Hopf algebras, Lie algebras, and analytic groups (Q1913936)

There are two parts: 1. Universal analytic Lie groups, and 2. Almost stable nuclei. The universal analytic group associated with a finite-dimensional real (or complex) Lie algebra \(L\) is a faithfully representable real (or complex) analytic Lie group \(G\) with Lie algebra \(L\), such that the finite-dimensional representations of \(L\) are the differentials of the representations of \(G\). The complex case is evident. Proposition 1.2 states the existence and uniqueness up to isomorphism in the real case. The universal analytic group \(G\) associated to a real finite-dimensional Lie algebra \(L\) can be characterized by the fact, that its complexification \(G^+\) is simply connected. Moreover, if \(R\) denotes the radical of \(G\) and \(H\) a maximal reductive Lie subgroup, then \(G^+\) is the product of the complexification \(R^+\) of \(R\) with the complexification \(H^+\) of \(H\) (Proposition 1.4). The considerations of the first part culminate in Theorem 1.6 proving isomorphy of the Hopf algebra of representative functions on the real (or complex) Lie algebra \(L\) with the Hopf algebra of representative functions on the universal analytic group \(G\) corresponding to \(L\). While the interesting part of the first point is the real case, the second point deals with faithfully representable complex analytic groups, its subgroups and automorphisms. A nucleus of a faithfully representable complex analytic Lie group is a closed simply connected solvable normal complex analytic subgroup, such that the quotient group is reductive. If \(G\) is such a group and \(T\) a maximal complex toroid of its radical, such that \(T\) meets the center of \(G\) trivially, then there exists a characteristic nucleus \(K\) in \(G\), i.e. a nucleus, which is invariant under all complex analytic automorphisms of \(G\) (Lemma 2.3). The general situation is described in Theorem 2.5: Let \(G\) be a faithfully representable complex analytic Lie group and \(A\) the maximal reductive complex Lie subgroup of its center, then there exists a nucleus \(K\) of \(G\), such that the image of \(K\) under any complex analytic automorphism of \(G\) is contained in the product of \(K\) with the connected component of the identity of \(A\).