Duality for Bethe algebras acting on polynomials in anticommuting variables

Letters in Mathematical Physics. 2020. No. 110. P. 3375–3400.

Tarasov V., Uvarov F.

We consider actions of the current Lie algebras $\mathfrak{gl}_{n}[t]$ and $\mathfrak{gl}_{k}[t]$ on the space of polynomials in $kn$ anticommuting variables. The actions depend on parameters $\bar{z}=(z_{1},\dots ,z_{k})$ and $\bar{\alpha}=(\alpha_{1},\dots ,\alpha_{n})$, respectively. We show that the images of the Bethe algebras $\mathcal{B}^{\langle n\rangle}_{\bar{\alpha}}\subset U(\mathfrak{gl}_{n}[t])$ and $\mathcal{B}^{\langle k\rangle}_{\bar{z}}\subset U(\mathfrak{gl}_{k}[t])$ under these actions coincide. To prove the statement, we use the Bethe ansatz description of eigenvalues of the actions of the Bethe algebras via spaces of quasi-exponentials and establish an explicit correspondence between these spaces for the actions of $\mathcal{B}^{\langle n\rangle}_{\bar{\alpha}}$ and $\mathcal{B}^{\langle k\rangle}_{\bar{k}}$.

Keywords: Bethe subalgebra, (gl(k), gl(n))-duality, Quantum Gaudin model