On the structure of quantum affine superalgebra $U_{v}(A(0,2)^{(4)})$

TL;DR

This paper investigates the positive part of a specific quantum affine superalgebra, proving its isomorphism to a Nichols algebra, and determines its root multiplicities and PBW basis.

Contribution

It establishes the isomorphism between the algebra defined by quantum Serre relations and a Nichols algebra, and explicitly computes root multiplicities and a PBW basis.

Findings

Proves $U_{v}(A(0,2)^{(4)})^{+}$ is isomorphic to a Nichols algebra.

Determines all root multiplicities.

Provides a PBW basis for the algebra.

Abstract

We research $U_{v}(A(0,2)^{(4)})^{+}$ defined by quantum Serre relations, when $v$ is not a root of unity. We prove that $U_{v}(A(0,2)^{(4)})^{+}$ is isomorphic to a Nichols algebra. In other words, it is equivalent to define $U_{v}(A(0,2)^{(4)})^{+}$ by quantum Serre relations and by the radical of the bilinear form. We determine all the root multiplicities and give a PBW basis of $U_{v}(A(0,2)^{(4)})^{+}$.