论文标题
Grassmann代数的无限层状的谎言超级代表
Representations of the Lie superalgebra of superderivations of the Grassmann algebra at infinity
论文作者
论文摘要
Lie superalgebra $ w（\ infty）$被定义为简单有限维纸箱类型的直接限制，lie superalgebras $ w（n）$ a $ n $ a $ to to infinity to in infinity，其中$ w（n）$表示Lie lie lie superalgebra是Grassmann Algebra $λ$λ（n）$的Superalgebra。 $ w（\ infty）$的零组件在其天然$ \ mathbb {z} $中 - 分级是同构对$ \ mathfrak {gl}（\ infty）$。 在本文中，我们启动了$ W（\ infty）$的表示理论的研究。我们研究$ \ mathbb {z} $ - 分级$ w（\ infty）$ - 模块，我们引入了与Koszul类别$ \ Mathbb {t} _ {t} _ {\ Mathfrak {\ Mathfrak {\ Mathfrak {\ Mathfrak {\ Mathfrak {\ Mathfrak {\ Mathfrak {\ Mathfrak {\ sl {sl} $ ats and} $与Koszul类别$ \ MATHBB类别$ \ Mathbb cantery $ \ Mathbb {t} _W $ $ \ mathfrak {sl}（\ infty）$ - 由Dan-Cohen，Serganova和Penkov引入和研究的模块。我们对$ \ mathbb {t} _W $的简单对象进行了分类（最重要的是同构）。我们证明，$ \ mathbb {t} _w $中的每个简单模块都是同构的，与$ \ m \ m \ mathbb {t} _ {\ mathfrak {\ mathfrak {gl}（\ infty）} $和vice versa的唯一简单的模块的唯一简单商，以及$ w（ns vens）$ w（n ser）。作为推论，我们发现$ \ mathbb {t} _W $中的所有简单模块相对于某个Borel SubeGerbra都是最高的权重模块。我们将每个简单的模块从$ \ mathbb {t} _W $作为张量字段的模块，将伯恩斯坦和leites的工作概括为$ w（n）$。我们证明类别$ \ mathbb {t} _W $具有足够的注入对象，对于每个简单的模块，我们在$ \ mathbb {t} _W $中提供一个明确的注射模块，其中包含它。
The Lie superalgebra $W(\infty)$ is defined to be the direct limit of the simple finite-dimensional Cartan type Lie superalgebras $W(n)$ as $n$ goes to infinity, where $W(n)$ denotes the Lie superalgebra of superderivations of the Grassmann algebra $Λ(n)$. The zeroth component of $W(\infty)$ in its natural $\mathbb{Z}$-grading is isomorphic to $\mathfrak{gl}(\infty)$. In this paper, we initiate the study of the representation theory of $W(\infty)$. We study $\mathbb{Z}$-graded $W(\infty)$-modules, and we introduce a category $\mathbb{T}_W$ that is closely related to the Koszul category $\mathbb{T}_{\mathfrak{sl}(\infty)}$ of tensor $\mathfrak{sl}(\infty)$-modules introduced and studied by Dan-Cohen, Serganova and Penkov. We classify the simple objects of $\mathbb{T}_W$ (up to isomorphism). We prove that each simple module in $\mathbb{T}_W$ is isomorphic to the unique simple quotient of a module induced from a simple module in $\mathbb{T}_{\mathfrak{gl}(\infty)}$, and vice versa, which is analogous to the case for $W(n)$ studied by Serganova. As a corollary, we find that all simple modules in $\mathbb{T}_W$ are highest weight modules with respect to a certain Borel subalgebra. We realize each simple module from $\mathbb{T}_W$ as a module of tensor fields, generalizing work of Bernstein and Leites for $W(n)$. We prove that the category $\mathbb{T}_W$ has enough injective objects, and for each simple module, we provide an explicit injective module in $\mathbb{T}_W$ that contains it.
