Algebraic expression - Revision history

en>Jochen Burghardt: /* top */ e is a non-algebraic constant; deleted 2nd 'numbers'; 'variety' is understatement (almost all numbers are non-algebraic); don't know what 'higher-order expr.s' shoyld mean

2014-12-20T18:34:58Z

top: e is a non-algebraic constant; deleted 2nd 'numbers'; 'variety' is understatement (almost all numbers are non-algebraic); don't know what 'higher-order expr.s' shoyld mean

← Older revision		Revision as of 20:34, 20 December 2014
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en>Bgwhite: Broken tag using AWB (9957)

2014-03-04T01:49:26Z

Broken tag using AWB (9957)

← Older revision		Revision as of 03:49, 4 March 2014
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	~~In [[mathematics]]~~, ~~a '''Lie bialgebra''' is the Lie-theoretic case of a~~ [~~[bialgebra]]~~: ~~it's a set with a [[Lie algebra]~~] and ~~a [[Lie coalgebra]] structure which are compatible~~.		fee to watch your webinar at their convenience. Once it is in place, [http://www.comoganhardinheiro101.com/?p=14 ganhando dinheiro na internet] promotion and possibly answering questions will be your como conseguir dinheiro only tasks. <br><br><br>

	It is ~~a [[bialgebra]] where~~ the ~~[[comultiplication]] is [[skew-symmetric]] and satisfies a dual [[Jacobi identity]], so~~ that the ~~dual vector space is a~~ [~~[Lie algebra]], whereas the comultiplication is a 1-[[cocycle]~~], so ~~that the multiplication and comultiplication~~ are ~~compatible. The cocycle condition implies that~~, ~~in practice, one studies only classes~~ of ~~bialgebras that are cohomologous to a Lie bialgebra on a coboundary~~.		It's easy to make money online. There is truth to the fact that you can start making money on the Internet as soon as you're done with this [http://ganhandodinheironainternet.comoganhardinheiro101.com/ article]. After all, so many others are making money online, why not you? Keep your mind open and you can make a lot of money. As you [http://www.comoganhardinheiro101.com/?p=66 ganhar dinheiro pela internet] can see, there are many ways to approach the world of online income. With various streams of income available, you are sure to find one, or two, that can help you with your income needs. Take this information to heart, put it to use and build your own online success story. <br><br><br>Make money online by selling your talents. Good music is always in demand and with today's technological advances, anyone with musical talent can make music and offer it for sale [http://www.comoganhardinheiro101.com/como-ganhar-dinheiro-pela-internet/ como ganhar dinheiro] to a broad audience. By setting up your own website and using social media for promotion, you can share your music with others and sell downloads with a free PayPal account.<br><br>Getting paid money to work online isn't the easiest thing to do in the world, but it is possible. If this is something you wish to work with, then the tips presented above should have helped you. Take some time, do things the right way and then you can succeed. Start your online [http://www.comoganhardinheiro101.com/category/opcoes-binarias/ como ficar rico] earning today by following the great advice discussed in this article. Here is more information on [http://comoficarrico.comoganhardinheiro101.com/ ganhar dinheiro] look into http://comoficarrico.comoganhardinheiro101.com/ Earning money is not as hard as it may seem, you just need to know [http://comoganhardinheironainternet.comoganhardinheiro101.com/ como conseguir dinheiro] how to get started. By choosing to put your right foot forward, you are heading off to a great start earning money to make ends meet.

	~~They are also called '''Poisson-Hopf algebras'''~~, ~~and~~ are the ~~[[Lie algebra]]~~ of ~~a [[Poisson-Lie group]]~~.

