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

2022/03/07 by Calixto, Lucas, Hoyt, Crystal
#17B10 #17B55 #17B65 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2203.03511

Abstract

The Lie superalgebra W(∞) 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(∞) in its natural ℤ-grading is isomorphic to \mathfrakgl(∞). In this paper, we initiate the study of the representation theory of W(∞). We study ℤ-graded W(∞)-modules, and we introduce a category \mathbbT_W that is closely related to the Koszul category \mathbbT_\mathfraksl(∞) of tensor \mathfraksl(∞)-modules introduced and studied by Dan-Cohen, Serganova and Penkov. We classify the simple objects of \mathbbT_W (up to isomorphism). We prove that each simple module in \mathbbT_W is isomorphic to the unique simple quotient of a module induced from a simple module in \mathbbT_\mathfrakgl(∞), 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 \mathbbT_W are highest weight modules with respect to a certain Borel subalgebra. We realize each simple module from \mathbbT_W as a module of tensor fields, generalizing work of Bernstein and Leites for W(n). We prove that the category \mathbbT_W has enough injective objects, and for each simple module, we provide an explicit injective module in \mathbbT_W that contains it.

Related

• Tensor representations of \mathfrak q(∞)
• bounded weight modules over the Lie superalgebra of Cartan W-type
• Bounded weight modules for basic classical Lie superalgebras at infinity
• Representations of a class of infinite-dimensional primitive Lie superalgebras
• Simple weight modules with finite-dimensional weight spaces over Witt superalgebras