Hessenberg varieties, Slodowy slices, and integrable systems
Abstract This work is intended to contextualize and enhance certain well-studied relationships between Hessenberg varieties and the Toda lattice, thereby building on the results of Kostant, Peterson, and others. One such relationship is the fact that every Lagrangian leaf in the Toda lattice is comp...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Abe, Hiraku [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2019 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer-Verlag GmbH Germany, part of Springer Nature 2019 |
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| Übergeordnetes Werk: |
Enthalten in: Mathematische Zeitschrift - Springer Berlin Heidelberg, 1918, 291(2019), 3-4 vom: 21. Jan., Seite 1093-1132 |
|---|---|
| Übergeordnetes Werk: |
volume:291 ; year:2019 ; number:3-4 ; day:21 ; month:01 ; pages:1093-1132 |
| Links: |
|---|
| DOI / URN: |
10.1007/s00209-019-02235-7 |
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OLC2039747796 |
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| 520 | |a Abstract This work is intended to contextualize and enhance certain well-studied relationships between Hessenberg varieties and the Toda lattice, thereby building on the results of Kostant, Peterson, and others. One such relationship is the fact that every Lagrangian leaf in the Toda lattice is compactified by a suitable choice of Hessenberg variety. It is then natural to imagine the Toda lattice as extending to an appropriate union of Hessenberg varieties. We fix a simply-connected complex semisimple linear algebraic group G and restrict our attention to a particular family of Hessenberg varieties, a family that includes the Peterson variety and all Toda leaf compactifications. The total space of this family, $$X(H_0)$$, is shown to be a Poisson variety with a completely integrable system defined in terms of Mishchenko–Fomenko polynomials. This leads to a natural embedding of completely integrable systems from the Toda lattice to $$X(H_0)$$. We also show $$X(H_0)$$ to have an open dense symplectic leaf isomorphic to $$G/Z \times S_{\text {reg}}$$, where Z is the centre of G and $$S_{\text {reg}}$$ is a regular Slodowy slice in the Lie algebra of G. This allows us to invoke results about integrable systems on $$G\times S_{\text {reg}}$$, as developed by Rayan and the second author. Lastly, we witness some implications of our work for the geometry of regular Hessenberg varieties. | ||
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10.1007/s00209-019-02235-7 doi (DE-627)OLC2039747796 (DE-He213)s00209-019-02235-7-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn Abe, Hiraku verfasserin aut Hessenberg varieties, Slodowy slices, and integrable systems 2019 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany, part of Springer Nature 2019 Abstract This work is intended to contextualize and enhance certain well-studied relationships between Hessenberg varieties and the Toda lattice, thereby building on the results of Kostant, Peterson, and others. One such relationship is the fact that every Lagrangian leaf in the Toda lattice is compactified by a suitable choice of Hessenberg variety. It is then natural to imagine the Toda lattice as extending to an appropriate union of Hessenberg varieties. We fix a simply-connected complex semisimple linear algebraic group G and restrict our attention to a particular family of Hessenberg varieties, a family that includes the Peterson variety and all Toda leaf compactifications. The total space of this family, $$X(H_0)$$, is shown to be a Poisson variety with a completely integrable system defined in terms of Mishchenko–Fomenko polynomials. This leads to a natural embedding of completely integrable systems from the Toda lattice to $$X(H_0)$$. We also show $$X(H_0)$$ to have an open dense symplectic leaf isomorphic to $$G/Z \times S_{\text {reg}}$$, where Z is the centre of G and $$S_{\text {reg}}$$ is a regular Slodowy slice in the Lie algebra of G. This allows us to invoke results about integrable systems on $$G\times S_{\text {reg}}$$, as developed by Rayan and the second author. Lastly, we witness some implications of our work for the geometry of regular Hessenberg varieties. Hessenberg variety Integrable system Slodowy slice Toda lattice Crooks, Peter aut Enthalten in Mathematische Zeitschrift Springer Berlin Heidelberg, 1918 291(2019), 3-4 vom: 21. Jan., Seite 1093-1132 (DE-627)129474193 (DE-600)203014-7 (DE-576)014852047 0025-5874 nnns volume:291 year:2019 number:3-4 day:21 month:01 pages:1093-1132 https://doi.org/10.1007/s00209-019-02235-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2007 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_4027 GBV_ILN_4126 GBV_ILN_4277 GBV_ILN_4306 GBV_ILN_4320 AR 291 2019 3-4 21 01 1093-1132 |
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10.1007/s00209-019-02235-7 doi (DE-627)OLC2039747796 (DE-He213)s00209-019-02235-7-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn Abe, Hiraku verfasserin aut Hessenberg varieties, Slodowy slices, and integrable systems 2019 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany, part of Springer Nature 2019 Abstract This work is intended to contextualize and enhance certain well-studied relationships between Hessenberg varieties and the Toda lattice, thereby building on the results of Kostant, Peterson, and others. One such relationship is the fact that every Lagrangian leaf in the Toda lattice is compactified by a suitable choice of Hessenberg variety. It is then natural to imagine the Toda lattice as extending to an appropriate union of Hessenberg varieties. We fix a simply-connected complex semisimple linear algebraic group G and restrict our attention to a particular family of Hessenberg varieties, a family that includes the Peterson variety and all Toda leaf compactifications. The total space of this family, $$X(H_0)$$, is shown to be a Poisson variety with a completely integrable system defined in terms of Mishchenko–Fomenko polynomials. This leads to a natural embedding of completely integrable systems from the Toda lattice to $$X(H_0)$$. We also show $$X(H_0)$$ to have an open dense symplectic leaf isomorphic to $$G/Z \times S_{\text {reg}}$$, where Z is the centre of G and $$S_{\text {reg}}$$ is a regular Slodowy slice in the Lie algebra of G. This allows us to invoke results about integrable systems on $$G\times S_{\text {reg}}$$, as developed by Rayan and the second author. Lastly, we witness some implications of our work for the geometry of regular Hessenberg varieties. Hessenberg variety Integrable system Slodowy slice Toda lattice Crooks, Peter aut Enthalten in Mathematische Zeitschrift Springer Berlin Heidelberg, 1918 291(2019), 3-4 vom: 21. Jan., Seite 1093-1132 (DE-627)129474193 (DE-600)203014-7 (DE-576)014852047 0025-5874 nnns volume:291 year:2019 number:3-4 day:21 month:01 pages:1093-1132 https://doi.org/10.1007/s00209-019-02235-7 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_2007 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_4027 GBV_ILN_4126 GBV_ILN_4277 GBV_ILN_4306 GBV_ILN_4320 AR 291 2019 3-4 21 01 1093-1132 |
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Hessenberg varieties, Slodowy slices, and integrable systems |
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Hessenberg varieties, Slodowy slices, and integrable systems |
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Abe, Hiraku |
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Abe, Hiraku Crooks, Peter |
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hessenberg varieties, slodowy slices, and integrable systems |
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Hessenberg varieties, Slodowy slices, and integrable systems |
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Abstract This work is intended to contextualize and enhance certain well-studied relationships between Hessenberg varieties and the Toda lattice, thereby building on the results of Kostant, Peterson, and others. One such relationship is the fact that every Lagrangian leaf in the Toda lattice is compactified by a suitable choice of Hessenberg variety. It is then natural to imagine the Toda lattice as extending to an appropriate union of Hessenberg varieties. We fix a simply-connected complex semisimple linear algebraic group G and restrict our attention to a particular family of Hessenberg varieties, a family that includes the Peterson variety and all Toda leaf compactifications. The total space of this family, $$X(H_0)$$, is shown to be a Poisson variety with a completely integrable system defined in terms of Mishchenko–Fomenko polynomials. This leads to a natural embedding of completely integrable systems from the Toda lattice to $$X(H_0)$$. We also show $$X(H_0)$$ to have an open dense symplectic leaf isomorphic to $$G/Z \times S_{\text {reg}}$$, where Z is the centre of G and $$S_{\text {reg}}$$ is a regular Slodowy slice in the Lie algebra of G. This allows us to invoke results about integrable systems on $$G\times S_{\text {reg}}$$, as developed by Rayan and the second author. Lastly, we witness some implications of our work for the geometry of regular Hessenberg varieties. © Springer-Verlag GmbH Germany, part of Springer Nature 2019 |
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Abstract This work is intended to contextualize and enhance certain well-studied relationships between Hessenberg varieties and the Toda lattice, thereby building on the results of Kostant, Peterson, and others. One such relationship is the fact that every Lagrangian leaf in the Toda lattice is compactified by a suitable choice of Hessenberg variety. It is then natural to imagine the Toda lattice as extending to an appropriate union of Hessenberg varieties. We fix a simply-connected complex semisimple linear algebraic group G and restrict our attention to a particular family of Hessenberg varieties, a family that includes the Peterson variety and all Toda leaf compactifications. The total space of this family, $$X(H_0)$$, is shown to be a Poisson variety with a completely integrable system defined in terms of Mishchenko–Fomenko polynomials. This leads to a natural embedding of completely integrable systems from the Toda lattice to $$X(H_0)$$. We also show $$X(H_0)$$ to have an open dense symplectic leaf isomorphic to $$G/Z \times S_{\text {reg}}$$, where Z is the centre of G and $$S_{\text {reg}}$$ is a regular Slodowy slice in the Lie algebra of G. This allows us to invoke results about integrable systems on $$G\times S_{\text {reg}}$$, as developed by Rayan and the second author. Lastly, we witness some implications of our work for the geometry of regular Hessenberg varieties. © Springer-Verlag GmbH Germany, part of Springer Nature 2019 |
| abstract_unstemmed |
Abstract This work is intended to contextualize and enhance certain well-studied relationships between Hessenberg varieties and the Toda lattice, thereby building on the results of Kostant, Peterson, and others. One such relationship is the fact that every Lagrangian leaf in the Toda lattice is compactified by a suitable choice of Hessenberg variety. It is then natural to imagine the Toda lattice as extending to an appropriate union of Hessenberg varieties. We fix a simply-connected complex semisimple linear algebraic group G and restrict our attention to a particular family of Hessenberg varieties, a family that includes the Peterson variety and all Toda leaf compactifications. The total space of this family, $$X(H_0)$$, is shown to be a Poisson variety with a completely integrable system defined in terms of Mishchenko–Fomenko polynomials. This leads to a natural embedding of completely integrable systems from the Toda lattice to $$X(H_0)$$. We also show $$X(H_0)$$ to have an open dense symplectic leaf isomorphic to $$G/Z \times S_{\text {reg}}$$, where Z is the centre of G and $$S_{\text {reg}}$$ is a regular Slodowy slice in the Lie algebra of G. This allows us to invoke results about integrable systems on $$G\times S_{\text {reg}}$$, as developed by Rayan and the second author. Lastly, we witness some implications of our work for the geometry of regular Hessenberg varieties. © Springer-Verlag GmbH Germany, part of Springer Nature 2019 |
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