Derivation-invariance is transitive
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Revision as of 22:27, 8 October 2008 by Vipul (talk | contribs) (New page: ==Statement== A derivation-invariant Lie subring of a derivation-invariant Lie subring is a derivation-invariant Lie subring. ==Related facts== * [[Derivation-invariant subring of i...)
Statement
A derivation-invariant Lie subring of a derivation-invariant Lie subring is a derivation-invariant Lie subring.
Related facts
Proof
Given: A Lie ring with Lie subrings
.
is a derivation-invariant Lie subring of
.
To prove: is a derivation-invariant subring of
.
Proof: Suppose is a derivation of
.
Since is a derivation-invariant subring of
,
restricts to a map from
to itself. Let
be the restriction of
to
. Clearly,
is a derivation of
.
Since is derivation-invariant in
,
restricts to a map from
to itself. Thus,
restricts to a map from
to
.