Adelic and Rational Grassmannians for finite dimensional algebras

TL;DR

This paper develops a comprehensive theory connecting adelic and rational Grassmannians for finite dimensional complex algebras, linking algebraic structures with geometric and analytical classifications of transformations.

Contribution

It introduces new descriptions of adelic Grassmannians in terms of algebraic modules and classifies bispectral Darboux transformations, establishing a canonical embedding into rational Grassmannians.

Findings

Classifies bispectral Darboux transformations of R-valued exponential functions.

Provides algebraic descriptions of adelic Grassmannians via projective modules.

Establishes an embedding of adelic into rational Grassmannians using a perfect pairing.

Abstract

We develop a theory of Wilson's adelic Grassmannian ${\mathrm{Gr}}^{\mathrm{ad}}(R)$ and Segal-Wilson's rational Grasssmannian ${\mathrm{Gr}}^ {\mathrm{rat}}(R)$ associated to an arbitrary finite dimensional complex algebra $R$. We provide several equivalent descriptions of the former in terms of the indecomposable projective modules of $R$ and its primitive idempotents, and prove that it classifies the bispectral Darboux transformations of the $R$-valued exponential function. The rational Grasssmannian $ {\mathrm{Gr}}^{\mathrm{rat}}(R)$ is defined by using certain free submodules of $R(z)$ and it is proved that it can be alternatively defined via Wilson type conditions imposed in a representation theoretic settings. A canonical embedding ${\mathrm{Gr}}^{\mathrm{ad}}(R) \hookrightarrow {\mathrm{Gr}}^{\mathrm{rat}}(R)$ is constructed based on a perfect pairing between the $R$-bimodule of quasiexponentials with values in $R$ and the $R$-bimodule $R[z]$.