Connection is module structure over connection algebra - Revision history

Vipul: /* From a connection to a module structure */

2009-07-24T00:42:16Z

From a connection to a module structure

← Older revision		Revision as of 00:42, 24 July 2009
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	* <math>f_\nabla</math> is <math>\R</math>-bilinear: This is obvious.		* <math>f_\nabla</math> is <math>\R</math>-bilinear: This is obvious.
	* Left module map property: For any element <math>X + m(g)</math> in <math>\mathcal{D}^1(M)</math> and any <math>h \in C^\infty(M)</math>, we have <math>f_\nabla(m(g) \cdot (X + m(h))(s) = m(g) \~~cdot~~ f_\nabla(X + m(h))(s)</math>. This essentially follows from the fact that a connection is [[tensorial map\|tensorial]] in the direction of differentiation:		* Left module map property: For any element <math>X + m(g)</math> in <math>\mathcal{D}^1(M)</math> and any <math>h \in C^\infty(M)</math>, we have <math>f_\nabla(m(g) \cdot (X + m(h))(s) = (m(g) \circ f_\nabla(X + m(h)))(s)</math>. This essentially follows from the fact that a connection is [[tensorial map\|tensorial]] in the direction of differentiation:

	<math>f_\nabla(m(g) \cdot (X + m(h)))(s) = f_\nabla(gX + m(gh))(s) = \nabla_{gX}(s) + (gh)(s)= g\nabla_X(s) + (gh)(s) = g(\nabla_X(s) + hs) = m(g)f_\nabla(X + m(h))(s)</math>.		<math>f_\nabla(m(g) \cdot (X + m(h)))(s) = f_\nabla(gX + m(gh))(s) = \nabla_{gX}(s) + (gh)(s)= g\nabla_X(s) + (gh)(s) = g(\nabla_X(s) + hs) = m(g)f_\nabla(X + m(h))(s)</math>.
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	<math>f_\nabla((X + m(h)) \cdot m(g))(s) = f_\nabla(m(Xg) +g\nabla_X + m(gh))(s) = (Xg)(s) + g\nabla_X(s) + (gh)s = \nabla_X(gs) + (gh)(s) = (f_\nabla(X + m(h)) \cdot m(g))(s)</math>.		<math>f_\nabla((X + m(h)) \cdot m(g))(s) = f_\nabla(m(Xg) +g\nabla_X + m(gh))(s) = (Xg)(s) + g\nabla_X(s) + (gh)s = \nabla_X(gs) + (gh)(s) = (f_\nabla(X + m(h)) \cdot m(g))(s)</math>.

			Since <math>\nabla \mapsto f_\nabla</math> is a <math>C^\infty</math>-bimodule map, it extends to a unique <math>C^\infty</math>-bimodule map from the <math>C^\infty</math>-tensor algebra over <math>\mathcal{D}^1(M)</math>. By definition, the element <math>m(1) - 1</math> induces the zero map on <math>\Gamma(E)</math>, so the map descends to a homomorphism <math>\mathcal{C}(M) \to B</math>, as desired.

			===From a module structure to a connection===

			'''Given''': A manifold <math>M</math>, a vector bundle <math>E</math> over <math>M</math>. <math>B</math> is the algebra of smooth fiber-preserving maps from <math>\Gamma(E)</math> to <math>\Gamma(E)</math>. <math>\mathcal{D}^1(M)</math> is the Lie algebra of first-order differential operators on <math>M</math> and <math>\mathcal{C}(M)</math> is the connection algebra on <math>M</math>. A module structure of <math>\Gamma(E)</math> over <math>\mathcal{C}(M)</math>.

	==References==		==References==

Vipul: /* From a connection to a module structure */

2009-07-24T00:26:27Z

From a connection to a module structure

← Older revision		Revision as of 00:26, 24 July 2009
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	First observe that the map sends <math>C^\infty(M) \subset \mathcal{D}^1(M)</math> to <math>C^\infty(M) \subset B</math>, and is the identity restricted to that subset. In other words, the differential operator of multiplication by a function <math>f</math>, goes to the operator of multiplication by the function <math>f</math>.		First observe that the map sends <math>C^\infty(M) \subset \mathcal{D}^1(M)</math> to <math>C^\infty(M) \subset B</math>, and is the identity restricted to that subset. In other words, the differential operator of multiplication by a function <math>f</math>, goes to the operator of multiplication by the function <math>f</math>.

