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Abstract Given a finite dimensional algebra over a perfect field the text introduces covering functors over the mesh category of any modulated Auslander–Reiten component of the algebra. This is applied to study the composition of irreducible morphisms between indecomposable modules in relation with the powers of the radical of the module category. [ABSTRACT FROM AUTHOR]

We present a novel construction of linear deformations for Lie algebras and use it to prove the non-rigidity of several classes of Lie algebras in different varieties. In particular, we address the problem of k -rigidity for k -step nilpotent Lie algebras and k -solvable Lie algebras. We show that Lie algebras with an abelian factor are not rigid, even for the case of a 1-dimensional abelian factor. This holds in the more restricted case of k -rigidity. We also prove that the k -step free nilpotent Lie algebras are not (k + 1) -rigid, but however they are k -rigid. [ABSTRACT FROM AUTHOR]

This paper is a continued investigation of the structure of Lie algebras in relation to their chief factors, using concepts that are analogous to corresponding ones in group theory. The first section investigates the structure of Lie algebras with a core-free maximal subalgebra. The results obtained are then used in section two to consider the relationship of two chief factors of L being L -connected, a weaker equivalence relation on the set of chief factors than that of being isomorphic as L -modules. A strengthened form of the Jordan–Hölder Theorem in which Frattini chief factors correspond is also established for every Lie algebra. The final section introduces the concept of a crown, a notion introduced in group theory by Gaschütz, and shows that it gives much information about the chief factors. [ABSTRACT FROM AUTHOR]