The subalgebra of graded central polynomials of an associative algebra (1409.7937v1)

Abstract: Let $F$ be a field and let $F \langle X \rangle$ 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 $F \langle X \rangle$ is called a $T$-subalgebra (a $T$-subspace) if $\phi (V) \subseteq V$ for all endomorphisms $\phi$ of $F \langle X \rangle$. For an algebra $G$, its central polynomials form a $T$-subalgebra $C(G)$ in $F \langle X \rangle$. 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 $F \langle X \rangle$. 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 $C_2 (B)$ of all $2$-graded central polynomials is not finitely generated as a $T_2$-subalgebra in the free $2$-graded unital associative $F$-algebra $F \langle Y,Z \rangle$. Here $Y = { y_1, y_2, \dots }$ and $Z = { z_1, z_2, \dots }$ are sets of even and odd free generators of $F \langle Y,Z \rangle$, 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 $F \langle X \rangle$.