Divisible Non-injective Modules over Leavitt Path Algebras

My research concerns the representation theory of Leavitt path algebras associated with finite directed graphs. In particular, my thesis focuses on the construction of modules that are divisible but not injective, a problem related to the gap between these two notions in module theory. The work studies modules generated by sinks in finite graphs containing source cycles, where the combinatorial structure of the graph plays a crucial role. The injective envelopes of these modules can be described in terms of formal series in paths ending at the sink, and this perspective motivates the main construction developed in the thesis. Inspired by the theory of formal power series and by the generating function of the Catalan numbers, I identify an explicit obstruction to injectivity arising from the impossibility of extending certain homomorphisms from non finitely generated ideals. This provides concrete examples of divisible non injective modules in the setting of Leavitt path algebras.