Symplectic groups over noncommutative algebras

• Verfasst von: Alessandrini, Daniele [VerfasserIn]
• Verfasst von: Berenstein, Arkady [VerfasserIn]
• Verfasst von: Retakh, Vladimir [VerfasserIn]
• Verfasst von: Rogozinnikov, Eugen [VerfasserIn]
• Verfasst von: Wienhard, Anna [VerfasserIn]
• Titel: Symplectic groups over noncommutative algebras
• Verf.angabe: Daniele Alessandrini, Arkady Berenstein, Vladimir Retakh, Eugen Rogozinnikov, Anna Wienhard
• E-Jahr: 2022
• Jahr: 12 September 2022
• Umfang: 119 S.
• Fussnoten: Gesehen am 23.09.2022
• Titel Quelle: Enthalten in: Selecta mathematica
• Ort Quelle: Basel [u.a.] : Birkhäuser, 1995
• Jahr Quelle: 2022
• Band/Heft Quelle: 28(2022), 4, Artikel-ID 82, Seite 1-119
• ISSN Quelle: 1420-9020
• DOI: doi:10.1007/s00029-022-00787-x
• Datenträger: Online-Ressource
• Sprache: eng
• Sach-SW: 16W10
• Sach-SW: 32M15
• Sach-SW: 46L05
• Sach-SW: 53C35
• Sach-SW: Hermitian algebra
• Sach-SW: Hermitian Lie group
• Sach-SW: Hermitian symmetric space
• Sach-SW: Involutive algebra
• Sach-SW: Jordan algebra
• K10plus-PPN: 1817283103

Abstract

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.