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

The Lie superalgebra W(infinity) 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(infinity) in its natural Z-grading is isomorphic to gl(infinity). In this paper, we initiate the study of the representation theory of W(infinity). We study Z-graded W(infinity)-modules, and we introduce a category T-W that is closely related to the Koszul category T-sl(infinity) of tensor sl(infinity)-modules introduced and studied by Dan-Cohen, Serganova and Penkov. We classify the simple objects of T-W (up to isomorphism). We prove that each simple module in T-W is isomorphic to the unique simple quotient of a module induced from a simple module in T-gl(infinity), 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 T-W are highest weight modules with respect to a certain Borel subalgebra. We realize each simple module from TW as a module of tensor fields, generalizing work of Bernstein and Leites for W(n). We prove that the category T-W has enough injective objects, and for each simple module, we provide an explicit injective module in T-W that contains it. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.