Approaching quantum queer supergroups using finite dimensional superalgebras (Preliminary version)

TL;DR

This paper extends the algebraic approach of finite dimensional algebras to construct quantum queer supergroups, building on previous work with quantum linear and super groups using convolution algebras.

Contribution

It introduces a new algebraic framework for quantum queer supergroups using finite dimensional queer q-Schur superalgebras, expanding the geometric and algebraic methods.

Findings

Constructs finite dimensional queer q-Schur superalgebras.

Provides a new realization of quantum queer supergroups.

Extends algebraic approaches to supergroup quantum structures.

Abstract

The idea of using a sequence of finite dimensional algebras to approach a quantum linear group (i.e., a quantum $\mathfrak{gl}_n$) was first introduced by Beilinson-Lusztig-MacPherson [BLM]. In their work, the algebras are convolution algebras of some finite partial flag varieties whose certain structure constants relative to the orbital basis satisfy a stabilization property. This property leads to the definition of an infinite dimensional idempotented algebra. Finally, taking a limit process yields a new realization for the quantum $\mathfrak{gl}_n$. Since then, this work has been modified [DF2] and generalized to quantum affine $\mathfrak{gl}_n$ (see [GV, L] for the geometric approach and [DDF, DF] for the algebraic approach and a new realization) and quantum super $\mathfrak{gl}_{m|n}$ [DG], and, more recently, to convolution algebras arising from type $B/C$ geometry and $i$-quantum groups $\boldsymbol U^\jmath$ and $\boldsymbol U^\imath$; see [BKLW, DWu1, DWu2]. This paper extends the algebraic approach to the quantum queer supergroup $U_{v}(\mathfrak{q}_n)$ via finite dimensional queer $q$-Schur superalgebras.