PBWD bases and shuffle algebra realizations for \(U_v(L\mathfrak{sl}_n)\), \(U_{v_1,v_2} (L\mathfrak{sl}_n)\), \(U_v (L\mathfrak{sl}(m|n))\) and their integral forms (Q2036284)

The quantum loop algebras associated to a simple finite dimensional Lie algebra $\mathfrak{g}$ admit two presentations: the Drinfeld-Jimbo realization $U^{DJ}_v(L\mathfrak{g})$ and the Drinfeld realization $U_v(L\mathfrak{g})$ (cf. [\textit{V. G. Drinfel'd}, Sov. Math., Dokl. 36, No. 2, 212--216 (1988; Zbl 0667.16004); translation from Dokl. Akad. Nauk SSSR 269, 13--17 (1987)]). The central aim of the paper under review is constructing a family of PBWD (Poincaré-Birkhoff-Witt-Drinfeld) bases for the quantum loop algebras $U_v(L\mathfrak{sl}_n)$, $U_{v_1,v_2}(L\mathfrak{sl}n)$, and $U_v(L\mathfrak{sl}(m|n))$, in the Drienfeld realization. Moreover, the author provides a family of PBWD bases for certain integral forms $\mathfrak{U}_v(L\mathfrak{sl}_n)$, $\mathfrak{U}_{v_1,v_2} (L\mathfrak{sl}_n)$, $\mathfrak{U}_v(L\mathfrak{sl}(m|n))$, defined over $\mathbb{C}[v, v^{-1}]$ and $\mathbb{C}[v_1, v_2, v_1^{-1}, v_2^{-1}]$, respectively.