The $i$-quantum group $\boldsymbol U^\imath(n)$

Jie Du and Yadi Wu

Vol. 320 (2022), No. 1, 61–101

DOI: 10.2140/pjm.2022.320.61

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.

Keywords

quantum linear group, $q$-Schur algebra, $\imath$-quantum group, quantum Schur–Weyl duality, finite symplectic group

Mathematical Subject Classification

Primary: 16T20, 17B37, 20C08, 20C33, 20G43

Milestones

Received: 16 November 2021

Revised: 16 July 2022

Accepted: 30 July 2022

Published: 16 October 2022