Abstract

This paper reveals some new structural property for the $i$-quantum group $U^{ı}(n)$ and constructs a certain hyperalgebra from the new structure which has connections to finite symplectic groups at the modular representation level. This work is built on certain finite dimensional $\mathbb{Q}(\upsilon )$-algebras ${\mathcal{𝒮}}^{ı}(n,r)$ whose integral form ${\mathcal{𝒮}}_{\mathcal{𝒵}}^{ı}(n,r)$ is investigated as a convolution algebra arising from the geometry of type $C$ in Bao et al. (2018). Similar to the approach in Lai and Luo 2021 or Luo and Wang 2022, we investigate ${\mathcal{𝒮}}_{\mathcal{𝒵}}^{ı}(n,r)$ as an endomorphism algebra of a certain $q$-permutation module over the Hecke algebra of type $C_r$ and interpret the convolution product as a composition of module homomorphisms. We then prove that the action of $U^{ı}(n)$ on the $r$-fold tensor space of the natural representation of $U({\mathfrak{𝔤}\mathfrak{𝔩}}_{2n})$ (via an embedding $U^{ı}(n)\hookrightarrow U({\mathfrak{𝔤}\mathfrak{𝔩}}_{2n})$) coincides with an action given by multiplications in ${\mathcal{𝒮}}^{ı}(n,r)$. In this way, we reestablish the surjective homomorphism from $U^{ı}(n)$ to ${\mathcal{𝒮}}^{ı}(n,r)$ due to Bao and Wang (2018). We then embed $U^{ı}(n)$ into the direct product of ${\mathcal{𝒮}}^{ı}(n,r)$ and completely determine its image. This gives a new realisation for $U^{ı}(n)$ and, as an application, the aforementioned hyperalgebra is an easy consequence of this new construction.

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