Symplectic groups over noncommutative algebras
We introduce the symplectic group $${{\,\mathrm{Sp}\,}}_2(A,\sigma )$$over a noncommutative algebra A with an anti-involution $$\sigma $$. We realize several classical Lie groups as $${{\,\mathrm{Sp}\,}}_2$$over various noncommutative algebras, which provides new insights into their structure theory...
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| Main Authors: | , , , , |
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| Format: | Article (Journal) |
| Language: | English |
| Published: |
12 September 2022
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| In: |
Selecta mathematica
Year: 2022, Volume: 28, Issue: 4, Pages: 1-119 |
| ISSN: | 1420-9020 |
| DOI: | 10.1007/s00029-022-00787-x |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1007/s00029-022-00787-x
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| Author Notes: | Daniele Alessandrini, Arkady Berenstein, Vladimir Retakh, Eugen Rogozinnikov, Anna Wienhard |
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| 520 | |a We introduce the symplectic group $${{\,\mathrm{Sp}\,}}_2(A,\sigma )$$over a noncommutative algebra A with an anti-involution $$\sigma $$. We realize several classical Lie groups as $${{\,\mathrm{Sp}\,}}_2$$over various noncommutative algebras, which provides new insights into their structure theory. We construct several geometric spaces, on which the groups $${{\,\mathrm{Sp}\,}}_2(A,\sigma )$$act. We introduce the space of isotropic A-lines, which generalizes the projective line. We describe the action of $${{\,\mathrm{Sp}\,}}_2(A,\sigma )$$on isotropic A-lines, generalize the Kashiwara-Maslov index of triples and the cross ratio of quadruples of isotropic A-lines as invariants of this action. When the algebra A is Hermitian or the complexification of a Hermitian algebra, we introduce the symmetric space $$X_{{{\,\mathrm{Sp}\,}}_2(A,\sigma )}$$, and construct different models of this space. Applying this to classical Hermitian Lie groups of tube type (realized as $${{\,\mathrm{Sp}\,}}_2(A,\sigma )$$) and their complexifications, we obtain different models of the symmetric space as noncommutative generalizations of models of the hyperbolic plane and of the three-dimensional hyperbolic space. We also provide a partial classification of Hermitian algebras in Appendix A. | ||
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