Fusion Product of Positive Level Representations and Lie Algebra Homology

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

Last updated: 26 March 2015

This is an excerpt of the paper Fusion Product of Positive Level Representations and Lie Algebra Homology by Shrawan Kumar, Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250, USA.

Preliminaries and Notation

Let $𝔤$ be a finite dimensional complex simple Lie algebra. (We also fix a Borel subalgebra $𝔟$ and a Cartan subalgebra $𝔥\subset 𝔟$ of $𝔤\text{.)}$ Then the associated affine Kac-Moody Lie algebra is by definition the space $$\stackrel{\sim }{𝔤}≔𝔤{\otimes }_{ℂ}ℂ\left[t^{\pm 1}\right]\oplus ℂK$$ together with the Lie bracket (for $X,Y\in 𝔤$ and $P,Q\in ℂ\left[t^{\pm 1}\right]\text{)}$ $$\begin{array}{c}[X\otimes P,Y\otimes Q]=[X,Y]\otimes PQ+\left(⟨X,Y)\underset{t=0}{\text{Res}}\left(\frac{dP}{dt}Q\right)\right)K,\text{and}\\ [\stackrel{\sim }{𝔤},K]=0,\end{array}$$ where $⟨·,·)$ is the Killing form on $𝔤,$ normalized so that $⟨\theta ,\theta ⟩=2$ for the highest root $\theta$ of $𝔤\text{.}$

The Lie algebra $𝔤$ sits as a Lie subalgebra of $\stackrel{\sim }{𝔤}$ as $𝔤\otimes t^0\text{.}$ The Lie algebra $\stackrel{\sim }{𝔤}$ admits a distinguished "parabolic" subalgebra $$\stackrel{\sim }{𝔭}≔𝔤\otimes ℂ\left[t\right]\oplus ℂK\text{.}$$ We also define its "nil-radical" $\stackrel{\sim }{𝔲}$ (which is an ideal of $\stackrel{\sim }{𝔭}\text{)}$ by $$\stackrel{\sim }{𝔲}≔𝔤\otimes tℂ\left[t\right],$$ and its "Levi component" (which is a Lie subalgebra of $\stackrel{\sim }{𝔭}\text{)}$ $${\stackrel{\sim }{𝔭}}^0≔𝔤\otimes t^0\oplus ℂK\text{.}$$ Clearly (as a vector space) $$\stackrel{\sim }{𝔭}=\stackrel{\sim }{𝔲}\oplus {\stackrel{\sim }{𝔭}}^0\text{.}$$

Also define ${\stackrel{\sim }{𝔲}}^{-}=𝔤\otimes t^{-1}ℂ\left[t^{-1}\right]\subseteq \stackrel{\sim }{𝔤},$ and the Cartan subalgebra $\stackrel{\sim }{𝔥}=𝔥\otimes t^0\oplus ℂK$ of $\stackrel{\sim }{𝔤}\text{.}$ Let $W$ (resp. $\stackrel{\sim }{W}\text{)}$ be the Weyl group of $𝔤$ (resp. $\stackrel{\sim }{𝔤}\text{).}$ Then $\stackrel{\sim }{W}$ acts on the dual space ${\stackrel{\sim }{𝔤}}^{*}$ by linear automorphisms. Let $\rho \in {𝔥}^{*}$ be half the sum of the positive roots of $𝔤$ and define $\stackrel{\sim }{\rho }\in {\stackrel{\sim }{𝔥}}^{*}$ by $\stackrel{\sim }{\rho }{|}_{𝔥}=\rho$ and $$\stackrel{\sim }{\rho }\left(K\right)=1+⟨\rho ,{\theta }^{\vee }⟩=\text{dual Coxeter number of}\hspace{0.17em}𝔤\text{,}$$ where $\theta$ (as earlier) is the highest root of $𝔤$ and ${\theta }^{\vee }$ is the associated coroot. Define the shifted action of $\stackrel{\sim }{W}$ on ${\stackrel{\sim }{𝔥}}^{*}$ by $w*\beta =w(\beta +\stackrel{\sim }{\rho })-\stackrel{\sim }{\rho },$ for $\beta \in {\stackrel{\sim }{𝔥}}^{*}$ and $w\in \stackrel{\sim }{W}\text{.}$ Fix a positive integer $\ell \text{.}$ Let $P^{+}\subset {𝔥}^{*}$ be the set of dominant integral weights of $𝔤$ and let $P_{\ell }^{+}≔\{\lambda \in P^{+}:⟨\lambda ,{\theta }^{\vee }⟩\le \ell \}$ be the fundamental alcove.

