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.

A New Geometric Definition of Fusion Product

Fix a positive integer $\ell \text{.}$ We first recall the definition of fusion product for positive level representations.

For any two $\lambda ,\mu \in P_{\ell }^{+},$ there is a fusion product $$L(V\left(\lambda \right),\ell ){\otimes }^{•}L(V\left(\mu \right),\ell ),$$ which is again an integrable representation of $\stackrel{\sim }{𝔤}$ with the same central charge $\ell \text{.}$ It is given as: $$L(V\left(\lambda \right),\ell ){\otimes }^{•}L(V\left(\mu \right),\ell )≔\underset{\nu \in P_{\ell }^{+}}{⨁}{n}_{\lambda ,\mu }\left(\nu \right)L(V\left(\nu \right),\ell ),$$ where $n_{\lambda ,\mu }\left(\nu \right)$ is the dimension of the space of vacua for the Riemann sphere ${\mathbb{P}}^1$ with three punctures 0, 1, $\infty$ and the representations $V\left(\lambda \right),$ $V\left(\mu \right)$ and $V{\left(\nu \right)}^{•}$ attached to them respectively (cf. [TUY1989]).

Let $\stackrel{\sim }{𝒢}$ be the affine Kac-Moody group associated to the Lie algebra $\stackrel{\sim }{𝔤}$ and $\stackrel{\sim }{𝒫}$ its parabolic subgroup (corresponding to the Lie subalgebra $\stackrel{\sim }{𝔭}\text{)}$ (cf. [Kum1987, §1]). Then $X=\stackrel{\sim }{𝒢}/\stackrel{\sim }{𝒫}$ is a projective ind-variety. Now, given a finite-dimensional algebraic representation $V$ of $\stackrel{\sim }{𝒫},$ we can consider the associated homogeneous vector bundle $𝒱$ on $X$ and the corresponding Euler-Poincaré characteristic (which is a virtual $\stackrel{\sim }{𝒢}\text{-module)}$ $$\chi (X,𝒱)≔\sum _i{(-1)}^i{H}^i(X,𝒱)\text{.}$$

Recall that $H^i(X,𝒱)$ is determined in [Kum1987, Corollary 3.11] (and also in [Mat1988]).

We give a new definition of a fusion product ${\otimes }^F$ in the following.

For any positive integer $\ell ,$ and $\lambda ,\mu \in P_{\ell }^{+},$ define $${\left[L(V\left(\lambda \right),\ell ){\otimes }^{F}L(V\left(\mu \right),\ell )\right]}^{•}\cong \chi (X,V),$$ as virtual $\stackrel{\sim }{𝒢}\text{-modules,}$ where the $\stackrel{\sim }{𝒫}\text{-module}$ $V≔{\left(I_{\ell }(V\left(\lambda \right)\otimes V\left(\mu \right))\right)}^{•}$ (cf. §1.1 for the notation $I_{\ell }\text{).}$

Let $R_{\ell }\left(G\right)$ be the free Abelian group generated by $\{L(V\left(\nu \right),\ell ):\nu \in P_{\ell }^{+}\}\text{.}$ Then ${\otimes }^F$ gives rise to a product in $R_{\ell }\left(G\right)\text{.}$

Let $G$ be the (simple) simply-connected complex algebraic group with Lie algebra $𝔤,$ and let $R\left(G\right)$ be its representation ring, i.e., $R\left(G\right)$ is the free Abelian group generated by the $G\text{-modules}$ $\{V\left(\lambda \right):\lambda \in P^{+}\},$ which is a ring under the usual tensor product of $G\text{-modules.}$ Define the $\mathbb{Z}\text{-linear}$ map $$\beta :R\left(G\right)⟶R_{\ell }\left(G\right)$$ by $\beta {\left(W\right)}^{•}=\chi (X,{𝒲}^{*}),$ where $𝒲$ is the homogeneous vector bundle on $X$ associated to the $\stackrel{\sim }{𝒫}\text{-module}$ $I_{\ell }\left(W\right)$ and ${𝒲}^{*}$ is the dual vector bundle on $X\text{.}$

We have the following lemma.

The kernel of $\beta$ is an ideal of $R\left(G\right)\text{.}$ Moreover, $\beta$ is a homomorphism with respect to the product ${\otimes }^F$ in $R_{\ell }\left(G\right)\text{.}$ In particular, $R_{\ell }\left(G\right)$ is an associative (and commutative) algebra under ${\otimes }^F\text{.}$

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