Let G be a reductive algebraic group and H a closed subgroup of G. Explicit constructions of G-invariant ideals in the algebra K[G/H] are given. This allows to obtain an elementary proof of Matsushima’s criterion: a homogeneous space G/H is an affine variety if and only if H is reductive. 1. Algebraic homogeneous spaces Let G be an affine algebraic group over an algebraically closed field K. A ...

In this paper, we investigate some properties of inverse limits of linear algebraic groups. For example, we show that if G = lim ←−Gi, where (Gi, πji)i,j∈I is an inverse system of algebraic groups over an algebraically closed field. Then each canonical projection πi : G → Gi maps closed subgroups of G onto closed subgroups of Gi. Furthermore, we prove directly that the inverse limit of an inver...

If M is a simple, closed, orientable 3-manifold such that π1(M) contains a genus-g surface group, and if H1(M ;Z2) has rank at least 4g−1, we show that M contains an embedded closed incompressible surface of genus at most g. As an application we show that if M is a closed orientable hyperbolic 3-manifold of volume at most 3.08, then the rank of H1(M ;Z2) is at most 6.

The ground field k is algebraically closed and of characteristic zero. Let g be a simple Lie algebra over k. The goal of this note is to prove a closed formula for the Dynkin index of a principal sl2-subalgebra of g, see Theorem 3.2. The key step in the proof uses the “strange formula” of Freudenthal–de Vries. As an application, we (1) compute the Dynkin index any simple g-module regarded as sl...

Hart, K.P. and J. van Mill, Discrete sets and the maximal totally bounded group topology, Journal of Pure and Applied Algebra 70 (1991) 73-80. If G is an Abelian group, then G # is G with its maximal totally bounded group topology. We prove that every A c G# contains a closed (in G#) and discrete subset B such that lB1 = IAl. This answers a question posed by Eric van Douwen. We also present an ...

We shall describe an algebraic scheme giving rise to a group P2 that can be identified as the second homotopy group when the elements entering the scheme are suitably interpreted. We consider particularly the group p2(X) associated with an abstract local group X. Let G be a groupoid.I We denote by it(g) and i,(g) the leftand rightidentities of elements g in G. We call g closed if iK(g) = ir(g) ...

Let G = 〈G, ·〉 be a group definable in an o-minimal structure M. A subset H of G is G-definable if H is definable in the structure 〈G, ·〉 (while definable means definable in the structure M). Assume G has no Gdefinable proper subgroup of finite index. In this paper we prove that if G has no nontrivial abelian normal subgroup, then G is the direct product of G-definable subgroups H1, . . . ,Hk s...