The subalgebra of graded central polynomials of an associative algebra

JOURNAL OF ALGEBRA

卷:425

Article

Deryabina, Galina¹  Krasilnikov, Alexei² 

[1] Bauman Moscow State Tech Univ, Dept Computat Math & Math Phys, Moscow 105005, Russia

[2] Univ Brasilia, Dept Matemat, BR-70910900 Brasilia, DF, Brazil

关键词: Free associative algebra;    Polynomial identity;    Central polynomial;    Graded associative algebra;   

DOI  :  10.1016/j.jalgebra.2014.11.022

来源: Elsevier

【 摘 要 】

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 sub-algebra (a vector subspace) V in F < X > is called a T-subalgebra (a T-subspace) if phi(V) subset of V for all endomorphisms phi 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 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 < Y, Z >. 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 >. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free