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Invariants of Domains under the Actions of Restricted Lie Algebras

✍ Scribed by J. Bergen

Publisher
Elsevier Science
Year
1995
Tongue
English
Weight
790 KB
Volume
177
Category
Article
ISSN
0021-8693

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✦ Synopsis

The goal of this paper is to examine the relationship between a domain (R) and its subring of invariants (R^{L}), under the action of a finite-dimensional restricted Lie algebra (L). We first show that there exists a nondegenerate (\left(R^{L}, R^{L}\right))-bimodule "trace-like" map (g: R \rightarrow R^{L}) and therefore (I \cap R^{L} \neq 0), for every ideal (I \neq 0) of (R). Next, we show that there exists a right (R^{L})-module embedding (\phi) of (R) into a finite direct sum of copies of (R^{L}). Using the maps (g) and (\phi), we obtain the following, which is a combination of several of the main results of this paper:
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Let F be a field of characteristic p ) 0, L a generalized restricted Lie algebra Ε½ . over F, and P L the primitive p-envelope of L. A close relation between Ε½ . L-representations and P L -representations is established. In particular, the irreducible -reduced modules of L for any g L\* coincide with