An envelope in a category is a construction that generalizes the operations of “exterior completion”, like completion of a locally convex space, or the Stone–Čech compactification of a topological space, or the universal enveloping algebra of a Lie algebra. Dually, a refinement generalizes the operations of “interior enrichment”, like bornologification (or saturation) of a locally convex space, or simply connected covering of a Lie group. In this paper we define envelopes and refinements in abstract categories and discuss conditions under which these constructions exist and are functors. The aim of the exposition is to lay the foundations for duality theories of non-commutative groups based on the idea of envelope. The advantage of this approach is that in the arising theories the analogs of group algebras are Hopf algebras. At the same time the classical Fourier and Gelfand transforms are interpreted as envelopes with respect to certain classes of algebras.
A smooth envelope of a topological algebra is introduced, and the following result is announced: the smooth envelope of a given subalgebra A in C∞(M) coincides with C∞(M) if and only if A has the same tangent bundle as M.
Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.
Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.