Oka’s principle for special Fréchet Lie groups and homogeneous manifolds in topological algebras of the microlocal analysis

Abstract We consider a wide class of unital involutive topological algebras provided with a C*-norm and which are inverse limits of sequences of unital involutive Banach algebras; these algebras are taking a prominent position in noncommutative differential geometry, where they are often called unital smooth algebras. In this paper we prove that the group of invertible elements of such a unital solution smooth algebra and the subgroup of its unitary elements are regular analytic Fréchet–Lie groups of Campbell–Baker–Hausdorff type and fulfill a nice infinite-dimensional version of Lie's second fundamental theorem.

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INVARIANT FUNCTION SPACES ON HOMOGENEOUS MANIFOLDS OF LIE GROUPS AND APPLICATIONS

Bulletin of the London Mathematical Society ◽

10.1112/blms/28.1.99 ◽

1996 ◽

Vol 28 (1) ◽

pp. 99-102

Author(s):

Carlos A. Berenstein

Keyword(s):

Lie Groups ◽

Invariant Function ◽

Function Spaces ◽

Homogeneous Manifolds

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Naturally reductive homogeneous (α,β) spaces

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887820501170 ◽

2020 ◽

Vol 17 (08) ◽

pp. 2050117

Author(s):

Parisa Bahmandoust ◽

Dariush Latifi

Keyword(s):

Lie Groups ◽

Explicit Formula ◽

Flag Curvature ◽

Homogeneous Manifolds ◽

Randers Metrics ◽

Naturally Reductive ◽

Special Case

In this paper, we study naturally reductive [Formula: see text]-metrics on homogeneous manifolds. We show that naturally reductive [Formula: see text]-metrics arise only when [Formula: see text] is naturally reductive and some conditions on [Formula: see text] is satisfied. We give an explicit formula for the flag curvature of naturally reductive [Formula: see text]metrics which improves the flag curvature formula of naturally reductive Randers metrics given in [D. Latifi, Naturally reductive homogeneous Randers spaces, J. Geom. Phys. 60 (2010) 1968–1973]. As a special case, we give an explicit formula for the flag curvature of bi-invariant [Formula: see text]-metrics on Lie groups.

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