Non-zero admissible representation of $sl_\infty$

Question

Does someone has an example of a non-zero admissible representation of $sl_\infty$ ?

Answer 1

I think I found one.

We consider a vector space $V_m = \mathbb{C}V_0 \oplus \cdot\cdot\cdot \oplus \mathbb{C} V_m$ and the vector space $V^{(\mathbb{Z})}_n$ of $\mathbb{Z}$ indexed sequences of $V_m$ with a finite number of non zero terms.

Then, for $N$ in $\mathbb{N}$ we introduce the basis $(v^i_p)$ where $0 \leq p \leq n$ and $-N \leq i \leq N$ of the subspace of $V^{(\mathbb{Z})}_n$ for which all terms vanish under $-N$ and above $N$. We define the action of the generators on that basis by :

$$X_iv^k_p = (n-p+1)\delta_{ik}v^k_{p-1}$$ $$Y_iv^k_p = (p+1)\delta_{ik}v^k_{p+1}$$ $$H_iv^k_p = (n-2p)\delta_{ik}v^k_p$$

We can see this satisfies $[X_i, Y_j] = \delta_{ij}H_j$ and similarly for $[H_i, X_j]$ and $[H_i, Y_j]$. Thus, this action defines a structure of representation of $sl_\infty$ on $V^{(\mathbb{Z})}_n$.

This representation is admissible and non zero by definitions of $V^{(\mathbb{Z})}_n$ and admissible representations.