Inner automorphism of a Lie ring: Difference between revisions

Latest revision as of 03:25, 15 May 2012

Definition

Suppose $L$ is a Lie ring. We call an automorphism $σ$ of $L$ a basic inner automorphism of $L$ if $σ = \exp (d)$ where $d$ is an inner derivation of $L$ (i.e., the adjoint action by some element) that is also an exponentiable derivation.

We call an automorphism of $L$ an inner automorphism if it can be expressed as a product of basic inner automorphisms. Note that the inverse of a basic inner automorphism is basic inner, so we see that the inner automorphisms are precisely the automorphisms in the subgroup generated by the basic inner automorphisms.

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 ==Definition==
-Suppose <math>L</math> is a [[Lie ring]]. We call an automorphism <math>\sigma</math> of <math>L</math> an '''inner automorphism''' of <math>L</math> if <math>\sigma = \exp(d)</math> where <matH>d</math> is an [[defining ingredient::inner derivation]] of <math>L</math> (i.e., the adjoint action by some element) that is also an [[defining ingredient::exponentiable derivation]], so that [[exponential of derivation is automorphism under suitable nilpotency assumptions|its exponential is an automorphism]].
+Suppose <math>L</math> is a [[Lie ring]]. We call an automorphism <math>\sigma</math> of <math>L</math> a '''basic inner automorphism''' of <math>L</math> if <math>\sigma = \exp(d)</math> where <matH>d</math> is an [[defining ingredient::inner derivation]] of <math>L</math> (i.e., the adjoint action by some element) that is also an [[defining ingredient::exponentiable derivation]].
+We call an automorphism of <math>L</math> an '''inner automorphism''' if it can be expressed as a product of basic inner automorphisms. Note that the inverse of a basic inner automorphism is basic inner, so we see that the inner automorphisms are precisely the automorphisms in the subgroup generated by the basic inner automorphisms.