A derivation is a linear map $D$ of the algebra into itself that satisfies the Leibniz rule:
$$D(ab) = aD(b) + D(a)b$$
where $a$ and $b$ are elements of the algebra.
An inner derivation is a special kind of derivation that is defined on an associative algebra with a fixed unit element. An inner derivation is associated with an element of the algebra, and is defined as follows:
For any element $a$ in the algebra, the inner derivation associated with $a$, denoted by $\operatorname{ad}(a)$, is defined as the linear map on the algebra given by
$$ \operatorname{ad}(a)(b) = [b,a] = ba-ab $$where $[a,b]$ denotes the commutator of $a$ and $b$. It can be also denoted by $[-,a]$.
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Author of the notes: Antonio J. Pan-Collantes
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