nLab
hyper-envelope of a Lie algebra

Let LL be a finite dimensional Lie algebra over real or complex numbers with basis x^ 1,,x^ n\hat{x}_1,\ldots,\hat{x}_n and let x 1,,x nx_1,\ldots, x_n be the corresponding basis of the symmetric algebra S(L)S(L) via the identity and U(L)U(L) the universal enveloping algebra of LL. Given any yLU(L)y\in L\hookrightarrow U(L) by y^\hat{y} we denote the same element understood in LS(L)L\hookrightarrow S(L). This correspondence extends to a coalgebra isomorphism e:S(L)U(L)e: S(L)\to U(L), the coexponential map (or symmetrization map)

e:y 1y k1k! σΣ(k)y^ σ1y^ σk 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,,y nLS(L)y_1,\ldots, y_n\in L\hookrightarrow S(L) (not necessarily basis elements). The coexponential map is the only functorial in LL isomorphism of coalgebras and it fixes kLk\otimes L where kk is the ground field. It is also characterized by y n(y^) ny^n\mapsto (\hat{y})^n, where yLS(L)y\in L\hookrightarrow S(L).

The Raševskii hyper-envelope of a Lie algebra LL is a completion of U(L)U(L) by means of a countable family of norms f^f^ ϵ\hat{f}\mapsto \|\hat{f}\|_{\epsilon} for all ϵ\epsilon in an arbitrary fixed family of positive numbers having 00 as an accumulation points, where

f^ ϵ=max s 1,,s nϵ (s 1+s 2++s n)|f s 1s n|, \| \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++s n=ss_1+\cdots + s_n = s, and where f s 1,,s nf_{s_1,\ldots,s_n} is the Taylor coefficient in front of x 1 s 1x n s nx_1^{s_1}\cdots x_n^{s_n} of the commutative polynomial f=e 1(f^)f = e^{-1}(\hat{f}), i.e.

f^=e( s 1s nf s 1s nx 1 s 1x n s ns 1!s 2!s n!) \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,,x nx_1,\ldots, x_n is a fixed basis of LL, viewed as commutative coordinates.

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

  • 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 ee replaced by another coalgebra isomorphism S(L)U(L)S(L)\to U(L).

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(L)U(L) 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)