Editing Derivation (differential algebra) (section)

== Graded derivations ==
{{Anchor|Homogeneous derivation|Graded derivation}}

Given a [[graded algebra]] ''A'' and a homogeneous linear map ''D'' of grade {{abs|''D''}} on ''A'', ''D'' is a '''homogeneous derivation''' if 
:<math>{D(ab)=D(a)b+\varepsilon^{|a||D|}aD(b)}</math>
for every homogeneous element ''a'' and every element ''b'' of ''A'' for a commutator factor {{nowrap|1=''ε'' = ±1}}.  A '''graded derivation''' is sum of homogeneous derivations with the same ''ε''.

If {{nowrap|1=''ε'' = 1}}, this definition reduces to the usual case.  If {{nowrap|1=''ε'' = &minus;1}}, however, then
:<math>{D(ab)=D(a)b+(-1)^{|a||D|}aD(b)}</math> 
for odd {{abs|''D''}}, and ''D'' is called an '''anti-derivation'''.

Examples of anti-derivations include the [[exterior derivative]] and the [[interior product]] acting on [[differential form]]s.

Graded derivations of [[superalgebra]]s (i.e. '''Z'''<sub>2</sub>-graded algebras) are often called '''superderivations'''.