Abstract
In this paper, we construct a Grobner-Shirshov bases for the finite dimensional irreducible module $$V_q(\lambda )$$ of the quantum group $$U_q(F_4)$$ by using the double free module method and the known Grobner-Shirshov bases of $$U_q(F_4).$$ Then, by specializing a suitable version of $$U_q(F_4)$$ at $$q=1,$$ we get a Grobner-Shirshov bases of the universal enveloping algebra $$U(F_4)$$ of the simple Lie algebra of type $$F_4$$ and the finite dimensional irreducible $$U(F_4)-$$ module $$V(\lambda )$$ .
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