	~~Lie bialgebras occur naturally in the study~~ of ~~the [[Yang-Baxter equation]]s.~~

	~~==Definition==~~
	~~More precisely~~, ~~comultiplication on the algebra~~, ~~<math>\delta:\mathfrak{g} \~~to ~~\mathfrak{g} \otimes \mathfrak{g}</math>, is called the '''cocommutator'''~~, and ~~must satisfy two properties~~. ~~The dual~~

	:<~~math~~>\delta^:\mathfrak{g}^ \otimes \mathfrak{g}^* \to \mathfrak{g}^*<~~/math~~>
	~~must be a Lie bracket on~~ <~~math>\mathfrak{g}^*</math~~>, and it ~~must be a cocycle~~:

	~~:<math>\delta([X,Y]) = \left(~~
	~~\operatorname{ad}_X \otimes 1 + 1 \otimes \operatorname{ad}_X~~
	~~\right) \delta(Y)~~ - ~~\left(~~
	~~\operatorname{ad}_Y \otimes 1 + 1 \otimes \operatorname{ad}_Y~~
	~~\right) \delta(X)~~
	</~~math>~~

	~~where <math>\operatorname{ad}_XY=[X,Y~~]~~</math> is the adjoint.~~

	~~==Relation~~ to ~~Poisson-Lie groups==~~
	~~Let ''G'' be~~ a ~~Poisson-Lie group~~, with ~~<math>f_1,f_2 \in C^\infty(G)</math> being two smooth functions on the group manifold~~. ~~Let~~ <~~math~~>~~\xi= (df)_e~~<~~/math~~> be the ~~differential at~~ the ~~identity element~~. ~~Clearly~~, ~~<math>\xi \in \mathfrak{g}^</math>~~. ~~The [[Poisson structure]] on~~ the ~~group~~ then ~~induces a bracket on <math>\mathfrak{g}^</math>, as~~

	:~~<math>[\xi_1,\xi_2]=(d\{f_1,f_2\})_e\,<~~/~~math>~~

	~~where <math>\{,\}<~~/~~math> is the [[Poisson bracket]]~~. ~~Given <math>\eta<~~/~~math> be the [[Poisson bivector]] on the manifold, define <math>\eta^R<~~/~~math> to be the right~~-~~translate of~~ the ~~bivector to the identity element~~ in ~~''G''~~. ~~Then one has that~~

	~~:<math>\eta^R:G\to \mathfrak{g} \otimes \mathfrak{g}</math>~~

	~~The cocommutator~~ is ~~then the tangent map:~~

	:~~<math>\delta = T_e \eta^R\,<~~/~~math>~~

	~~so that~~

	~~:<math>[\xi_1,\xi_2]= \delta^*(\xi_1 \otimes \xi_2)<~~/~~math>~~

	~~is the dual of the cocommutator~~.

	~~==See also==~~

	*[[Lie coalgebra]]
	*[[Manin triple]]

	~~==References==~~
	* H.-D. ~~Doebner~~, J.-D. ~~Hennig, eds, ''Quantum groups, Proceedings of the 8th International Workshop on Mathematical Physics, Arnold Sommerfeld Institute, Claausthal, FRG, 1989'', Springer-Verlag Berlin, ISBN 3-540-53503-9~~.
	* Vyjayanthi Chari and Andrew Pressley, ~~''A Guide~~ to ~~Quantum Groups'', (1994), Cambridge University Press, Cambridge ISBN 0-521-55884-0~~.

	~~[[Category:Lie algebras]]~~
	~~[[Category:Coalgebras]]~~
	~~[[Category:Symplectic geometry]]~~

2601:2:2180:287:60C6:49EA:8296:886D at 23:28, 3 February 2014

2014-02-03T23:28:53Z

← Older revision		Revision as of 01:28, 4 February 2014
Line 1:		Line 1:
	Ed is ~~what individuals contact me~~ and ~~my wife doesn't like it at all~~. ~~Alaska~~ is ~~exactly~~ where ~~I've always been residing. The preferred pastime for him~~ and ~~his children~~ is ~~style~~ and he'~~ll be beginning something else along with it~~. ~~Distributing production~~ is ~~where my main income comes from~~ and ~~it's something I truly appreciate~~.<br><br>~~my website~~ :: ~~best psychics~~, [~~http:~~//1.~~234~~.36.~~240~~/~~fxac~~/~~m001_2~~/~~7330 http:~~//1.~~234.36.240~~/~~fxac~~/~~m001_2~~/~~7330~~],		In [[mathematics]], a '''Lie bialgebra''' is the Lie-theoretic case of a [[bialgebra]]: it's a set with a [[Lie algebra]] and a [[Lie coalgebra]] structure which are compatible.

