The subalgebra of graded central polynomials of an associative algebra

Let F   be a field and let F〈X〉F〈X〉 be the free unital associative F  -algebra on the free generating set X={x1,x2,…}X={x1,x2,…}. A subalgebra (a vector subspace) V   in F〈X〉F〈X〉 is called a T-subalgebra (a T-subspace  ) if ϕ(V)⊆Vϕ(V)⊆V for all endomorphisms ϕ   of F〈X〉F〈X〉. For an algebra G, its central polynomials form a T  -subalgebra C(G)C(G) in F〈X〉F〈X〉. Over a field of characteristic p>2p>2 there are algebras G   whose algebras of all central polynomials C(G)C(G) are not finitely generated as T-subspaces   in F〈X〉F〈X〉. However, no example of an algebra G   such that C(G)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>2p>2 whose algebra C2(B)C2(B) of all 2-graded central polynomials is not finitely generated as a T2T2-subalgebra in the free 2-graded unital associative F  -algebra F〈Y,Z〉F〈Y,Z〉. We hope that our example will help to construct an algebra G   whose algebra C(G)C(G) of (ordinary) central polynomials is not finitely generated as a T  -subalgebra in F〈X〉F〈X〉.