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1
C o m m u t a t i v e a l g e b r a
T o g g l e C o m m u t a t i v e a l g e b r a s u b s e c t i o n
1 . 1
F i r s t d e f i n i t i o n
1 . 2
S e c o n d d e f i n i t i o n
2
G r a d e d c o m m u t a t i v e a l g e b r a
3
N o n c o m m u t a t i v e a l g e b r a
4
S e e a l s o
5
N o t e s
6
R e f e r e n c e s
7
E x t e r n a l l i n k s
T o g g l e t h e t a b l e o f c o n t e n t s
C o n n e c t i o n ( a l g e b r a i c f r a m e w o r k )
2 l a n g u a g e s
● Р у с с к и й
● У к р а ї н с ь к а
E d i t l i n k s
● A r t i c l e
● T a l k
E n g l i s h
● R e a d
● E d i t
● V i e w h i s t o r y
T o o l s
T o o l s
A c t i o n s
● R e a d
● E d i t
● V i e w h i s t o r y
G e n e r a l
● W h a t l i n k s h e r e
● R e l a t e d c h a n g e s
● U p l o a d f i l e
● S p e c i a l p a g e s
● P e r m a n e n t l i n k
● P a g e i n f o r m a t i o n
● C i t e t h i s p a g e
● G e t s h o r t e n e d U R L
● D o w n l o a d Q R c o d e
● W i k i d a t a i t e m
P r i n t / e x p o r t
● D o w n l o a d a s P D F
● P r i n t a b l e v e r s i o n
A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Commutative algebra [ edit ]
Let
A
{\displaystyle A}
be a commutative ring
and
M
{\displaystyle M}
an A -module . There are different equivalent definitions
of a connection on
M
{\displaystyle M}
.[2]
First definition [ edit ]
If
k
→
A
{\displaystyle k\to A}
is a ring homomorphism, a
k
{\displaystyle k}
-linear connection is a
k
{\displaystyle k}
-linear morphism
∇
:
M
→
Ω
A
/
k
1
⊗
A
M
{\displaystyle \nabla :M\to \Omega _{A/k}^{1}\otimes _{A}M}
which satisfies the identity
∇
(
a
m
)
=
d
a
⊗
m
+
a
∇
m
{\displaystyle \nabla (am )=da\otimes m+a\nabla m}
A connection extends, for all
p
≥
0
{\displaystyle p\geq 0}
to a unique map
∇
:
Ω
A
/
k
p
⊗
A
M
→
Ω
A
/
k
p
+
1
⊗
A
M
{\displaystyle \nabla :\Omega _{A/k}^{p}\otimes _{A}M\to \Omega _{A/k}^{p+1}\otimes _{A}M}
satisfying
∇
(
ω
⊗
f
)
=
d
ω
⊗
f
+
(
−
1
)
p
ω
∧
∇
f
{\displaystyle \nabla (\omega \otimes f)=d\omega \otimes f+(-1)^{p}\omega \wedge \nabla f}
. A connection is said to be integrable if
∇
∘
∇
=
0
{\displaystyle \nabla \circ \nabla =0}
, or equivalently, if the curvature
∇
2
:
M
→
Ω
A
/
k
2
⊗
M
{\displaystyle \nabla ^{2}:M\to \Omega _{A/k}^{2}\otimes M}
vanishes.
Second definition [ edit ]
Let
D
(
A
)
{\displaystyle D(A )}
be the module of derivations of a ring
A
{\displaystyle A}
. A
connection on an A -module
M
{\displaystyle M}
is defined
as an A -module morphism
∇
:
D
(
A
)
→
D
i
f
f
1
(
M
,
M
)
;
u
↦
∇
u
{\displaystyle \nabla :D(A )\to \mathrm {Diff} _{1}(M,M);u\mapsto \nabla _{u}}
such that the first order differential operators
∇
u
{\displaystyle \nabla _{u}}
on
M
{\displaystyle M}
obey the Leibniz rule
∇
u
(
a
p
)
=
u
(
a
)
p
+
a
∇
u
(
p
)
,
a
∈
A
,
p
∈
M
.
{\displaystyle \nabla _{u}(ap )=u(a )p+a\nabla _{u}(p ),\quad a\in A,\quad p\in M.}
Connections on a module over a commutative ring always exist.
The curvature of the connection
∇
{\displaystyle \nabla }
is defined as
the zero-order differential operator
R
(
u
,
u
′
)
=
[
∇
u
,
∇
u
′
]
−
∇
[
u
,
u
′
]
{\displaystyle R(u,u')=[\nabla _{u},\nabla _{u'}]-\nabla _{[u,u']}\,}
on the module
M
{\displaystyle M}
for all
u
,
u
′
∈
D
(
A
)
{\displaystyle u,u'\in D(A )}
.
