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

TL;DR

This paper explicitly describes the lowest positive almost central elements in certain quantum affine superalgebras and their multi-parameter versions, extending previous work for the case t=1 to t=2 and t=4.

Contribution

It provides explicit formulas for elements corresponding to the kernel basis in quantum affine superalgebras for t=2 and t=4, generalizing earlier results for t=1.

Findings

Explicit description of elements in $U_q(sl^{(t)}(n|n))$ for t=2,4

Extension of previous t=1 results to multi-parameter cases

Identification of lowest positive almost central elements

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 $\pi^{(t)}:sl^{(t)}(n|n)\to A^{(t)}(n-1,n-1)$ be the natural epimorphism, where $t=1,2,4$. Let $\{e_k|k\in{\mathbb{Z}}\}$ be the basis of $\ker\pi^{(t)}$ with $e_k\in sl^{(t)}(n|n)_{(a_tk+b_t)\delta}$, 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 $U_q(sl^{(t)}(n|n))$ (and its multi-parameter version) corresponding to $e_1$ (i.e., $k=1$). As for $U_q(sl^{(1)}(n|n))$ (i.e., $t=1$), the author had already had explicit description for every $k$ in 1999.