The subalgebra of graded central polynomials of an associative algebra

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Let F be a field and let F X be the free unital associative F-algebra on the free generating set X = { x₁, x₂, }. A subalgebra (a vector subspace) V in F X is called a T-subalgebra (a T-subspace) if ϕ(V) V for all endomorphisms ϕ 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 X. However, no example of an algebra G such that 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 C₂ (B) of all 2-graded central polynomials is not finitely generated as a T₂-subalgebra in the free 2-graded unital associative F-algebra F Y,Z. Here Y = { y₁, y₂, } and Z = { z₁, z₂, } are sets of even and odd free generators of F Y,Z, 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.