If
E
→
X
{\displaystyle E\to X}
is a vector bundle, there is one-to-one
correspondence between linear
connections
Γ
{\displaystyle \Gamma }
on
E
→
X
{\displaystyle E\to X}
and the
connections
∇
{\displaystyle \nabla }
on the
C
∞
(
X
)
{\displaystyle C^{\infty }(X )}
-module of sections of
E
→
X
{\displaystyle E\to X}
. Strictly speaking,
∇
{\displaystyle \nabla }
corresponds to
the covariant differential of a
connection on
E
→
X
{\displaystyle E\to X}
.
Graded commutative algebra [ edit ]
The notion of a connection on modules over commutative rings is
straightforwardly extended to modules over a graded
commutative algebra .[3] This is the case of
superconnections in supergeometry of graded manifolds and supervector bundles .
Superconnections always exist.
Noncommutative algebra [ edit ]
If
A
{\displaystyle A}
is a noncommutative ring, connections on left
and right A -modules are defined similarly to those on
modules over commutative rings.[4] However
these connections need not exist.
In contrast with connections on left and right modules, there is a
problem how to define a connection on an
R -S -bimodule over noncommutative rings
R and S . There are different definitions
of such a connection.[5] Let us mention one of them. A connection on an
R -S -bimodule
P
{\displaystyle P}
is defined as a bimodule
morphism
∇
:
D
(
A
)
∋
u
→
∇
u
∈
D
i
f
f
1
(
P
,
P
)
{\displaystyle \nabla :D(A )\ni u\to \nabla _{u}\in \mathrm {Diff} _{1}(P,P)}
which obeys the Leibniz rule
∇
u
(
a
p
b
)
=
u
(
a
)
p
b
+
a
∇
u
(
p
)
b
+
a
p
u
(
b
)
,
a
∈
R
,
b
∈
S
,
p
∈
P
.
{\displaystyle \nabla _{u}(apb)=u(a )pb+a\nabla _{u}(p )b+apu(b ),\quad a\in R,\quad b\in S,\quad p\in P.}
See also [ edit ]
^ (Bartocci, Bruzzo & Hernández-Ruipérez 1991 ), (Mangiarotti & Sardanashvily 2000 )
^ (Landi 1997 )
^ (Dubois-Violette & Michor 1996 ),(Landi 1997 )
References [ edit ]
Koszul, J. L. (1986). Lectures on Fibre Bundles and Differential Geometry (Tata University, Bombay, 1960) . doi :10.1007/978-3-662-02503-1 (inactive 2024-04-26). ISBN 978-3-540-12876-2 . S2CID 51020097 . Zbl 0244.53026 . {{cite book }}
: CS1 maint: DOI inactive as of April 2024 (link )
Bartocci, Claudio; Bruzzo, Ugo; Hernández-Ruipérez, Daniel (1991). The Geometry of Supermanifolds . doi :10.1007/978-94-011-3504-7 . ISBN 978-94-010-5550-5 .
Dubois-Violette, Michel; Michor, Peter W. (1996). "Connections on central bimodules in noncommutative differential geometry". Journal of Geometry and Physics . 20 (2–3): 218–232. arXiv :q-alg/9503020 . doi :10.1016/0393-0440(95 )00057-7 . S2CID 15994413 .
Landi, Giovanni (1997). An Introduction to Noncommutative Spaces and their Geometries . Lecture Notes in Physics. Vol. 51. arXiv :hep-th/9701078 . doi :10.1007/3-540-14949-X . ISBN 978-3-540-63509-3 . S2CID 14986502 .
Mangiarotti, L.; Sardanashvily, G. (2000). Connections in Classical and Quantum Field Theory . doi :10.1142/2524 . ISBN 978-981-02-2013-6 .
External links [ edit ]
Sardanashvily, G. (2009). "Lectures on Differential Geometry of Modules and Rings". arXiv :0910.1515 [math-ph ].
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Connection_(algebraic_framework)&oldid=1220876780 "
C a t e g o r i e s :
● C o n n e c t i o n ( m a t h e m a t i c s )
● N o n c o m m u t a t i v e g e o m e t r y
H i d d e n c a t e g o r y :
● C S 1 m a i n t : D O I i n a c t i v e a s o f A p r i l 2 0 2 4
● T h i s p a g e w a s l a s t e d i t e d o n 2 6 A p r i l 2 0 2 4 , a t 1 3 : 3 8 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
● P r i v a c y p o l i c y
● A b o u t W i k i p e d i a
● D i s c l a i m e r s
● C o n t a c t W i k i p e d i a
● C o d e o f C o n d u c t
● D e v e l o p e r s
● S t a t i s t i c s
● C o o k i e s t a t e m e n t
● M o b i l e v i e w