A)，一个以泛函方式与任何非交换代数A相关联的Fock空间。微分算子的相应代数D（F(A))是过滤的，而相关分次代数gr D（F(A))在某种扭曲意义上是交换的。由此得到的gr D（F(A)）上的双泊松结构与Van den Bergh提出的双泊松结构密切相关。具体地，我们证明了当A是光滑的时，gr D(F(A))=F(T_A(DDer(A))。Fock空间F(A)中带有一个额外的轮架结构，这是一个与轮式支柱概念密切相关的新概念，对我们的构造至关重要。也有谎言的概念，等等。在这种语言中，D(F(A))成为一个双导子的李轴子的泛包络轴子。在本文的第二部分，我们将Koszul的一个经典构造推广到非对易环境中，证明了DDer(a)上任何Ricci-平坦的、无扭转的双模连接都会产生一个二阶（轮式）微分算子，它是BV-算子的非对易模拟。
---
英文标题：
《Differential operators and BV structures in noncommutative geometry》
---
作者：
Victor Ginzburg and Travis Schedler
---
最新提交年份：
2010
---
分类信息：
一级分类：Mathematics 数学
二级分类：Quantum Algebra 量子代数
分类描述：Quantum groups, skein theories, operadic and diagrammatic algebra, quantum field theory
量子群，skein理论，运算代数和图解代数，量子场论
--
一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
--
一级分类：Mathematics 数学
二级分类：Rings and Algebras 环与代数
分类描述：Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups
非交换环与代数，非结合代数，泛代数与格论，线性代数，半群
--
---
英文摘要：
We introduce a new formalism of differential operators for a general associative algebra A. It replaces Grothendieck's notion of differential operator on a commutative algebra in such a way that derivations of the commutative algebra are replaced by DDer(A), the bimodule of double derivations. Our differential operators act not on the algebra A itself but rather on F(A), a certain `Fock space' associated to any noncommutative algebra A in a functorial way. The corresponding algebra D(F(A)), of differential operators, is filtered and gr D(F(A)), the associated graded algebra, is commutative in some `twisted' sense. The resulting double Poisson structure on gr D(F(A)) is closely related to the one introduced by Van den Bergh. Specifically, we prove that gr D(F(A))=F(T_A(DDer(A)), provided A is smooth. It is crucial for our construction that the Fock space F(A) carries an extra-structure of a wheelgebra, a new notion closely related to the notion of a wheeled PROP. There are also notions of Lie wheelgebras, and so on. In that language, D(F(A)) becomes the universal enveloping wheelgebra of a Lie wheelgebroid of double derivations. In the second part of the paper we show, extending a classical construction of Koszul to the noncommutative setting, that any Ricci-flat, torsion-free bimodule connection on DDer(A) gives rise to a second order (wheeled) differential operator, a noncommutative analogue of the BV-operator.
---
PDF链接：
https://arxiv.org/pdf/0710.3392
京公网安备 11010802022788号
