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Closed forms for the exponential mapping on matrix Lie groups based on Putzer’s method. (English) Zbl 0942.22009

By Putzer’s method closed forms are found for the exponential of the infinitesimal generators of some Lie groups. First, the authors define a class of Lie algebras having a symmetric spectrum. They consider the Lie algebras defined by \( L = \{ A \in Gl (n, \mathbb{R}): A^T P = -PA \} \) and the Jordan algebra defined by \( J = \{ A \in Gl (n, \mathbb{R}): A^T P =PA \} \), where \(P\) is any \(n\times n\) orthogonal matrix. If \(P=I\) and \( L\) and \(J\) are the sets of skew-symmetric and symmetric matrices respectively it is proved that \(P\)-skew symmetric matrices have a symmetric spectrum. Particular cases of \(P\)-skew symmetric matrices are considered. Closed forms are found for the exponential of the infinitesimal generators of the orthogonal group \(SO(2,4)\) and the symplectic group \(SP(3, \mathbb{R})\).

MSC:

22E15 General properties and structure of real Lie groups
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