Representations of the Lie superalgebra of superderivations of the Grassmann algebra at infinity

Abstract

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 $\Lambda(n)$. 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.