Isomorphism between the R-matrix and Drinfeld presentations of quantum affine superalgebra for type $\mathbf{A}$

Abstract

In this paper, we establish an explicit isomorphism between Drinfeld and R-matrix presentations of the quantum affine superalgebra associated with the general linear Lie superalgebra $\mathfrak{gl}(m|n)$. This result can be viewed as a supersymmetric analogue of the isomorphism between two presentations of the quantum affine algebra $U_q(\widehat{\mathfrak{gl}(n)})$ proposed by Frenkel-Ding. Furthermore, employing the quantum Berezinian of $U_q(\widehat{\mathfrak{gl}(m|n))}$, we construct the R-matrix presentation of the quantum affine superalgebra $U_q(\widehat{\mathfrak{sl}(m|n))}$ and prove the isomorphism between its R-matrix and Drinfeld presentations. Our proof based on the commutative relation between Gaussian generators, $\mathbb{A}$-form and quantum Berezinian.

arXiv comment29 pages