Lowest positive almost central elements of $U_q(s{l}^{(1)}(n|n))$ $(n\ge 2)$, $U_q(s{l}^{(2)}(2n|2n))$ $(n\ge 2)$ and $U_q(s{l}^{(4)}(2n+1|2n+1))$ $(n\ge 1)$ and their multi-parameter quantum affine superalgebras

Abstract

Let $\pi :sl(n|n)\to A(n-1,n-1)$ be the natural epimorphism of Lie superalgebra. Then $\dim \ker \pi =1$. Let ${\pi }^{(t)}:s{l}^{(t)}(n|n)\to A^{(t)}(n-1,n-1)$ be the natural epimorphism, where $t=1,2,4$. Let $\{e_k|k\in \mathbb{Z}\}$ be the basis of $\ker {\pi }^{(t)}$ with $e_k\in s{l}^{(t)}(n|n{)}_{(a_{t}k+b_t)\delta }$, where $(a_1,b_1)=(1,0)$, $(a_2,b_2)=(2,-1)$ and $(a_4,b_4)=(4,-2)$. The main result of this paper is to explicitly describe an element of $U_q(s{l}^{(t)}(n|n))$ (and its multi-parameter version) corresponding to $e_1$ (i.e., $k=1$). As for $U_q(s{l}^{(1)}(n|n))$ (i.e., $t=1$), the author had already had explicit description for every $k$ in 1999.

arXiv comment13 pages, 3 figures, submitted to Proceedings of Meeting for Study of Number theory, Hopf algebras and related topics, 2017 February 12-15, the room B121 of Faculty of Science, University of Toyama, Japan