The double super Yangians in type A for arbitrary $0^m 1^n$-sequences and their bosonic representations

In this paper, we introduce the double super Yangian $\mathrm{DY}_{h}(\mathfrak{gl}_{m|n}^{\mathfrak{s}})$ and $\mathrm{DY}_{h}(\mathfrak{sl}_{m|n}^{\mathfrak{s}})$ associated with any fixed $0^{m}1^{n}$--sequence $\mathfrak{s}$. First, we establish an explicit isomorphism between the Drinfeld and R--matrix presentations of $\mathrm{DY}_{h}(\mathfrak{gl}^{\mathfrak{s}}_{m|n})$. We then generalize the notion of the quantum Berezinian to $\mathrm{DY}_{h}(\mathfrak{gl}_{m|n}^{\mathfrak{s}})$, and employ it to construct the R--matrix presentation of $\mathrm{DY}_{h}(\mathfrak{sl}_{m|n}^{\mathfrak{s}})$ and prove that it is isomorphic to the Drinfeld presentation. As an application, we present level--1 bosonic representations for $\mathrm{DY}_{h}(\mathfrak{gl}_{m|n}^{\mathfrak{s}})$ and $\mathrm{DY}_{h}(\mathfrak{sl}_{m|n}^{\mathfrak{s}})$ in terms of their Drinfeld current generators.