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The idea of using a sequence of finite dimensional algebras to approach aquantum linear group (i.e., a quantum $\mathfrak{gl}_n$) was first introducedby Beilinson-Lusztig-MacPherson [BLM]. In their work, the algebras areconvolution algebras of some finite partial flag varieties whose certainstructure constants relative to the orbital basis satisfy a stabilizationproperty. This property leads to the definition of an infinite dimensionalidempotented algebra. Finally, taking a limit process yields a new realizationfor 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 thegeometric approach and [DDF, DF] for the algebraic approach and a newrealization) 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]. Thispaper extends the algebraic approach to the quantum queer supergroup$U_{v}(\mathfrak{q}_n)$ via finite dimensional queer $q$-Schur superalgebras.

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