The Structure of the Algebra of Weak Jacobi Forms for the Root System $F_4$

We prove the polynomiality of the bigraded ring $J_{*,*}^{w, W}(F_4)$ of weak Jacobi forms for the root system $F_4$ which are invariant with respect to the corresponding Weyl group. This work is a continuation of a joint article with V. A. Gritsenko, where the structure of the algebras of weak Jacobi forms related to the root systems of $D_n$ type for $2\leqslant n \leqslant 8$ was studied.