Study on the Group-Theoretic and Ring-Theoretic Properties of the Centroid of a Class of Filiform Lie Algebras

This paper investigates the centroid structure and algebraic properties of a class of n-dimensional filiform Lie algebras Ln. The structural characteristics of the group G and the ring R formed by centroids are analyzed. The invertible linear transformations of the centroid form a mixed group G, and it is proved that G can be decomposed into the internal direct product of two commuting subgroups G1 and G2, where G2 is a torsion-free group. The centroid ring R is a commutative ring with a unit element and a zero-divisor if and only if the element on the main diagonal is zero. By constructing a ring homomorphism Ψ:R⟶R1, it is proved that ker Ψ is a maximal ideal of R, and all its nonzero elements are zero-divisors.