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.

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