TY - JOUR

T1 - Symplectic groups over noncommutative algebras

AU - Alessandrini, Daniele

AU - Berenstein, Arkady

AU - Retakh, Vladimir

AU - Rogozinnikov, Eugen

AU - Wienhard, Anna

N1 - Funding Information:
D.A, E.R and A.W. acknowledge support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network). E.R. and A.W. were supported by the National Science Foundation under Grant No. 1440140 and the Clay Foundation (A.W.), while they were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall Semester 2019. A.W acknowledges funding by the Deutsche Forschungsgemeinschaft within the Priority Program SPP 2026 “Geometry at Infinity”, by the European Research Council under ERC-Consolidator grant 614733, and by the Klaus-Tschira-Foundation. E.R. acknowledges funding by the Deutsche Forschungsgemeinschaft within the RTG 2229 “Asymptotic invariants and limits of groups and spaces” and the Priority Program SPP 2026 “Geometry at Infinity”, and thanks the Labex IRMIA of the Université de Strasbourg for support during the finishing of this project. A.B. was partially supported by Simons Foundation Collaboration Grant No. 636972. This work has been supported under Germany’s Excellence Strategy EXC-2181/1 - 390900948 (the Heidelberg STRUCTURES Cluster of Excellence).
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

PY - 2022/9

Y1 - 2022/9

N2 - We introduce the symplectic group Sp2(A,σ) over a noncommutative algebra A with an anti-involution σ. We realize several classical Lie groups as Sp2 over various noncommutative algebras, which provides new insights into their structure theory. We construct several geometric spaces, on which the groups Sp2(A,σ) act. We introduce the space of isotropic A-lines, which generalizes the projective line. We describe the action of Sp2(A,σ) 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 XSp2(A,σ), and construct different models of this space. Applying this to classical Hermitian Lie groups of tube type (realized as Sp2(A,σ)) 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.

AB - We introduce the symplectic group Sp2(A,σ) over a noncommutative algebra A with an anti-involution σ. We realize several classical Lie groups as Sp2 over various noncommutative algebras, which provides new insights into their structure theory. We construct several geometric spaces, on which the groups Sp2(A,σ) act. We introduce the space of isotropic A-lines, which generalizes the projective line. We describe the action of Sp2(A,σ) 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 XSp2(A,σ), and construct different models of this space. Applying this to classical Hermitian Lie groups of tube type (realized as Sp2(A,σ)) 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.

KW - Hermitian algebra

KW - Hermitian Lie group

KW - Hermitian symmetric space

KW - Involutive algebra

KW - Jordan algebra

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U2 - 10.1007/s00029-022-00787-x

DO - 10.1007/s00029-022-00787-x

M3 - Article

AN - SCOPUS:85138255621

SN - 1022-1824

VL - 28

JO - Selecta Mathematica, New Series

JF - Selecta Mathematica, New Series

IS - 4

M1 - 82

ER -