In various sub-areas of mathematics , images are called derivatives if they meet Leibniz's rule . The concept of derivations is a generalization of the derivation known from school mathematics .
definition
Let it be a commutative ring with one , for example a body like or . Besides, be a - algebra . A ( -linear) derivation (also -derivation) of is a -linear mapping that
R.
{\ displaystyle R}
R.
{\ displaystyle \ mathbb {R}}
C.
{\ displaystyle \ mathbb {C}}
A.
{\ displaystyle A}
R.
{\ displaystyle R}
R.
{\ displaystyle R}
R.
{\ displaystyle R}
A.
{\ displaystyle A}
R.
{\ displaystyle R}
D.
:
A.
→
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{\ displaystyle D \ colon A \ to A}
D.
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a
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a
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)
=
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+
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{\ displaystyle D (a_ {1} a_ {2}) = D (a_ {1}) a_ {2} + a_ {1} D (a_ {2})}
for all
a
1
,
a
2
∈
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{\ displaystyle a_ {1}, a_ {2} \ in A}
Fulfills. The -linear property says that for all and the equations
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{\ displaystyle R}
a
1
,
a
2
∈
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{\ displaystyle a_ {1}, a_ {2} \ in A}
r
∈
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{\ displaystyle r \ in R}
D.
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+
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=
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+
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{\ displaystyle D (a_ {1} + a_ {2}) = D (a_ {1}) + D (a_ {2})}
and
D.
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r
a
1
)
=
r
D.
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a
1
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{\ displaystyle D (ra_ {1}) = rD (a_ {1})}
be valid. The definition includes rings by considering them as -algebras. If it is mapped into a module or bimodule , the definition can be given analogously.
A.
{\ displaystyle A}
Z
{\ displaystyle \ mathbb {Z}}
D.
{\ displaystyle D}
General properties
Is an algebra with identity element , the following applies . This also applies to everyone .
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{\ displaystyle A}
1
A.
{\ displaystyle 1_ {A}}
D.
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1
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)
=
0
{\ displaystyle D (1_ {A}) = 0}
D.
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r
)
=
0
{\ displaystyle D (r) = 0}
r
∈
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{\ displaystyle r \ in R}
The core of a derivation is a sub-algebra.
The set of derivations of with values in forms a Lie algebra with the commutator : are and derivations, so too
A.
{\ displaystyle A}
A.
{\ displaystyle A}
D.
1
{\ displaystyle D_ {1}}
D.
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{\ displaystyle D_ {2}}
[
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]
=
D.
1
∘
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2
-
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2
∘
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1
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{\ displaystyle [D_ {1}, D_ {2}] = D_ {1} \ circ D_ {2} -D_ {2} \ circ D_ {1}.}
For an item is , a derivation. Derivations of this type are called internal derivatives . The Hochschild cohomology is the quotient of the module of the derivatives after the sub-module of the inner derivatives.
b
∈
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{\ displaystyle b \ in A}
D.
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:
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→
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{\ displaystyle D_ {b} \ colon A \ to A}
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-
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{\ displaystyle D_ {b} (a) = ba-ab}
H
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{\ displaystyle H ^ {1} (A, A)}
In commutative algebra holds true for all and all non-negative integers .
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{\ displaystyle A}
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=
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{\ displaystyle D (a ^ {n}) = na ^ {n-1} D (a)}
a
∈
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{\ displaystyle a \ in A}
n
{\ displaystyle n}
Examples
The derivation of real functions is a derivation. This is what the product rule says .
f
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⊆
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→
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{\ displaystyle f \ colon D \ subseteq \ mathbb {R} \ to \ mathbb {R}}
For is the formal derivation
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=
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{\ displaystyle A = R [X]}
∑
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i
↦
∑
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a
i
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i
-
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{\ displaystyle \ sum a_ {i} X ^ {i} \ mapsto \ sum ia_ {i} X ^ {i-1}}
a -linear derivation of with values in .
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{\ displaystyle R}
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{\ displaystyle A}
A.
{\ displaystyle A}
Be a manifold. Then the is Cartan deriving a -linear Derivation of values in the space of 1-forms on .
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{\ displaystyle X}
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{\ displaystyle \ mathbb {R}}
C.
∞
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{\ displaystyle C ^ {\ infty} (X)}
A.
1
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{\ displaystyle A ^ {1} (X)}
X
{\ displaystyle X}
One of the reformulations of the Jacobi identity for Lie algebras says that the adjoint representation operates through derivatives:
[
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,
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=
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,
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,
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]
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{\ displaystyle [X, [A, B]] = [[X, A], B] + [A, [X, B]].}
Derivatives and Kähler differentials
By definition -linear derivatives of a commutative algebra are classified by the module of Kähler differentials , i.e. That is, there is a natural bijection between the -linear derivatives of with values in a module and the -linear mappings . Every derivation arises as a concatenation of the universal derivation with a linear mapping .
R.
{\ displaystyle R}
A.
{\ displaystyle A}
Ω
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/
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{\ displaystyle \ Omega _ {A / R}}
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{\ displaystyle R}
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{\ displaystyle A}
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{\ displaystyle A}
M.
{\ displaystyle M}
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{\ displaystyle A}
Ω
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/
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→
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{\ displaystyle \ Omega _ {A / R} \ to M}
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:
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→
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{\ displaystyle D \ colon A \ to M}
d
:
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→
Ω
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/
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{\ displaystyle \ mathrm {d} \ colon A \ to \ Omega _ {A / R}}
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{\ displaystyle A}
Ω
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/
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→
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{\ displaystyle \ Omega _ {A / R} \ to M}
Anti-derivatives
definition
If a - or - graduated -algebra, then a -linear graduated mapping is called an antiderivation , if
A.
{\ displaystyle A}
Z
{\ displaystyle \ mathbb {Z}}
Z
/
2
Z
{\ displaystyle \ mathbb {Z} / 2 \ mathbb {Z}}
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{\ displaystyle R}
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{\ displaystyle R}
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:
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→
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{\ displaystyle D \ colon A \ to A}
D.
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=
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+
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|
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⋅
a
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{\ displaystyle D (a_ {1} a_ {2}) = D (a_ {1}) a_ {2} + (- 1) ^ {| a_ {1} |} \ cdot a_ {1} D (a_ { 2})}
holds for all homogeneous elements ; where denotes the degree of .
a
1
,
a
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∈
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{\ displaystyle a_ {1}, a_ {2} \ in A}
|
a
1
|
{\ displaystyle | a_ {1} |}
a
1
{\ displaystyle a_ {1}}
Examples
d
(
ω
∧
η
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=
d
ω
∧
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+
(
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|
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⋅
ω
∧
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η
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{\ displaystyle \ mathrm {d} (\ omega \ wedge \ eta) = \ mathrm {d} \ omega \ wedge \ eta + (- 1) ^ {| \ omega |} \ cdot \ omega \ wedge \ mathrm {d } \ eta.}
literature
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">