Arens-Michael enveloping algebras and analytic smash products

Abstract

Abstract. Let g be a finite-dimensional complex Lie algebra, and let U (g) be its universal enveloping algebra. We prove that if U (g), the Arens-Michael envelope of U (g) is stably flat over U (g) (i.e., if the canonical homomorphism U (g) → U (g) is a localization in the sense of , then g is solvable.To this end, given a cocommutative Hopf algebra H and an H-module algebra A, we explicitly describe the Arens-Michael envelope of the smash product A#H as an "analytic smash product" of their completions w.r.t. certain families of seminorms.The Arens-Michael envelope of a complex associative algebra A is defined as the completion of A w.r.t. the family of all submultiplicative seminorms on A. This notion (under a different name) was introduced by Taylor [15], and the terminology "Arens-Michael envelope" is due to Helemskii [5]. An important example is the polynomial algebra C[t 1 , . . . , t n ] whose Arens-Michael envelope is isomorphic to the algebra O(C n ) of entire functions endowed with the compact-open topology. Thus the Arens-Michael envelope of a noncommutative finitely generated algebra can be viewed as an "algebra of noncommutative entire functions" (cf. [16,17]).Given an algebra A, it is natural to ask to what extent homological properties of its Arens-Michael envelope A (considered as a topological algebra) are related to those of A. To handle this problem, it is convenient to use the notion of localization. Roughly speaking, a topological algebra homomorphism A → B is a localization if it identifies the category of topological B-modules with a full subcategory of the category of topological A-modules, and if the homological relations between B-modules do not change when the modules are considered as A-modules. Localizations were introduced by Taylor [16] in connection with the functional calculus problem for several commuting Banach space operators. A purely algebraic counterpart of this notion was studied by Neeman and Ranicki [10]. (Note that their terminology differs from Taylor's terminology; namely, a homomorphism A → B is a localization in Taylor's sense precisely when B is stably flat over A in the sense of Neeman and Ranicki.)Thus a natural question is whether or not A is stably flat over A. Taylor [16] proved that this is the case for A = C[t 1 , . . . , t n ] and for A = F n , the free algebra on n generators. In the case where A = U (g), the universal enveloping algebra of a