Translation Functors and the Shapovalov Determinant

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 10 February 2015

This is an excerpt from the PhD thesis Translation Functors and the Shapovalov Determinant by Emilie Wiesner, University of Wisconsin-Madison, 2005.

Introduction

In this dissertation, I consider translation functors for the Virasoro algebra and quantum groups. The main tool of investigation is the Shapovalov determinant. This work was motivated by Jantzen's study of translation functors for semisimple Lie algebras.

A finite-dimensional, semisimple Lie algebra $𝔤$ (over $ℂ\text{)}$ decomposes as a direct sum of subalgebras ${𝔤}_{<0}\oplus 𝔥\oplus {𝔤}_{>0},$ where $𝔥$ is the Cartan subalgebra. This decomposition of $𝔤$ is well-reflected in the category $𝒪$ of $𝔤\text{-modules,}$ which includes Verma modules $M\left(\lambda \right)$ and simple modules $L\left(\lambda \right)$ $\text{(}\lambda \in {𝔥}^{*}\text{).}$ A useful concept for studying modules in Category $𝒪$ is that of blocks. We can partition the simple modules $L\left(\lambda \right)$ (or equivalently the weights $\lambda \in {𝔥}^{*}\text{)}$ into blocks so that, for any indecomposable module $M\in 𝒪,$ a composition series of $M$ will have factors coming from only one block.

A useful characterization of blocks is in terms of Verma modules. If $M\left(\mu \right)\subseteq M\left(\lambda \right),$ then $L\left(\mu \right)$ and $L\left(\lambda \right)$ are in the same block, and a block is completely determined by such embeddings. Shapovalov [Sha1972] introduced a bilinear form $⟨,⟩:M\left(\lambda \right)\times M\left(\lambda \right)\to ℂ$ on $M\left(\lambda \right)$ which relies on the decomposition of $𝔤$ given above. He also defined an associated determinant $\text{det}\hspace{0.17em}M{\left(\lambda \right)}^{\mu }$ corresponding to each weight space $M{\left(\lambda \right)}^{\mu }$ of $M\left(\lambda \right)\text{.}$ The zeros in this determinant correspond to embeddings of Verma modules $M\left(\mu \right)$ in $M\left(\lambda \right)$ and thus provide a description of the blocks of $𝔤\text{.}$

Jantzen [Jan1973] used the Shapovalov determinant $\text{det}\hspace{0.17em}M{\left(\lambda \right)}^{\mu }$ as a tool for studying translation functors for semisimple Lie algebras $𝔤\text{.}$ He showed that for an appropriate choice of $\lambda$ and $\mu ,$ the module $M\left(\lambda \right)\otimes L\left(\mu \right)$ decomposes as a direct sum of Verma modules $M\left(\nu \right)\text{.}$ Each such Verma module inherits a contravariant form from $M\left(\lambda \right)\otimes L\left(\mu \right)\text{.}$ Jantzen showed that this form differs from the standard contravariant form $⟨,⟩$ defined on $M\left(\nu \right)$ by an error term $a_{\left[\nu \right]}^{\mu }\left(\lambda \right),$ which he computed explicitly. (Here $\left[\nu \right]$ is the block containing $\nu \text{.)}$

Semisimple Lie algebras can be generalized to Lie algebras with triangular decomposition, a class of Lie algebras which possess the structural properties of $𝔤$ mentioned above. The Virasoro algebra is a concrete example of a Lie algebra with triangular decomposition. The quantum groups $U_q\left(𝔤\right)$ of semisimple Lie algebras g also have a similar decomposition. A contravariant form and Shapovalov determinant can be defined for Verma modules for both Lie algebras with triangular decomposition and for quantum groups.

Kac [Kac1980] first found a formula for the determinant $\text{det}\hspace{0.17em}M{(h,c)}^{(h+n,c)},$ where $M(h,c)$ is a Verma module for the Virasoro algebra (Theorem 3.4.3). Feigin and Fuchs [FFu1990] gave an alternative proof of this theorem and used the determinant formula to completely describe the structure of Verma module embeddings (Theorem 3.5.1). DeConcini and Kac [DKa1992] computed the determinant $\text{det}\hspace{0.17em}{M}_q{\left(\omega \right)}^{\lambda }$ of the Verma module $M_q{\left(\omega \right)}^{\lambda }$ for $U_q\left(𝔤\right)\text{.}$ I use these results in order to extend the work of Jantzen to the Virasoro algebra and quantum groups.

This thesis has the following structure. As the name suggests, Chapter 1 (Basics and Background) contains background information. Chapter 2 presents a variety of results for Lie algebras with triangular decomposition. In this chapter, I also establish some preliminary results for translation functors in the general setting of Lie algebras with triangular decomposition. In order to take advantage of the results of Kac and Feigin and Fuchs, Chapter 3 focuses on the Virasoro algebra. In Proposition 3.5.3, I use the description of Verma module embeddings given by Feigin and Fuchs to give a new description of the blocks of the Virasoro algebra. I also obtain the following results on translation functors for the Virasoro algebra: explicit projection maps onto each block of $M(h,c)\otimes L(h\prime ,c\prime )$ (Proposition 3.6.2); determinant formulas for each block of $M(h,c)\otimes L(h\prime ,c\prime ),$ including formulas for the error terms $a_j^{(h\prime ,c\prime )}(h,c)$ (Proposition 3.6.4); and a representation-theoretic interpretation of $a_j^{(h\prime ,c\prime )}(h,c)$ (Proposition 3.6.6). In Chapter 4, I produce similar results on translation functors for quantum groups: explicit projection maps onto each block of $M_q\left(\omega \right)\otimes L_q\left({\omega }_0\right)$ for weights $\omega ,{\omega }_0\in P$ (Proposition 4.6.6) and determinant formulas for each block, including formulas for the error terms $a_{\mu }^{{\omega }_0}\left(\omega \right)$ (Proposition 4.6.7).

Acknowledgements

Thanks to my advisor, Arun Ram, for his help and guidance. Thanks to Aaron for his support and love.

My research was supported in part by NSF #DMS-0245082 and NSA #MDA904-03-1-0068.

Notes and References

This is an excerpt from the PhD thesis Translation Functors and the Shapovalov Determinant by Emilie Wiesner, University of Wisconsin-Madison, 2005.

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