U(h)-free modules and coherent families

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Abstract

We investigate the category of U(h)-free g-modules. Using a functor from this category to the category of coherent families, we show that U(h)-free modules only can exist when g is of type A or C. We then proceed to classify isomorphism classes of U(h)-free modules of rank 1 in type C, which includes an explicit construction of new simple sp(2n)-modules. The classification is then extended to higher ranks via translation functors.

Introduction

In order to understand the structure of a given module category, a classification of simple modules is very helpful. However, when g is a finite-dimensional simple complex Lie algebra, a classification of simple modules seems beyond reach; only when g=sl2 a weak version of such a classification exists, see [5], [13]. However, some classes of simple g-modules are well understood. For example, simple weight modules with finite-dimensional weight spaces are completely classified, they fall into two categories: parabolically induced modules and cuspidal modules. Parabolically induced modules include simple finite-dimensional modules [6], [7], and more generally simple highest weight modules [7], [10], [3]. Simple cuspidal modules were classified by Mathieu in 2000, see [12]. Other well studied classes of simple modules include Whittaker modules [11], and Gelfand–Zetlin modules [8].

Another natural class of modules are the ones where the Cartan subalgebra acts freely. Specifically, we let M be the full subcategory of g-Mod consisting of modules M such that ReshgMU(h)U(h). In other words, M consists of the modules which are free of rank 1 as U(h)-modules. In the paper [14], isomorphism classes of the objects of M were classified for g=sln which led to several new families of simple sln-modules. Some of these modules were also studied in connection to the Witt algebra in [16], and classifications of U(h)-free modules over different Witt algebras were obtained in [17].

In the present paper, we focus on a similar classification of the category M in type C. We start by explicitly constructing an object M0 of M, and we proceed to show that every other isomorphism class of M can be obtained by twisting M0 by an automorphism. This result is achieved by considering the connection between M and the coherent families of degree 1. We construct a functor between these categories, and can then rely on the classification of irreducible semisimple coherent families from [12] to classify M. This line of argument also directly shows that the category M is empty for finite-dimensional simple complex Lie algebras of all types other than A and C. This then completes the classification of M for all such Lie algebras. To summarize, we have the following results about the category M for simple complex finite-dimensional Lie algebras:

  • The category M is empty unless g is of type A or type C.

  • When g is of type C, there exists an object M0M (definition in Theorem 12) such that any object of M is isomorphic to M0φ (twist by automorphism) for some explicitly given φAut(g). See Theorem 22.

  • When g is of type An, a classification of M was obtained in [14]. In the context of this paper, this would be formulated as: there exists an explicitly given family of modules {MbS} parametrized by bC and S{1,,n} such that for any object M of M, there exists φAut(g) such that Mφ is isomorphic to some MbS. See [14] for details.

Here follows a brief summary of the paper. Section 2 deals with the relationship between U(h)-free modules and coherent families. In Section 2.1 we briefly discuss the category M and give an example of one of its objects. Section 2.2 reminds the reader of the notion of a coherent family, and it lists some known results about these. In Section 2.3 we construct an endofunctor W on g-Mod and prove that its image of M lies in the set of coherent families of rank 1, which proves the first point above. Section 3 deals with the classification of M in type C. In Section 3.2 we explicitly construct a simple object M0 of M, and in Section 3.3 we proceed by describing the submodule structure of W(M0) and of its semisimplification W(M0)ss. Section 3.4 discusses twisting modules by a family of automorphisms of g, which eventually leads to the proof in Section 3.5 of the second point above. Finally, in Section 3.6 we show that by applying the correct translation functor (see [2]) to M in type C, we can obtain any simple module which is U(h)-free of any finite rank. This provides a more general but less explicit classification of a larger category of modules.

Section snippets

Modules where the Cartan acts freely

Let g be a finite-dimensional simple complex Lie algebra with a fixed Cartan subalgebra h. Denote by Δ the root system and let Q:=ZΔ be the root lattice. We denote the category of all U(g)-modules by U(g)-Mod or sometimes just g-Mod. Denote by M the full subcategory of U(g)-Mod consisting of modules whose restriction to U(h) is free of rank one. When g is realized as a Lie algebra of matrices, we use the notation ei,j to denote the matrix with a single 1 in position (i,j) and zeroes everywhere

Classification of U(h)-free modules of rank 1 in type Cn

From here on, we fix g:=sp(2n); the complex symplectic Lie algebra of rank n.

Acknowledgements

I am very thankful to Volodymyr Mazorchuk for his many helpful remarks and ideas. I am also particularly grateful to Professor Olivier Mathieu who, during his visit to Uppsala, suggested the connection between U(h)-free modules and coherent families. The results of this paper are based on this connection. The remarks of the referee were also appreciated.

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