Define the loop algebra $\mathrm{\Omega }\left(𝔤\right)≔𝔤{\otimes }_{ℂ}ℂ\left[t^{\pm 1}\right]$ with Lie bracket $[X\otimes P,Y\otimes Q]=[X,Y]\otimes PQ,$ for $X,Y\in 𝔤$ and $P,Q\in ℂ\left[t^{\pm 1}\right]\text{.}$ Then $\stackrel{\sim }{𝔤}$ can be viewed as a one-dimensional central extension of $\mathrm{\Omega }\left(𝔤\right)\text{:}$ $$\begin{array}{cc}0⟶ℂK⟶\stackrel{\sim }{𝔤}\stackrel{\pi }{⟶}\mathrm{\Omega }\left(𝔤\right)⟶0,& \text{(1.1)}\end{array}$$ where the Lie algebra homomorphism $\pi$ is defined by $\pi (X\otimes P)=X\otimes P$ and $\pi \left(K\right)=0\text{.}$

Irreducible Representations of $\stackrel{\sim }{𝔤}$

Fix an irreducible (finite-dimensional) representation $V$ of $𝔤$ and a number $\ell \in ℂ$ (to be called the level or central charge). Then we define the associated generalized Verma module for $\stackrel{\sim }{𝔤}$ as $$M(V,\ell )=U\left(\stackrel{\sim }{𝔤}\right){\otimes }_{U\left(\stackrel{\sim }{𝔭}\right)}{I}_{\ell }\left(V\right),$$ where the $\stackrel{\sim }{𝔭}\text{-module}$ $I_{\ell }\left(V\right)$ has the same underlying vector space as $V$ on which $\stackrel{\sim }{𝔲}$ acts trivially, the central element $K$ acts via the scalar $\ell$ and the action of $𝔤=𝔤\otimes t^0$ is via the $𝔤\text{-module}$ structure on $V\text{.}$

In the case when $\ell$ is a positive integer (in fact, it suffices to assume that $\ell \ne -h,$ where $h$ is the dual Coxeter number of $𝔤\text{),}$ $M(V,\ell )$ has a unique irreducible quotient, denoted $L(V,\ell )\text{.}$

It is easy to see that any vector $v\in M(V,\ell )$ is contained in a finite dimensional $𝔤\text{-submodule}$ of $M(V,\ell )\text{.}$ In particular, the same property holds for any vector in $L(V,\ell )\text{.}$

Consider the Lie subalgebra ${𝔯}^0$ of $\stackrel{\sim }{𝔤}$ spanned by $\{Y_{\theta }\otimes t,{\theta }^{\vee }\otimes 1,X_{\theta }\otimes t^{-1}\},$ where $Y_{\theta }$ (resp. $X_{\theta }\text{)}$ is a non-zero root vector of $𝔤$ corresponding to the root $-\theta$ (resp. $\theta \text{)}$ and the coroot ${\theta }^{\vee }$ is to be thought of as an element of $𝔥\text{.}$ Then the Lie algebra ${𝔯}^0$ is isomorphic to $sl\left(2\right)\text{.}$

A $\stackrel{\sim }{𝔤}\text{-module}$ $W$ is said to be integrable if every vector $v\in W$ is contained in a finite-dimensional $𝔤\text{-submodule}$ of $W$ and also $v$ is contained in a finite-dimensional ${𝔯}^0\text{-submodule}$ of $W\text{.}$

Then it follows easily from the $sl\left(2\right)\text{-theory}$ that the irreducible module $L(V,\ell )$ is integrable if and only if $\ell$ is an integer and $\ell \ge (\lambda ,{\theta }^{\vee }),$ where $\lambda$ is the highest weight of $V\text{.}$

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