The quantum group $$U_q^+(F_4)$$ as a quadratic-linear algebra

Abstract Let $$U_q^+(\mathfrak {g})$$ U q + ( g ) be the multiparameter quantum Borel algebra, where $$\mathfrak {g}$$ g is a simple Lie algebra. The purpose of this paper is to prove that, in the case that $$\mathfrak {g}$$ g is a simple Lie algebra of type $$F_4$$ F 4 , the algebra $$U_q^+(\mathfrak {g})$$ U q + ( g ) can be constructed as a quadratic-linear algebra, but it is not a quadratic-linear Koszul algebra.