2024/01/16 by de França, Antonio
#16W10 #16W22 #17B30 #FOS: Mathematics #Primary 16R10 #Rings and Algebras (math.RA) #Secondary 16W50
paper · doi:10.48550/arxiv.2401.08074
This paper is devoted to the study of graded associative algebras that satisfy a graded polynomial identity of degree 2. % Let G be a finite abelian group, \mathbbF a field of characteristic zero and \mathfrakA a G-graded \mathbbF-algebra. % We prove that, for \mathbbF algebraically closed, if \mathfrakAe satisfies a polynomial identity g=g(x1(e), …, xn(e))∈\mathbbF⟨ XG ⟩ of degree 2, then \mathfrakA is either nilpotent or has commutative neutral component, % and we ensure that the G-graded variety \mathfrakWG determined by g is equal to either varG([x(e),y(e)]) or varG(N) for some nilpotent G-graded algebra N. % Posteriorly, we investigate the implications of \mathfrakAe being central in \mathfrakA. The results obtained allow us to prove that, when G is finite cyclic, if \mathfrakA is finitely generated and \mathfrakAe is central in \mathfrakA, then the commutator ideal of \mathfrakA is nilpotent, and the algebra \mathfrakA(-)=(\mathfrakA,[ , ]) is a solvable Lie algebra, % and, if G has odd order, then [x1,x2][x3,x4]⋯[x2d-1,x2d]≡0 in \mathfrakA, for some d∈ℕ.