Monoidal functor - Revision history

en>IkamusumeFan: Use SVG diagrams instead of PNGs

2014-11-22T08:23:17Z

Use SVG diagrams instead of PNGs

← Older revision		Revision as of 10:23, 22 November 2014
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	~~{{Orphan\|date=March 2013}}~~		Wilber Berryhill is what his wife loves to call him and he totally loves this name. Invoicing is what I do. To play lacross is something he would never give up. Her family members lives in Ohio but her husband wants them to transfer.<br><br>my site; [http://Www.weddingwall.Com.au/groups/easy-advice-for-successful-personal-development-today/ clairvoyants]

	~~In [[mathematics]], a '''Poisson ring'''~~ is ~~a [[commutative ring]] on which an [[anticommutativity\|anticommutative]]~~ and ~~[[distributivity\|distributive]] [[binary operation]] <math>[\cdot,\cdot]</math> satisfying the [[Jacobi identity]] and the [[product rule]]~~ is ~~defined~~. ~~Such an operation~~ is ~~then known as the [[Poisson bracket]] of the Poisson ring~~.

	~~Many important operations and results of [[symplectic geometry]] and [[Hamiltonian mechanics]] may be formulated~~ in ~~terms of the Poisson bracket and, hence, apply~~ to ~~[[Poisson algebra]]s as well~~. This observation is important in studying the [[classical limit]] of [[quantum mechanics]]—the [[non-commutative algebra]] of [[Operator (mathematics)\|operators]] on a [[Hilbert space]] has the Poisson algebra of functions on a [[symplectic manifold]] as a singular limit, and properties of the non-commutative algebra pass over to corresponding properties of the Poisson algebra.

	~~==Definition==~~
	~~The Poisson bracket must satisfy the identities~~
	* <math>[f,g] = -[g,f]<~~/math~~> ~~(skew symmetry)~~
	* <~~math~~>[~~f + g, h] = [f,h] + [g,h] <~~/~~math> (distributivity)~~
	* <math>[fg,h] = f[g,h] + [f,h]g</~~math> ([[derivation (abstract algebra)\|derivation]])~~
	* <math>[f,[g,h]] + [g,[h,f]] + [h,[f,g]] = 0</math> ([[Jacobi identity]])
	~~for all <math>f,g,h</math> in the ring~~.

	~~A [[Poisson algebra]] is a Poisson ring that is also an [[algebra over a field]]~~. ~~In this case, add the extra requirement~~

	~~:<math>[sf,g] = s[f,g]</math>~~

	~~for all scalars ''s''~~.

	~~For each ''g'' in a Poisson ring ''A'', the operation <math>ad_g<~~/~~math> defined as <math>ad_g(f) = [f,g]<~~/~~math> is a [[derivation (abstract algebra)\|derivation]]. If the set <math>\{ ad_g \| g \in A \}</math> generates the set of derivations of ''A'', then ''A'' is said to be '''non~~-~~degenerate'''.~~

	~~If a non~~-degenerate Poisson ring is [[ring isomorphism\|isomorphic as a commutative ring]] to the [[algebra of smooth functions]] on a manifold ''M'', then ''M'' must be a [[symplectic manifold]] and <math>[\cdot,\cdot]</~~math> is the Poisson bracket defined by the [[symplectic form]].~~

	~~==References==~~
	* {{planetmath reference\|id=6422\|title=If the algebra of functions on a manifold is a Poisson ring then the manifold is symplectic}}

	~~{{PlanetMath attribution\|id=6414\|title=Poisson Ring}}~~

	~~{{DEFAULTSORT:Poisson Ring}}~~
	~~[[Category:Ring theory]]~~
	~~[[Category:Symplectic geometry]~~]

72.93.32.62: It is unclear what "properties" was doing there.

2013-12-09T00:25:56Z

It is unclear what "properties" was doing there.

← Older revision		Revision as of 02:25, 9 December 2013
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	~~Jayson Berryhill~~ is ~~how I'm~~ known as and ~~my spouse doesn't like it at~~ all. ~~Playing badminton~~ is a ~~thing~~ that he is ~~totally addicted to. I am presently~~ a ~~journey agent~~. ~~Alaska is~~ the ~~only place I~~'~~ve been residing~~ in ~~but now I~~'~~m contemplating other choices.~~<br><br>~~Also visit my blog post;~~ [~~http:~~//~~conniecolin~~.~~com~~/~~xe/community/24580 free psychic~~]		{{Orphan\|date=March 2013}}

			In [[mathematics]], a '''Poisson ring''' is a [[commutative ring]] on which an [[anticommutativity\|anticommutative]] and [[distributivity\|distributive]] [[binary operation]] <math>[\cdot,\cdot]</math> satisfying the [[Jacobi identity]] and the [[product rule]] is defined. Such an operation is then known as the [[Poisson bracket]] of the Poisson ring.

			Many important operations and results of [[symplectic geometry]] and [[Hamiltonian mechanics]] may be formulated in terms of the Poisson bracket and, hence, apply to [[Poisson algebra]]s as well. This observation is important in studying the [[classical limit]] of [[quantum mechanics]]—the [[non-commutative algebra]] of [[Operator (mathematics)\|operators]] on a [[Hilbert space]] has the Poisson algebra of functions on a [[symplectic manifold]] as a singular limit, and properties of the non-commutative algebra pass over to corresponding properties of the Poisson algebra.

			==Definition==
			The Poisson bracket must satisfy the identities
			* <math>[f,g] = -[g,f]</math> (skew symmetry)
			* <math>[f + g, h] = [f,h] + [g,h] </math> (distributivity)
			* <math>[fg,h] = f[g,h] + [f,h]g</math> ([[derivation (abstract algebra)\|derivation]])
			* <math>[f,[g,h]] + [g,[h,f]] + [h,[f,g]] = 0</math> ([[Jacobi identity]])
			for all <math>f,g,h</math> in the ring.

			A [[Poisson algebra]] is a Poisson ring that is also an [[algebra over a field]]. In this case, add the extra requirement

			:<math>[sf,g] = s[f,g]</math>

			for all scalars ''s''.

			For each ''g'' in a Poisson ring ''A'', the operation <math>ad_g</math> defined as <math>ad_g(f) = [f,g]</math> is a [[derivation (abstract algebra)\|derivation]]. If the set <math>\{ ad_g \| g \in A \}</math> generates the set of derivations of ''A'', then ''A'' is said to be '''non-degenerate'''.

			If a non-degenerate Poisson ring is [[ring isomorphism\|isomorphic as a commutative ring]] to the [[algebra of smooth functions]] on a manifold ''M'', then ''M'' must be a [[symplectic manifold]] and <math>[\cdot,\cdot]</math> is the Poisson bracket defined by the [[symplectic form]].

			==References==
			* {{planetmath reference\|id=6422\|title=If the algebra of functions on a manifold is a Poisson ring then the manifold is symplectic}}

			{{PlanetMath attribution\|id=6414\|title=Poisson Ring}}

			{{DEFAULTSORT:Poisson Ring}}
			[[Category:Ring theory]]
			[[Category:Symplectic geometry]]

en>Daniel5Ko: /* Definition */ Typography.

2012-08-23T23:42:10Z

Definition: Typography.

New page

Jayson Berryhill is how I'm known as and my spouse doesn't like it at all. Playing badminton is a thing that he is totally addicted to. I am presently a journey agent. Alaska is the only place I've been residing in but now I'm contemplating other choices.<br><br>Also visit my blog post; [http://conniecolin.com/xe/community/24580 free psychic]