			It is a [[bialgebra]] where the [[comultiplication]] is [[skew-symmetric]] and satisfies a dual [[Jacobi identity]], so that the dual vector space is a [[Lie algebra]], whereas the comultiplication is a 1-[[cocycle]], so that the multiplication and comultiplication are compatible. The cocycle condition implies that, in practice, one studies only classes of bialgebras that are cohomologous to a Lie bialgebra on a coboundary.

			They are also called '''Poisson-Hopf algebras''', and are the [[Lie algebra]] of a [[Poisson-Lie group]].

			Lie bialgebras occur naturally in the study of the [[Yang-Baxter equation]]s.

			==Definition==
			More precisely, comultiplication on the algebra, <math>\delta:\mathfrak{g} \to \mathfrak{g} \otimes \mathfrak{g}</math>, is called the '''cocommutator''', and must satisfy two properties. The dual

			:<math>\delta^:\mathfrak{g}^ \otimes \mathfrak{g}^* \to \mathfrak{g}^*</math>
			must be a Lie bracket on <math>\mathfrak{g}^*</math>, and it must be a cocycle:

			:<math>\delta([X,Y]) = \left(
			\operatorname{ad}_X \otimes 1 + 1 \otimes \operatorname{ad}_X
			\right) \delta(Y) - \left(
			\operatorname{ad}_Y \otimes 1 + 1 \otimes \operatorname{ad}_Y
			\right) \delta(X)
			</math>

			where <math>\operatorname{ad}_XY=[X,Y]</math> is the adjoint.

			==Relation to Poisson-Lie groups==
			Let ''G'' be a Poisson-Lie group, with <math>f_1,f_2 \in C^\infty(G)</math> being two smooth functions on the group manifold. Let <math>\xi= (df)_e</math> be the differential at the identity element. Clearly, <math>\xi \in \mathfrak{g}^</math>. The [[Poisson structure]] on the group then induces a bracket on <math>\mathfrak{g}^</math>, as

			:<math>[\xi_1,\xi_2]=(d\{f_1,f_2\})_e\,</math>

			where <math>\{,\}</math> is the [[Poisson bracket]]. Given <math>\eta</math> be the [[Poisson bivector]] on the manifold, define <math>\eta^R</math> to be the right-translate of the bivector to the identity element in ''G''. Then one has that

			:<math>\eta^R:G\to \mathfrak{g} \otimes \mathfrak{g}</math>

			The cocommutator is then the tangent map:

			:<math>\delta = T_e \eta^R\,</math>

			so that

			:<math>[\xi_1,\xi_2]= \delta^*(\xi_1 \otimes \xi_2)</math>

			is the dual of the cocommutator.

			==See also==

			*[[Lie coalgebra]]
			*[[Manin triple]]

			==References==
			* H.-D. Doebner, J.-D. Hennig, eds, ''Quantum groups, Proceedings of the 8th International Workshop on Mathematical Physics, Arnold Sommerfeld Institute, Claausthal, FRG, 1989'', Springer-Verlag Berlin, ISBN 3-540-53503-9.
			* Vyjayanthi Chari and Andrew Pressley, ''A Guide to Quantum Groups'', (1994), Cambridge University Press, Cambridge ISBN 0-521-55884-0.

			[[Category:Lie algebras]]
			[[Category:Coalgebras]]
			[[Category:Symplectic geometry]]

99.9.84.219: /* See also */

2012-09-02T15:33:21Z