	We now prove ~~some basic results about~~ <math>f_\nabla</math>:		We now prove that the map <math>\nabla \mapsto f_\nabla</math> is a <math>C^\infty(M)</math>-bimodule map from <math>D^1(M)</math> to <math>B</math>, i.e., left and right multiplication by <math>m(g)</math> can be ''pulled out'' of the <math>f_\nabla</math>:

	* <math>f_\nabla</math> is <math>\R</math>-bilinear: This is obvious.		* <math>f_\nabla</math> is <math>\R</math>-bilinear: This is obvious.
	* For any element <math>X + m(g)</math> in <math>\mathcal{D}^1(M)</math> and any <math>h \in C^\infty(M)</math>, we have <math>f_\nabla(m(g) \~~dot~~ (X + m(h))(s) = m(g)f_\nabla(X + m(h))(s)</math>. This essentially follows from the fact that a connection is [[tensorial map\|tensorial]] in the direction of differentiation:		* Left module map property: For any element <math>X + m(g)</math> in <math>\mathcal{D}^1(M)</math> and any <math>h \in C^\infty(M)</math>, we have <math>f_\nabla(m(g) \cdot (X + m(h))(s) = m(g) \cdot f_\nabla(X + m(h))(s)</math>. This essentially follows from the fact that a connection is [[tensorial map\|tensorial]] in the direction of differentiation:

	<math>f_\nabla(m(g) \~~dot~~ (X + m(h)))(s) = f_\nabla(gX + m(gh))(s) = \nabla_{gX}(s) + (gh)(s)= g\nabla_X(s) + (gh)(s) = g(\nabla_X(s) + hs) = m(g)f_\nabla(X + m(h))(s)</math>.		<math>f_\nabla(m(g) \cdot (X + m(h)))(s) = f_\nabla(gX + m(gh))(s) = \nabla_{gX}(s) + (gh)(s)= g\nabla_X(s) + (gh)(s) = g(\nabla_X(s) + hs) = m(g)f_\nabla(X + m(h))(s)</math>.

	* For any element <math>X + m(g)</math> in <math>\mathcal{D}^1(M)</math> and any <math>h \in C^\infty(M)</math>, we have <math>f_\nabla((X + m(h)) \~~dot~~ m(g))(s) = f_\nabla(X + m(h))(m(g)s)</math>. This essentially follows from the Leibniz rule property.		* For any element <math>X + m(g)</math> in <math>\mathcal{D}^1(M)</math> and any <math>h \in C^\infty(M)</math>, we have <math>(f_\nabla((X + m(h)) \cdot m(g))(s) = (f_\nabla(X + m(h)) \circ m(g))(s)</math>. This essentially follows from the Leibniz rule property.

	<math>f_\nabla((X + m(h)) \~~dot~~ m(g))(s) = f_\nabla(m(Xg) +g\nabla_X + m(gh))(s) = (Xg)(s) + g\nabla_X(s) + (gh)s = \nabla_X(gs) + (gh)(s)</math>.		<math>f_\nabla((X + m(h)) \cdot m(g))(s) = f_\nabla(m(Xg) +g\nabla_X + m(gh))(s) = (Xg)(s) + g\nabla_X(s) + (gh)s = \nabla_X(gs) + (gh)(s) = (f_\nabla(X + m(h)) \cdot m(g))(s)</math>.

	==References==		==References==

Vipul: /* From a connection to a module structure */

2009-07-24T00:19:25Z

From a connection to a module structure

← Older revision		Revision as of 00:19, 24 July 2009
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	* For any element <math>X + m(g)</math> in <math>\mathcal{D}^1(M)</math> and any <math>h \in C^\infty(M)</math>, we have <math>f_\nabla(m(g) \dot (X + m(h))(s) = m(g)f_\nabla(X + m(h))(s)</math>. This essentially follows from the fact that a connection is [[tensorial map\|tensorial]] in the direction of differentiation:		* For any element <math>X + m(g)</math> in <math>\mathcal{D}^1(M)</math> and any <math>h \in C^\infty(M)</math>, we have <math>f_\nabla(m(g) \dot (X + m(h))(s) = m(g)f_\nabla(X + m(h))(s)</math>. This essentially follows from the fact that a connection is [[tensorial map\|tensorial]] in the direction of differentiation:

