Derivation (Abstract Algebra)
In
abstract algebra
, a
derivation
on an
algebra
A
over a
field
k
is a
linear map
D
:
A
→
A
that satisfies
Leibniz' law
:
D
(
ab
) = (
Da
)
b
+
a
(
Db
).
As a consequence, if
A
is
unital
, then
D
(1) = 0 since
D
1 =
D
(1·1) =
D
1 +
D
1. Examples of derivations are
partial derivatives
,
Lie derivatives
, the
Pincherle derivative
, and the
commutator
with respect to an element of the algebra. All these examples are tightly related, with the concept of derivation as the major unifying theme. See also:
Khler differential
If we have a
Z
2
graded algebra
A
,
D
is an
antiderivation
if
D
(
ab
) = (
Da
)
b
+ (−1)
deg(
a
)
a
(
Db
).
The same proof showing
D
(1)=0 applies, if
A
is unital.
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