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Symplectic groups over noncommutative algebras

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Daniele Alessandrini, Arkady Berenstein, Vladimir Retakh, Eugen Rogozinnikov, Anna Wienhard

Differential Geometry · Submitted 16 Jun 2021

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

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arXiv ID2106.08736 (version 1)
Pith IDSRTR3QHF, Pith's permanent record of this paper and its review
CitationAnna Wienhard, Arkady Berenstein, Daniele Alessandrini, Eugen Rogozinnikov, Vladimir Retakh. "Symplectic groups over noncommutative algebras." arXiv:2106.08736 (2021). https://pith.science/paper/SRTR3QHF
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,466 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.50 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.20 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

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.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We introduce the symplectic group $\mathrm{Sp}_2(A,\sigma)$ over a noncommutative algebra $A$ with an anti-involution $\sigma$. contribution 1.00 0.95 0.09 asserted 0.99
s1 We realize several classical Lie groups as $\mathrm{Sp}_2$ over various noncommutative algebras, which provides new insights into their structure theory. contribution 0.56 0.94 0.20 asserted 1.00
s2 We construct several geometric spaces, on which the groups $\mathrm{Sp}_2(A,\sigma)$ act. contribution 1.00 0.96 0.06 asserted 1.00
s3 We introduce the space of isotropic $A$-lines, which generalizes the projective line. contribution 1.00 0.95 0.05 asserted 1.00
s4 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. contribution 0.95 0.94 0.14 asserted 1.00
s5 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. contribution 1.00 0.95 0.11 asserted 1.00
s6 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. result 0.66 0.94 0.25 asserted 1.00
s7 We also provide a partial classification of Hermitian algebras in Appendix A. contribution 0.92 0.93 0.03 asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.95 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original 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.

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  • An immediate application is that the construction of noncommutative coordinates for symplectic representations of fundamental groups of punctured surfaces developed in [ref] and its relation to noncommutative cluster algebras introduced in [ref] generalizes to representations into classical Hermitian Lie groups of tube type.

    Noncommutative coordinates for symplectic representations · Introduction · 0.99

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