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.

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 $$L(V\left(\lambda \right),\ell ){\otimes}^{\u2022}L(V\left(\mu \right),\ell ),$$ which is again an integrable representation of $\stackrel{\sim}{\U0001d524}$ with the same central charge $\ell \text{.}$ It is given as: $$L(V\left(\lambda \right),\ell ){\otimes}^{\u2022}L(V\left(\mu \right),\ell )\u2254\underset{\nu \in {P}_{\ell}^{+}}{\u2a01}{n}_{\lambda ,\mu}\left(\nu \right)L(V\left(\nu \right),\ell ),$$ 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 $V{\left(\nu \right)}^{\u2022}$ attached to them respectively (cf. [TUY1989]).

Let $\stackrel{\sim}{\mathcal{G}}$ be the affine Kac-Moody group associated to the Lie algebra $\stackrel{\sim}{\U0001d524}$ and $\stackrel{\sim}{\mathcal{P}}$ its parabolic subgroup (corresponding to the Lie subalgebra $\stackrel{\sim}{\U0001d52d}\text{)}$ (cf. [Kum1987, §1]). Then $X=\stackrel{\sim}{\mathcal{G}}/\stackrel{\sim}{\mathcal{P}}$ is a projective ind-variety. Now, given a finite-dimensional algebraic representation $V$ of $\stackrel{\sim}{\mathcal{P}},$ we can consider the associated homogeneous vector bundle $\mathcal{V}$ on $X$ and the corresponding Euler-Poincaré characteristic (which is a virtual $\stackrel{\sim}{\mathcal{G}}\text{-module)}$ $$\chi (X,\mathcal{V})\u2254\sum _{i}{(-1)}^{i}{H}^{i}(X,\mathcal{V})\text{.}$$

Recall that ${H}^{i}(X,\mathcal{V})$ 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 $${\left[L(V\left(\lambda \right),\ell ){\otimes}^{F}L(V\left(\mu \right),\ell )\right]}^{\u2022}\cong \chi (X,V),$$ as virtual $\stackrel{\sim}{\mathcal{G}}\text{-modules,}$ where the $\stackrel{\sim}{\mathcal{P}}\text{-module}$ $V\u2254{\left({I}_{\ell}(V\left(\lambda \right)\otimes V\left(\mu \right))\right)}^{\u2022}$ (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 $\U0001d524,$ 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 $$\beta :R\left(G\right)\u27f6{R}_{\ell}\left(G\right)$$ by $\beta {\left(W\right)}^{\u2022}=\chi (X,{\mathcal{W}}^{*}),$ where $\mathcal{W}$ is the homogeneous vector bundle on $X$ associated to the $\stackrel{\sim}{\mathcal{P}}\text{-module}$ ${I}_{\ell}\left(W\right)$ and ${\mathcal{W}}^{*}$ 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 ${\otimes}^{F}\text{.}$