On the combinatorial rank of quantum groups of type $F_4$

In this paper, we calculate the combinatorial rank of the positive part $u_q^{+}(𝔤)$ of the multiparameter version of the small Lusztig quantum group, where $𝔤$ is a simple Lie algebra of type $F_4$. For this purpose, we also present the basis and relations of the quantum algebra, calculate the coproduct of its PBW-generators and determine all skew-primitive elements of this quantum group. In conclusion, we obtain that the combinatorial rank of $u_q^{+}(F_4)$ is 4.