Adelic and Rational Grassmannians for finite dimensional algebras
| Autor: | Horozov, Emil, Yakimov, Milen |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 $R[z]$. Comment: 38 pages |
| Databáze: | arXiv |
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