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 $U_{ v}(A(0,2)^{ (4)})^{ +}$.
Categories: math.QA