2025/12/10 by Job Daisie Rock, Hugh Thomas, Rock, Job Daisie +1
Mathematics · #Algebraic structures and combinatorial models #Rings, Modules, and Algebras #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2512.09618
We introduce a continuous version of preprojective algebras of type A. In particular, we are interested in the preprojective category over an open, bounded subinterval \mathbbI of ℝ, denoted Λ_\mathbbI. We study the representable projective modules and define a useful type of sub- and quotient module called decorous modules. These are completely described by a function from the closure \mathbbI of \mathbbI to ℝ whose 'slopes' are not too steep anywhere. We later use these to describe permuton ideals, a generalization of the support τ-tilting ideals of preprojective algebras of type An, which we call permutation ideals. Once we have our generalization, we show that permutation ideals can be recovered from permuton ideals. Moreover, permutation ideals are τ-rigid and we show an analogous property for our permuton ideals. Along the way, we classify all the brick Λ_\mathbbI-modules.