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Symplectic groups over Lie subgroups of involutive algebras

Eugen Rogozinnikov

TL;DR

The paper develops a unified framework for symplectic groups over Lie subgroups of involutive algebras, introducing $\mathrm{Sp}_2(G,\sigma)$ and its Lie algebra $\mathfrak{sp}_2(B,\sigma)$ under Jordan-type and weakly Hermitian hypotheses. It constructs and analyzes $G$-isotropic lines, invariants such as positive triples and a generalized cross-ratio, and builds multiple models of the associated Riemannian symmetric spaces, including complex-structure, projective, precompact, and half-space realizations, with explicit boundary descriptions. A central contribution is realizing spin groups, notably $\mathrm{Spin}_0(m,n)$, as instances of $\mathrm{Sp}_2(G,\sigma)$ inside Clifford algebras, together with an explicit isomorphism between $\mathfrak{sp}_2(B(m,n),\sigma)$ and $\mathfrak{spin}(m+1,n+1)$ and a constructive embedding of $\mathrm{Spin}_0(m,n)$ into $\mathrm{Sp}_2(G,\sigma)$. The work provides concrete models (including an upper half-space model for Spin0(2,n)) and sets a foundation for dynamics, boundary maps, and Higgs-bundle contexts within a uniform algebraic-Lie framework.

Abstract

We introduce the symplectic group $\mathrm{Sp}_2(G, σ)$ associated to a Lie subgroup $G$ of a (possibly noncommutative) associative algebra $A$ equipped with an anti-involution $σ$. Our construction recovers several classical Lie groups as special cases, and in particular provides new realizations of spin groups as instances of $\mathrm{Sp}_2(G, σ)$ for suitable subgroups $G$ of the Clifford algebra. This case is not covered by the framework, which focuses on the specific situation $G = A^\times$, and is thus of particular interest. We construct and study geometric spaces on which $\mathrm{Sp}_2(G, σ)$ acts. In particular, we define the space of $G$-isotropic elements and the corresponding space of $G$-isotropic lines, which generalize the classical projective line. We analyze the group action on these spaces and introduce natural invariants, such as the notion of positive triples and quadruples of $G$-isotropic lines and a generalized cross-ratio of positive quadruples of $G$-isotropic lines. Finally, when the Lie algebra of $G$ is Hermitian, we define the associated Riemannian symmetric space of $\mathrm{Sp}_2(G,σ)$ and provide several models for it.

Symplectic groups over Lie subgroups of involutive algebras

TL;DR

The paper develops a unified framework for symplectic groups over Lie subgroups of involutive algebras, introducing and its Lie algebra under Jordan-type and weakly Hermitian hypotheses. It constructs and analyzes -isotropic lines, invariants such as positive triples and a generalized cross-ratio, and builds multiple models of the associated Riemannian symmetric spaces, including complex-structure, projective, precompact, and half-space realizations, with explicit boundary descriptions. A central contribution is realizing spin groups, notably , as instances of inside Clifford algebras, together with an explicit isomorphism between and and a constructive embedding of into . The work provides concrete models (including an upper half-space model for Spin0(2,n)) and sets a foundation for dynamics, boundary maps, and Higgs-bundle contexts within a uniform algebraic-Lie framework.

Abstract

We introduce the symplectic group associated to a Lie subgroup of a (possibly noncommutative) associative algebra equipped with an anti-involution . Our construction recovers several classical Lie groups as special cases, and in particular provides new realizations of spin groups as instances of for suitable subgroups of the Clifford algebra. This case is not covered by the framework, which focuses on the specific situation , and is thus of particular interest. We construct and study geometric spaces on which acts. In particular, we define the space of -isotropic elements and the corresponding space of -isotropic lines, which generalize the classical projective line. We analyze the group action on these spaces and introduce natural invariants, such as the notion of positive triples and quadruples of -isotropic lines and a generalized cross-ratio of positive quadruples of -isotropic lines. Finally, when the Lie algebra of is Hermitian, we define the associated Riemannian symmetric space of and provide several models for it.

Paper Structure

This paper contains 49 sections, 100 theorems, 258 equations.

Table of Contents

  1. Introduction
  2. Hermitian Lie algebras with anti-involution
  3. Lie subalgebras of involutive algebras
  4. Weakly Hermitian Lie subalgebras
  5. Classification of simple formally real Jordan algebras
  6. Spectral theorem
  7. Lie group corresponding to weakly Hermitian Lie algebra
  8. Minimal extension of G0
  9. Polar decomposition in G and maximal compact subgroup of G
  10. Symplectic group over G
  11. Symplectic group Sp2(A,sigma) over an involutive algebra
  12. Lie algebra sp2(B,sigma)
  13. Symplectic Lie algebra over Jordan algebras
  14. Indefinite orthogonal Lie group over involutive algebras
  15. First definition of Sp2(G,sigma)
  16. ...and 34 more sections

Key Result

Theorem 1.1

For a Lie subgroup $G\leq A^\times$ as above, assume that its Lie algebra $B$ is of Jordan type with $1\in B$. Then $\mathrm{Sp}_2(G,\sigma)$ is a real Lie group, and, moreover, the Lie algebra of $\mathrm{Sp}_2(G,\sigma)$ agrees with $\mathfrak{sp}_2(B,\sigma)$. If, in addition, $B$ is weakly Hermi

Theorems & Definitions (216)

  • Theorem 1.1: Theorem \ref{['thm:Sp_2-LieGroup']}, Proposition \ref{['prop:LieAlgOfSp2']}, Proposition \ref{['prop:Sp2Connected']}
  • Theorem 1.2
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Remark 2.7
  • ...and 206 more