	<math>f_\nabla(m(g) \dot (X + m(h)))(s) = f_\nabla(gX + m(gh))(s) = \nabla_{gX}(s) + (gh)(s)= g\nabla_X(s) + (gh)(s) = g(\nabla_X(s) + hs)</math>.		<math>f_\nabla(m(g) \dot (X + m(h)))(s) = f_\nabla(gX + m(gh))(s) = \nabla_{gX}(s) + (gh)(s)= g\nabla_X(s) + (gh)(s) = g(\nabla_X(s) + hs) = m(g)f_\nabla(X + m(h))(s)</math>.

	* For any element <math>X + m(g)</math> in <math>\mathcal{D}^1(M)</math> and any <math>h \in C^\infty(M)</math>, we have <math>f_\nabla((X + m(h)) \dot m(g))(s) = f_\nabla(X + m(h))(m(g)s)</math>. This essentially follows from the Leibniz rule property.		* For any element <math>X + m(g)</math> in <math>\mathcal{D}^1(M)</math> and any <math>h \in C^\infty(M)</math>, we have <math>f_\nabla((X + m(h)) \dot m(g))(s) = f_\nabla(X + m(h))(m(g)s)</math>. This essentially follows from the Leibniz rule property.

Vipul at 00:18, 24 July 2009

2009-07-24T00:18:25Z

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	==Proof==		==Proof==

	~~We start with~~ a connection ~~<math>\nabla</math> on <math>E</math> and show how <math>\nabla</math> naturally equips <math>E</math> with the structure of~~ a module ~~over <math>\mathcal{C}(M)</math>.~~		===From a connection to a module structure===

	~~Let <math>D^1(M)</math> denote~~ the ~~[[Lie algebra of first-order differential operators]] of <math>M</math>. For now, we're thinking of <math>D^1(M)</math>~~ as ~~a <math>C^\infty(M)</math>-bimodule. Let <math>B</math> be the algebra of all smooth fiber-preserving linear maps from <math>\Gamma(E)</math> to <math>\Gamma(E)</math>.~~		The outline of the proof is as follows:

	A connection gives a ~~map:~~		* We first show that a connection gives an action of the first-order differentiable operators on the space of sections.
			* Next, we show that the Leibniz rule property of connections allows us to extend this to a well-defined action of the connection algebra.

	<math>D^1(M) \~~to B~~</math>		'''Given''': A manifold <math>M</math>, a vector bundle <math>E</math> over <math>M</math>, a connection <math>\nabla</math> on <math>E</math>. <math>B</math> is the algebra of smooth fiber-preserving maps from <math>\Gamma(E)</math> to <math>\Gamma(E)</math>. <math>\mathcal{D}^1(M)</math> is the Lie algebra of first-order differential operators on <math>M</math> and <math>\mathcal{C}(M)</math> is the connection algebra on <math>M</math>.

	~~as follows~~:		'''To prove''': <math>\nabla</math> gives rise to a homomorphism from <math>\mathcal{C}(M)</math> to <math>B</math>.

	<math>X \~~mapsto (s \mapsto \nabla_X(s))~~</math>		'''Proof''': <math>\nabla</math> gives rise to a map:

	~~First observe that the map sends~~ <math>C^\~~infty(M) \subset~~ D^1(M)~~</math>~~ to ~~<math>C^\infty(M) \subset~~ B</math>, and is the identity restricted to that subset. In other words, the differential operator of multiplication by a function <math>f</math>, goes to the operator of multiplication by the function <math>f</math>.		<math>f_\nabla: D^1(M) \to B</math>

	~~We now argue that the map is in fact a <math>C^\infty(M)</math>-bimodule map. The fact that it is a left <math>C^\infty(M)</math>-module map~~ follows ~~from the fact that~~:		as follows:

	<math>\~~nabla_{fX}~~ = f \nabla_X</math>		<math>f_\nabla(X+m(g)) = s \mapsto \nabla_X(s) + (gs)</math>.

