On Weakly Complete Universal Enveloping Algebras: A Poincare-Birkhoff-Witt Theorem

摘要

The Poincare-Birkhoff-Witt Theorem deals with the structure and universal property of the universal enveloping algebra U(L) of a Lie algebra L, e.g., over R or C. K. H. Hofmann and L. Kramer (HK) On weakly complete group algebras of Compact Groups, J. Lie Theory 30 (2020) 407-426 recently introduced the weakly complete universal enveloping algebra U(g) of a profinite-dimensional topological Lie algebra g . Here it is shown that the classical universal enveloping algebra U(vertical bar g vertical bar) of the abstract Lie algebra underlying g is a dense subalgebra of U(g), algebraically generated by g subset of U(g). It is further shown that, inspite of U being a left adjoint functor, it nevertheless preserves projective limits in the form U(lim(i) g/i) congruent to lim(i) U(g/i), for profinite-dimensional Lie algebras g represented as projective limits of their finite-dimensional quotients. The required theory is presented in an appendix which is of independent interest. In a natural way, a weakly complete enveloping algebra U(g) is a weakly complete symmetric Hopf algebra with a Lie subalgebra P(U(g)) of primitive elements containing g (indeed properly if g not equal {0}), and with a nontrivial multiplicative pro-Lie group G(U(g)) of grouplike units, having P(U(g)) as its Lie algebra - in contrast with the classical Poincare-Birhoff-Witt environment of U(L), thus providing a new aspect of Lie's Third Fundamental Theorem: Indeed a canonical pro-Lie subgroup Gamma* (g ) of G(U(g)) is identified whose Lie algebra is naturally isomorphic to g. The structure of U(g) is described in detail for dim g = 1. The primitive and grouplike components and their mutual relationship are evaluated precisely. In (HK), cited above, and in the work of R. Dahmen and K. H. Hofmann The pro-Lie group aspect of weakly complete algebras and weakly complete group Hopf algebras, J. Lie Theory 29 (2019) 413-455 the real weakly complete group Hopf algebra RG of a compact group G was described. In particular, the set P(RG)) of primitive elements of RG was identified as the Lie algebra g of G. It is now shown that for any compact group G with Lie algebra g there is a natural morphism of weakly complete symmetric Hopf algebras omega(g): U(g) -> RG, implementing the identity on g and inducing a morphism of pro-Lie groups Gamma* (G) -> G(RG) congruent to G: yet another aspect of Sophus Lie's Third Fundamental Theorem!