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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2408.04355 |
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| _version_ | 1866911982231224320 |
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| author | Horozov, Emil Yakimov, Milen |
| author_facet | Horozov, Emil Yakimov, Milen |
| contents | 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 $R[z]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_04355 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Adelic and Rational Grassmannians for finite dimensional algebras Horozov, Emil Yakimov, Milen Classical Analysis and ODEs Mathematical Physics Representation Theory Spectral Theory Primary 14M15, Secondary 16S32, 13E10, 37K35 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 $R[z]$. |
| title | Adelic and Rational Grassmannians for finite dimensional algebras |
| topic | Classical Analysis and ODEs Mathematical Physics Representation Theory Spectral Theory Primary 14M15, Secondary 16S32, 13E10, 37K35 |
| url | https://arxiv.org/abs/2408.04355 |