Inner automorphism of a Lie ring: Difference between revisions
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==Definition== | ==Definition== | ||
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 | 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. | 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. | ||
Latest revision as of 03:25, 15 May 2012
Definition
Suppose is a Lie ring. We call an automorphism of a basic inner automorphism of if where is an inner derivation of (i.e., the adjoint action by some element) that is also an exponentiable derivation.
We call an automorphism of 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.