	~~while~~ the ~~fact that it is a right~~ <math>C^\infty(M)</math>~~-module map follows from~~ the ~~Leibniz rule~~.		First observe that the map sends <math>C^\infty(M) \subset \mathcal{D}^1(M)</math> to <math>C^\infty(M) \subset B</math>, and is the identity restricted to that subset. In other words, the differential operator of multiplication by a function <math>f</math>, goes to the operator of multiplication by the function <math>f</math>.

	~~The upshot is that the induced map~~:		We now prove some basic results about <math>f_\nabla</math>:

	<math>D^1(M) \~~to B~~</math>		* <math>f_\nabla</math> is <math>\R</math>-bilinear: This is obvious.
			* For any element <math>X + m(g)</math> in <math>\mathcal{D}^1(M)</math> and any <math>h \in C^\infty(M)</math>, we have <math>f_\nabla(m(g) \dot (X + m(h))(s) = m(g)f_\nabla(X + m(h))(s)</math>. This essentially follows from the fact that a connection is [[tensorial map\|tensorial]] in the direction of differentiation:

	~~is a~~ <math>C^\~~infty~~(M)~~</math>-bimodule map, and hence extends to a <math>~~\~~R</math>-linear homomorphism from the tensor algebra of <math>D^1~~(M)~~</math> over <math>C^~~\~~infty~~(M)~~</math> to <math>B</math>. Clearly, <math>m~~(1) ~~- 1~~</math> ~~acts trivially, so we obtain a homomorphism:~~		<math>f_\nabla(m(g) \dot (X + m(h)))(s) = f_\nabla(gX + m(gh))(s) = \nabla_{gX}(s) + (gh)(s)= g\nabla_X(s) + (gh)(s) = g(\nabla_X(s) + hs)</math>.

	<math>\mathcal{C}(M) \~~to B~~</math>		* For any element <math>X + m(g)</math> in <math>\mathcal{D}^1(M)</math> and any <math>h \in C^\infty(M)</math>, we have <math>f_\nabla((X + m(h)) \dot m(g))(s) = f_\nabla(X + m(h))(m(g)s)</math>. This essentially follows from the Leibniz rule property.

	~~giving~~ <math>\~~Gamma~~(E)~~</math> the structure of a module over the algebra <math>~~\~~mathcal{C}~~(M)</math>.		<math>f_\nabla((X + m(h)) \dot m(g))(s) = f_\nabla(m(Xg) +g\nabla_X + m(gh))(s) = (Xg)(s) + g\nabla_X(s) + (gh)s = \nabla_X(gs) + (gh)(s)</math>.

	==References==		==References==

Vipul: 6 revisions

2008-05-18T19:35:27Z

6 revisions

← Older revision	Revision as of 19:35, 18 May 2008
(No difference)

Vipul at 20:43, 6 April 2008

2008-04-06T20:43:34Z

← Older revision		Revision as of 20:43, 6 April 2008
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	==Statement==		==Statement==

	Let <math>E</math> be a [[vector bundle]] over a [[differential manifold]] <math>M</math>. Then, a [[connection]] on <math>E</math> is equivalent to giving <math>\mathcal{E}</math> (the [[sheaf of sections of a vector bundle\|sheaf of sections]] of <math>E</math>) the structure of a module over the [[connection ~~algebra~~]] over <math>M</math>.		Let <math>E</math> be a [[vector bundle]] over a [[differential manifold]] <math>M</math>. Then, a [[connection]] on <math>E</math> is equivalent to giving <math>\Gamma(E)</math> (the vector space of sections of <math>E</math>) the structure of a module over the [[connection algebra]] of <math>M</math>. Equivalently, it gives <math>\mathcal{E}</math> (the [[sheaf of sections of a vector bundle\|sheaf of sections]] of <math>E</math>) the structure of a module over the [[sheaf of connection algebras]] over <math>M</math>.

