[Paper Review] Translated simple modules for Lie algebras and simple supermodules for Lie superalgebras

Chih-Whi Chen, Kevin Coulembier+1 more|arXiv (Cornell University)|Jul 10, 2018

Algebraic structures and combinatorial models3 citations

TL;DR

This paper establishes that tensor products of simple and finite-dimensional modules over $ \mathfrak{sl}_n$ have finite-type socles, enabling a reduction of the classification of simple $ \mathfrak{q}(n)$-supermodules to that of simple $ \mathfrak{sl}_n$-modules. It further shows that the Kac induction functor preserves finite-type socles and radicals for type I Lie superalgebras, providing a structural framework for supermodule classification via category $ \mathcal{O}$ combinatorics.

ABSTRACT

We prove that the tensor product of a simple and a finite dimensional $\mathfrak{sl}_n$-module has finite type socle. This is applied to reduce classification of simple $\mathfrak{q}(n)$-supermodules to that of simple $\mathfrak{sl}_n$-modules. Rough structure of simple $\mathfrak{q}(n)$-supermodules, considered as $\mathfrak{sl}_n$-modules, is described in terms of the combinatorics of category $\mathcal{O}$.

Motivation & Objective

• To determine whether tensor products of simple and finite-dimensional $ \mathfrak{sl}_n$-modules have finite-type socles.
• To reduce the classification of simple $ \mathfrak{q}(n)$-supermodules to that of simple $ \mathfrak{sl}_n$-modules.
• To describe the rough structure of simple $ \mathfrak{q}(n)$-supermodules as $ \mathfrak{sl}_n$-modules using category $ \mathcal{O}$ combinatorics.
• To prove that the Kac induction functor preserves finite-type socles and radicals for Lie superalgebras of type I.
• To establish a connection between the representation theory of Lie superalgebras and the combinatorics of Kazhdan-Lusztig theory in type A.

Proposed method

• Prove that the socle of $S \otimes E$, where $S$ is a simple and $E$ a finite-dimensional $ \mathfrak{sl}_n$-module, has finite length.
• Use projective functors and Kazhdan-Lusztig combinatorics in type A to show that the socle of $S \otimes E$ is essential in the case of reductive Lie superalgebras over $ \mathbb{C}$.
• Apply the result on finite-type socles to show that any $ \mathfrak{g}$-supermodule is a quotient of an induced $ \mathfrak{g}_{ \bar{0}}$-module when $ \mathfrak{g}_{ \bar{0}}$ is of type A.
• Use the Kac induction functor $\mathrm{Ind}^{ \widetilde{{\mathfrak{g}}}}_{{\mathfrak{p}}}$ to relate $ \mathfrak{g}$-supermodules to $ \mathfrak{g}_{ \bar{0}}$-modules.
• Leverage the isomorphism $\mathrm{Hom}_{\widetilde{{\mathfrak{g}}}}(K(M), L) \cong \mathrm{Hom}_{{\mathfrak{g}}}(M, S_L)$ to relate simple modules and preserve socle/radical structure.
• Use the fact that $\mathrm{soc}(K(V))$ is generated by $\Lambda^d \widetilde{{\mathfrak{g}}}_{-1} \otimes \mathrm{soc}(V)$ to show preservation of socle finiteness and essentiality.

Experimental results

Research questions

• RQ1 Does the tensor product of a simple and a finite-dimensional $ \mathfrak{sl}_n$-module have a socle of finite length?
• RQ2 Is the socle of such a tensor product an essential submodule?
• RQ3 Can the classification of simple $ \mathfrak{q}(n)$-supermodules be reduced to the classification of simple $ \mathfrak{sl}_n$-modules?
• RQ4 Does the Kac induction functor preserve finite-type socles and radicals for Lie superalgebras of type I?
• RQ5 What is the rough structure of simple $ \mathfrak{q}(n)$-supermodules when viewed as $ \mathfrak{sl}_n$-modules?

Key findings

• The tensor product of a simple and a finite-dimensional $ \mathfrak{sl}_n$-module has a socle of finite length.
• The socle of such a tensor product is an essential submodule, implying finite-type socle in general for reductive Lie superalgebras over $ \mathbb{C}$ of type A.
• The classification of simple $ \mathfrak{q}(n)$-supermodules reduces to that of simple $ \mathfrak{sl}_n$-modules via the socle finiteness property.
• The Kac induction functor preserves finite-type socles and radicals for Lie superalgebras of type I, and preserves the length of the socle and radical.
• Verma supermodules with respect to distinguished Borel subalgebras over type I Lie superalgebras have simple socles.
• The socle of $K(V)$, where $V$ is a $ \mathfrak{g}$-module with finite-type socle, is isomorphic to $\Lambda^d \widetilde{{\mathfrak{g}}}_{-1} \otimes \mathrm{soc}(V)$ and has the same length as $\mathrm{soc}(V)$.