Symplectic groups over noncommutative algebras. (English) Zbl 1503.53111
The authors develop a special geometry whose main object is the symplectic group \(\mathrm{Sp}_2(A,\sigma)\) over a noncommutative algebra \(A\) with an anti-involution \(\sigma \). The group \(\mathrm{Sp}_2(A,\sigma )\) is the automorphism group of the symplectic form
\[
\omega (x,y)=\sigma (x)^T\Omega y,
\]
where \(x,y\in A\times A\) and
\[
\Omega = \begin{pmatrix} 0 & 1 \cr -1 & 0 \cr \end{pmatrix}.
\]
The authors consider mainly the case of a Hermitian algebra \(A\). In this case one introduces the unitary group \(\mathrm{U}_2(A,\sigma )\), the maximal compact subgroup \(\mathrm{KSp}_2(A,\sigma )\) of \(\mathrm{Sp}(A,\sigma)\), and the symmetric space \(\mathrm{Sp}_2(A,\sigma )/\mathrm{KSp}_2(A,\sigma )\). If \(A\) is a classical matrix algebra one recovers classical groups and classical symmetric spaces. If \(A=\mathrm{Mat}(n,\mathbb{R})\), and \(\sigma \) is the transposition, then \(\mathrm{Sp}_2(A,\sigma )\) is isomorphic to the classical group \(\mathrm{Sp}(2n,\mathbb{R})\). If \(A=\mathrm{Mat}(n,\mathbb{C})\), then one obtains \(\mathrm{U}(n,n)\). If \(A=\mathrm{Mat}(n,\mathbb{H})\), then one obtains \(\mathrm{SO}^*(4n)\). The corresponding symmetric spaces are classical tube-type Hermitian symmetric spaces. In the general case of a Hermitian algebra \(A\), one describes several geometric realizations of the symmetric space associated to \(\mathrm{Sp}_2(A,\sigma )\). For a classical matrix algebra one recovers classical geometric realizations. In this setting it is possible to define the Shilov boundary of the symmetric space associated to \(\mathrm{Sp}(A,\sigma )\).
Further one defines the complex extension \((A_{\mathbb{C}},\sigma _{\mathbb{C}})\) of \((A,\sigma )\) where \(A_{\mathbb{C}}=A+Ai\), and \(\sigma _{\mathbb{C}}(x+yi)=\sigma (x)-\sigma (y)i\). Then one considers the associated group \(\mathrm{Sp}_2(A_{\mathbb{C}},\sigma _{\mathbb{C}})\) and the associated symmetric space. If \(A\) is a classical matrix algebra, one obtains for \(\mathrm{Sp}_2(A_{\mathbb{C}},\sigma _{\mathbb{C}})\) the classical groups \(\mathrm{Sp}(2n,\mathbb{C})\), \(\mathrm{GL}(2n,\mathbb{C})\), \(\mathrm{O}(4n,\mathbb{C})\). Then it is possible to describe several geometric realizations of the symmetric space associated to \(\mathrm{Sp}_2(A_{\mathbb{C}},\sigma _{\mathbb{C}})\).
Further one defines the complex extension \((A_{\mathbb{C}},\sigma _{\mathbb{C}})\) of \((A,\sigma )\) where \(A_{\mathbb{C}}=A+Ai\), and \(\sigma _{\mathbb{C}}(x+yi)=\sigma (x)-\sigma (y)i\). Then one considers the associated group \(\mathrm{Sp}_2(A_{\mathbb{C}},\sigma _{\mathbb{C}})\) and the associated symmetric space. If \(A\) is a classical matrix algebra, one obtains for \(\mathrm{Sp}_2(A_{\mathbb{C}},\sigma _{\mathbb{C}})\) the classical groups \(\mathrm{Sp}(2n,\mathbb{C})\), \(\mathrm{GL}(2n,\mathbb{C})\), \(\mathrm{O}(4n,\mathbb{C})\). Then it is possible to describe several geometric realizations of the symmetric space associated to \(\mathrm{Sp}_2(A_{\mathbb{C}},\sigma _{\mathbb{C}})\).
Reviewer: Jacques Faraut (Paris)
MSC:
| 53C35 | Differential geometry of symmetric spaces |
| 22E10 | General properties and structure of complex Lie groups |
| 32M15 | Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) |
| 16W10 | Rings with involution; Lie, Jordan and other nonassociative structures |
| 15B30 | Matrix Lie algebras |
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