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Generalized polar decompositions on Lie groups with involutive automorphisms. (English) Zbl 0980.22010

The classical polar decomposition of \(n\times n\) matrices can be generalized by considering involutive automorphisms of Lie groups: Let \(\sigma: G\to G\) be an involutive automorphism of a Lie group \(G\), \(G^\sigma\) the set of fixed points of \(\sigma\), and \(G_\sigma\) the set of anti-fixed points of \(\sigma\). In this general setting the polar decomposition requires to decompose an element \(z\in G\) into a product \(xy\) with \(x\in G_\sigma\) and \(y\in G^\sigma\). By the Campbell-Hausdorff formalism such generalized (differentiable) polar decompositions, depending on the choice of the involutive automorphism \(\sigma\), always exist (and are unique) near the identity. Moreover, these decompositions can be extended to larger portions of the group under suitable assumptions. Results in this sense can be found, for instance, in a paper by J. D. Lawson [J. Reine Angew. Math. 448, 191-219 (1994; Zbl 0786.22012)].
The authors present an alternative proof for the local existence and uniqueness of the generalized polar decomposition in a Lie group \(G\). They derive the differential equations obeyed by the two factors of the decomposition and solve them analytically, thereby obtaining recurrence relations on the Lie algebra level for the coefficients of the series expansion of each factor. The paper deals further with certain optimality properties of the subgroup factor (i.e., the factor that belongs to \(G^\sigma\)) of the generalized polar decomposition in a semisimple Lie group \(G\). Under the assumption that the involutive automorphism \(\sigma\) defines a Cartan decomposition of the Lie algebra of \(G\) and that \(G\) is endowed with a right invariant metric inherited from the Killing form, the authors prove first a local optimality result: If in a sufficiently small identity neighborhood an element \(z\in G\) has the polar decomposition \(z=xy\), where \(x\in G_\sigma\) and \(y\in G^\sigma\), then \(y\) is the best approximant to \(z\) in the subgroup \(G^\sigma\) (in the domain of convergence of the generalized polar decomposition). It is shown further that the optimality of the subgroup factor holds globally whenever the polar decomposition is global as, for instance, in the case of connected semisimple real Lie groups with finite center. The step from the local optimality result to the global one is made via the Iwasawa decomposition, using also some background on Riemannian manifolds. Finally, the paper deals with some applications to computations.

MSC:

22E15 General properties and structure of real Lie groups
53C22 Geodesics in global differential geometry
65F25 Orthogonalization in numerical linear algebra
65Q05 Numerical methods for functional equations (MSC2000)
53C35 Differential geometry of symmetric spaces

Citations:

Zbl 0786.22012
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