Let be a finite dimensional Lie algebra over real or complex numbers with basis and let be the corresponding basis of the symmetric algebra via the identity and the universal enveloping algebra of . Given any by we denote the same element understood in . This correspondence extends to a coalgebra isomorphism , the coexponential map (or symmetrization map)
for any (not necessarily basis elements). The coexponential map is the only functorial in isomorphism of coalgebras and it fixes where is the ground field. It is also characterized by , where .
The Raševskii hyper-envelope of a Lie algebra is a completion of by means of a countable family of norms for all in an arbitrary fixed family of positive numbers having as an accumulation points, where
for , and where is the Taylor coefficient in front of of the commutative polynomial , i.e.
Here is a fixed basis of , viewed as commutative coordinates.
It is nontrivial and proved by Raševskii that the algebra multiplication in is continuous in this topology and hence that the completion of the as a countably normed vector space carries the unique structure of a topological algebra extending the algebra operations on .
It may be tried to use the same definition with replaced by another coalgebra isomorphism .
Remark: It is announced (by author of these lines ZŠ, June 11, 2011), that under mild conditions this modified definition results in the isomorphic completion of as a topological algebra.
There is also an non-Archimedean version of the notion in the literature.