Shuffle algebra realization of quantum affine superalgebra \(U_v( \widehat{\mathfrak{D}}(2, 1; \theta))\) (Q2659132)

Feigin-Odesskii shuffle algebras, first studied by \textit{B. Feigin} and \textit{A. Odesskii} in their series of papers including [Int. Math. Res. Not. 1997, No. 11, 531--539 (1997; Zbl 0923.16027)], are certain (skew)symmetric Laurent polynomials satisfying the so-called wheel conditions on the poles, equipped with an associative algebra structure by shuffle product. Currently they are attracting more and more attention with the expectation to give realizations of quantum affine and toroidal (super)algebras. The paper under review gives shuffle algebra realization of positive part of quantum affine superalgebra corresponding to the Lie superalgebra \(\mathfrak{D}(2,1;\theta)\). The realization is inspired by the works of \textit{A. Tsymbaliuk} including [Lett. Math. Phys. 110, No. 8, 2083--2111 (2020; Zbl 1464.17021)], and one of the main ideas is to use skew-symmetric Laurent polynomials instead of symmetric ones. The paper also gives the shuffle algebra realization of the positive part of \(U_v(\widehat{\mathfrak{sl}}(2 | 1))\) with odd root system when \(v\) is a primitive root of unity of even order, generalizing the results in [\textit{B. Feigin} et al., Int. Math. Res. Not. 2003, No. 18, 999--1014 (2003; Zbl 1090.05071)].