A generalization of dual symmetry and reciprocity for symmetric algebras

TL;DR

This paper generalizes the Landrock lemma, extending dual symmetry and reciprocity concepts from symmetric algebras to arbitrary finite dimensional algebras, relating radical and socle layers via the Nakayama functor.

Contribution

It introduces a generalized theorem that relates radical and socle layers in a broader class of algebras using new functorial tools.

Findings

Generalization of Landrock lemma to all finite dimensional algebras.

Introduction of socle and capital functors as key tools.

Establishment of relations between radical layers, duals, and socle layers.

Abstract

Slicing a module into semisimple ones is useful to study modules. Loewy structures provide a means of doing so. To establish the Loewy structures of projective modules over a finite dimensional symmetric algebra over a field $F$, the Landrock lemma is a primary tool. The lemma and its corollary relate radical layers of projective indecomposable modules to radical layers of the $F$-duals of those modules ("dual symmetry") and to socle layers of those modules ("reciprocity"). We generalize these results to an arbitrary finite dimensional algebra $A$. Our main theorem, which is the same as the Landrock lemma for finite dimensional symmetric algebras, relates radical layers of projective indecomposable modules $P$ to radical layers of the $A$-duals of those modules and to socle layers of injective indecomposable modules $\nu P$ where $\nu({-})$ is the Nakayama functor. A key tool to prove the main theorem is a pair of adjoint functors, which we call socle functors and capital functors.