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

Source abstract

This paper investigates the centroid structure and algebraic properties of a class of n ‐dimensional filiform Lie algebras L n . 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 G 1 and G 2 , where G 2 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 ⟶ R 1 , it is proved that ker Ψ is a maximal ideal of R , and all its nonzero elements are zero‐divisors.