# nLab hyper-envelope of a Lie algebra

Let $L$ be a finite dimensional Lie algebra over real or complex numbers with basis ${\stackrel{^}{x}}_{1},\dots ,{\stackrel{^}{x}}_{n}$ and let ${x}_{1},\dots ,{x}_{n}$ be the corresponding basis of the symmetric algebra $S\left(L\right)$ via the identity and $U\left(L\right)$ the universal enveloping algebra of $L$. Given any $y\in L↪U\left(L\right)$ by $\stackrel{^}{y}$ we denote the same element understood in $L↪S\left(L\right)$. This correspondence extends to a coalgebra isomorphism $e:S\left(L\right)\to U\left(L\right)$, the coexponential map (or symmetrization map)

$e:{y}_{1}\dots {y}_{k}↦\frac{1}{k!}\sum _{\sigma \in \Sigma \left(k\right)}{\stackrel{^}{y}}_{\sigma 1}\cdots {\stackrel{^}{y}}_{\sigma k}$e : y_1 \ldots y_k \mapsto \frac{1}{k!}\sum_{\sigma\in \Sigma(k)} \hat{y}_{\sigma 1}\cdots \hat{y}_{\sigma k}

for any ${y}_{1},\dots ,{y}_{n}\in L↪S\left(L\right)$ (not necessarily basis elements). The coexponential map is the only functorial in $L$ isomorphism of coalgebras and it fixes $k\otimes L$ where $k$ is the ground field. It is also characterized by ${y}^{n}↦\left(\stackrel{^}{y}{\right)}^{n}$, where $y\in L↪S\left(L\right)$.

The Raševskii hyper-envelope of a Lie algebra $L$ is a completion of $U\left(L\right)$ by means of a countable family of norms $\stackrel{^}{f}↦\parallel \stackrel{^}{f}{\parallel }_{ϵ}$ for all $ϵ$ in an arbitrary fixed family of positive numbers having $0$ as an accumulation points, where

$\parallel \stackrel{^}{f}{\parallel }_{ϵ}={\mathrm{max}}_{{s}_{1},\dots ,{s}_{n}}{ϵ}^{-\left({s}_{1}+{s}_{2}+\dots +{s}_{n}\right)}\mid {f}_{{s}_{1}\dots {s}_{n}}\mid ,$\| \hat{f}\|_{\epsilon} = max_{s_1,\ldots, s_n}\epsilon^{-(s_1+s_2+\ldots+s_n)} |f_{s_1\ldots s_n}|,

for ${s}_{1}+\cdots +{s}_{n}=s$, and where ${f}_{{s}_{1},\dots ,{s}_{n}}$ is the Taylor coefficient in front of ${x}_{1}^{{s}_{1}}\cdots {x}_{n}^{{s}_{n}}$ of the commutative polynomial $f={e}^{-1}\left(\stackrel{^}{f}\right)$, i.e.

$\stackrel{^}{f}=e\left(\sum _{{s}_{1}\dots {s}_{n}}{f}_{{s}_{1}\dots {s}_{n}}\frac{{x}_{1}^{{s}_{1}}\cdots {x}_{n}^{{s}_{n}}}{{s}_{1}!{s}_{2}!\cdots {s}_{n}!}\right)$\hat{f} = e \left(\sum_{s_1\ldots s_n} f_{s_1\ldots s_n}\frac{x_1^{s_1}\cdots x_n^{s_n}}{s_1! s_2!\cdots s_n!} \right)

Here ${x}_{1},\dots ,{x}_{n}$ is a fixed basis of $L$, viewed as commutative coordinates.

It is nontrivial and proved by Raševskii that the algebra multiplication in $U\left(L\right)$ is continuous in this topology and hence that the completion of the $U\left(L\right)$ as a countably normed vector space carries the unique structure of a topological algebra extending the algebra operations on $U\left(L\right)$.

• P. K. Raševskii, Associative hyper-envelopes of Lie algebras, their regular representations and ideals, Trudy Mosk. Mat. Ob.

It may be tried to use the same definition with $e$ replaced by another coalgebra isomorphism $S\left(L\right)\to U\left(L\right)$.

Remark: It is announced (by author of these lines , June 11, 2011), that under mild conditions this modified definition results in the isomorphic completion of $U\left(L\right)$ as a topological algebra.

There is also an non-Archimedean version of the notion in the literature.

Revised on October 4, 2011 01:22:51 by Zoran Škoda (161.53.130.104)