The interior and the exterior of the image of the exponential map in classical Lie groups
In Lie group theory it is of essential importance to understand the exponential function exp: \({\mathfrak g}\to G\). Its global behavior is not as simple as it may seem at first glance. For instance, it is not an obvious matter to determine the image \(E=\exp {\mathfrak g}\) and its boundary \(\partial E\) in G. Previously, the author established that \(E=G\) if \(G=GL(n,{\mathbb{C}})\), GL(n,\({\mathbb{H}})\), U(k,n-k), O *(2n), Sp(k,n-k). In the present article he completely clarifies the situation for \(G=SL(n,{\mathbb{C}})\), GL(n,\({\mathbb{R}})\), Sl(n,\({\mathbb{R}})\), SU(k,n-k), O(n,\({\mathbb{C}})\), O(k,n-k), Sp(2n,\({\mathbb{C}})\), Sp(2n,\({\mathbb{R}}).\) It is beautifully demonstrated that, in the case of these linear Lie groups, the issue is a question of linear algebra. Since, for the purposes of this review it would require too much space to render a complete report of the results, their general flavor is well represented by the following sample: Theorem 1. If \(G=GL(n,{\mathbb{R}})\), Sl(n,\({\mathbb{R}})\) or Sp(2n,\({\mathbb{R}})\), then \(g\in G\) is in the interior of E iff Spec g contains no negative number, and \(g\not\in \bar E\) iff Spec g contains at least one negative eigenvalue of odd multiplicity. Theorem 2. If \(G=O(p,q)\) then g is in the interior of E iff Spec g contains no elements \(<-1\) and if the generalized eigenspace for the eigenvalue -1 is positive or negative definite. Further \(g\not\in \bar E\) iff g has at least one negative eigenvalue of odd multiplicity or else -1 is an eigenvalue of even multiplicity and the numbers p and q of the signature of the generalized eigenspace of the eigenvalue -1 are both odd.
scientific article; zbMATH DE number 4040192
scientific article