	==Definitions used==		==Definitions used==

Vipul at 20:42, 6 April 2008

2008-04-06T20:42:51Z

← Older revision		Revision as of 20:42, 6 April 2008
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	==Statement==		==Statement==

	Let <math>E</math> be a [[vector bundle]] over a [[differential manifold]] <math>M</math>. Then, a [[connection]] on <math>E</math> is equivalent to giving <math>E</math> the structure of a module over the [[connection algebra]] over <math>M</math>.		Let <math>E</math> be a [[vector bundle]] over a [[differential manifold]] <math>M</math>. Then, a [[connection]] on <math>E</math> is equivalent to giving <math>\mathcal{E}</math> (the [[sheaf of sections of a vector bundle\|sheaf of sections]] of <math>E</math>) the structure of a module over the [[connection algebra]] over <math>M</math>.

	==Definitions used==		==Definitions used==
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	<math>\mathcal{C}(M) \to B</math>		<math>\mathcal{C}(M) \to B</math>

	giving <math>E</math> the structure of a module over <math>\mathcal{C}(M)</math>.		giving <math>\Gamma(E)</math> the structure of a module over the algebra <math>\mathcal{C}(M)</math>.

	==References==		==References==

Vipul: /* Textbook references */

2008-04-06T19:00:48Z

Textbook references

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	===Textbook references===		===Textbook references===

	* {{booklink\|Globalcalculus}}		* {{booklink\|Globalcalculus}}, Page 64

Vipul at 19:00, 6 April 2008

2008-04-06T19:00:06Z

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	giving <math>E</math> the structure of a module over <math>\mathcal{C}(M)</math>.		giving <math>E</math> the structure of a module over <math>\mathcal{C}(M)</math>.

			==References==

			===Textbook references===

			* {{booklink\|Globalcalculus}}

Vipul: /* Proof */

2008-04-05T21:50:33Z

Proof

← Older revision		Revision as of 21:50, 5 April 2008
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	We start with a connection <math>\nabla</math> on <math>E</math> and show how <math>\nabla</math> naturally equips <math>E</math> with the structure of a module over <math>\mathcal{C}(M)</math>.		We start with a connection <math>\nabla</math> on <math>E</math> and show how <math>\nabla</math> naturally equips <math>E</math> with the structure of a module over <math>\mathcal{C}(M)</math>.

	~~First, observe that a connection gives a rule for~~ the [[Lie algebra of first-order differential operators]] to ~~''act'' on~~ <math>E</math>, ~~hence~~ the ~~tensor algebra generated~~ by ~~it as~~ a ~~vector space~~, ~~acts on~~ <math>E</math>. We ~~need~~ to ~~check that under this action~~ <math>m(1) - 1</math> acts trivially on <math>E</math>.		Let <math>D^1(M)</math> denote the [[Lie algebra of first-order differential operators]] of <math>M</math>. For now, we're thinking of <math>D^1(M)</math> as a <math>C^\infty(M)</math>-bimodule. Let <math>B</math> be the algebra of all smooth fiber-preserving linear maps from <math>\Gamma(E)</math> to <math>\Gamma(E)</math>.

			A connection gives a map:

			<math>D^1(M) \to B</math>

			as follows:

			<math>X \mapsto (s \mapsto \nabla_X(s))</math>

			First observe that the map sends <math>C^\infty(M) \subset D^1(M)</math> to <math>C^\infty(M) \subset B</math>, and is the identity restricted to that subset. In other words, the differential operator of multiplication by a function <math>f</math>, goes to the operator of multiplication by the function <math>f</math>.

			We now argue that the map is in fact a <math>C^\infty(M)</math>-bimodule map. The fact that it is a left <math>C^\infty(M)</math>-module map follows from the fact that:

			<math>\nabla_{fX} = f \nabla_X</math>

			while the fact that it is a right <math>C^\infty(M)</math>-module map follows from the Leibniz rule.

			The upshot is that the induced map:

			<math>D^1(M) \to B</math>

			is a <math>C^\infty(M)</math>-bimodule map, and hence extends to a <math>\R</math>-linear homomorphism from the tensor algebra of <math>D^1(M)</math> over <math>C^\infty(M)</math> to <math>B</math>. Clearly, <math>m(1) - 1</math> acts trivially, so we obtain a homomorphism:

			<math>\mathcal{C}(M) \to B</math>

			giving <math>E</math> the structure of a module over <math>\mathcal{C}(M)</math>.