The subalgebra of graded central polynomials of an associative algebra

Let $F$ be a field and let $F⟨X⟩$ be the free unital associative $F$-algebra on the free generating set $X=\{x_1,x_2,\dots \}$. A subalgebra (a vector subspace) $V$ in $F⟨X⟩$ is called a $T$-subalgebra (a $T$-subspace) if $\varphi (V)\subseteq V$ for all endomorphisms $\varphi$ of $F⟨X⟩$. For an algebra $G$, its central polynomials form a $T$-subalgebra $C(G)$ in $F⟨X⟩$. Over a field of characteristic $p>2$ there are algebras $G$ whose algebras of all central polynomials $C(G)$ are not finitely generated as $T$-subspaces in $F \langle X \rangle$\mathrm{.}However,noexampleofanalgebra$G$suchthat$C(G)$ is not finitely generated as a $T$-subalgebra is known yet. In the present paper we construct the first example of a $2$-graded unital associative algebra $B$ over a field of characteristic $p>2$ whose algebra $C_2 (B)$ofall$2$-gradedcentralpolynomialsisnotfinitelygeneratedasa$T_2$-subalgebrainthefree$2$-gradedunitalassociative$F$-algebra$F \langle Y,Z \rangle$\mathrm{.}Here$Y = { y_1, y_2, \dots }$and$Z = { z_1, z_2, \dots }$aresetsofevenandoddfreegeneratorsof$F \langle Y,Z \rangle$, respectively. We hope that our example will help to construct an algebra $G$ whose algebra $C(G)$ of (ordinary) central polynomials is not finitely generated as a $T$-subalgebra in $F⟨X⟩$.