Arturo Rodas - Revision history

en>Ser Amantio di Nicolao: /* Scores */recat using AWB

2014-09-13T06:11:40Z

Scores: recat using AWB

← Older revision		Revision as of 08:11, 13 September 2014
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	~~{{Lie groups \|Semi-simple}}~~

	In the [[mathematical]] field of [[Lie theory]], a '''split Lie algebra''' is a pair <math>(\mathfrak{g}, \mathfrak{h})</math> where <math>\mathfrak{g}</math> is a [[Lie algebra]] and <math>\mathfrak{h} < \mathfrak{g}</math> is a '''splitting [[Cartan subalgebra]]''', where "splitting" means that for all <math>x \in \mathfrak{h},</math> <math>\operatorname{ad}_{\mathfrak{g}} x</math> is [[triangularizable]]. If a Lie algebra admits a splitting, it is called a '''splittable Lie algebra'''.<ref>{{Harv \|Bourbaki \|2005 \|loc=Chapter VIII, Section 2: Root System of a Split Semi-Simple Lie Algebra, [http://books.google.com/books?id=Yh1RHnYCDNsC&pg=PA77 p. 77]}}</ref> Note that for reductive Lie algebras, the Cartan subalgebra is required to contain the center.

	~~Over an [[algebraically closed field]] such as the [[complex numbers]], all [[semisimple Lie algebra]]s are splittable (indeed~~, the ~~Cartan subalgebra acts not only by triangularizable matrices but a fortiori by diagonalizable ones)~~ and ~~all splittings are conjugate; thus split Lie algebras are of most interest for non~~-~~algebraically closed fields.~~		By investing in a premium Word - Press theme, you're investing in the future of your website. This means you can setup your mailing list and auto-responder on your wordpress site and then you can add your subscription form to any other blog, splash page, capture page or any other site you like. Should you go with simple HTML or use a platform like Wordpress. In the recent years, there has been a notable rise in the number of companies hiring Indian Word - Press developers. In case you have almost any issues relating to where by in addition to how to utilize [http://www.ase.re/wordpress_dropbox_backup_1267139 wordpress dropbox backup], you possibly can call us from our web page. You can easily customize the titles of the posts in Word - Press blog in a way that only title comes in the new post link and not the date or category of posts. <br><br>Most Word - Press web developers can provide quality CMS website solutions and they price their services at reasonable rates. But as expected the level of support you get with them can be hit or miss based on the developer's free time and desire. With the free Word - Press blog, you have the liberty to come up with your own personalized domain name. E-commerce websites are meant to be buzzed with fresh contents, graphical enhancements, and functionalities. By using Word - Press, you can develop very rich, user-friendly and full-functional website. <br><br>Here are a few reasons as to why people prefer Word - Press over other software's. When a business benefits from its own domain name and a tailor-made blog, the odds of ranking higher in the search engines and being visible to a greater number of people is more likely. Possibly the most downloaded Word - Press plugin, the Google XML Sitemaps plugin but not only automatically creates a site map linking to everyone your pages and posts, it also notifies Google, Bing, Yahoo, and Ask. Our skilled expertise, skillfulness and excellence have been well known all across the world. Purchase these from our site, or bring your own, it doesn't matter, we will still give you free installation and configuration. <br><br>Word - Press has plenty of SEO benefits over Joomla and Drupal. As an example, if you are promoting a product that cures hair-loss, you most likely would not wish to target your adverts to teens. Exacting subjects in reality must be accumulated in head ahead of planning on your high quality theme. Can you imagine where you would be now if someone in your family bought an original painting from van Gogh during his lifetime. Wordpress template is loaded with lots of prototype that unite graphic features and content area. <br><br>Someone with a basic knowledge of setting up a website should be able to complete the process in a couple of minutes however even basic users should find they are able to complete the installation in around 20 minutes by following the step by step guide online. By using Word - Press MLM websites or blogs, an online presence for you and your MLM company can be created swiftly and simply. Must being, it's beneficial because I don't know about you, but loading an old website on a mobile, having to scroll down, up, and sideways' I find links being clicked and bounced around like I'm on a freaking trampoline. Word - Press is the most popular personal publishing platform which was launched in 2003. However, if you're just starting out your blog site or business site, you can still search for an ideal theme for it without breaking your bank account.

	~~Split Lie algebras are of interest both because they formalize the [[split real~~ form~~]] of a complex Lie algebra~~, ~~and because split semisimple Lie algebras (more generally~~, ~~split reductive Lie algebras) over~~ any ~~field share many properties~~ with semisimple Lie algebras over algebraically closed fields – having essentially the same representation theory, for instance – the splitting Cartan subalgebra playing the same role as the Cartan subalgebra plays over algebraically closed fields. ~~This is~~ the ~~approach followed in {{Harv\|Bourbaki\|2005}}~~, ~~for instance.~~

	~~== Properties ==~~
	* Over an algebraically closed field, all Cartan subalgebras are conjugate. Over a ~~non-algebraically closed fields, not all Cartan subalgebras are conjugate~~ in ~~general; however, in a splittable semisimple Lie algebra all ''splitting'' Cartan algebras are conjugate.~~
	* Over an algebraically closed field, all semisimple Lie algebras are splittable.
	* Over a non-~~algebraically closed field, there exist non-splittable semisimple Lie algebras~~.~~<ref>{{Harv\|Bourbaki\|2005\|loc=Chapter VIII, Section 2: Root System of a Split Semi-Simple Lie Algebra, Exercise 2 ''a''~~ [http://~~books~~.~~google~~.~~com~~/~~books?id=Yh1RHnYCDNsC&pg=PA77 p~~. ~~77]}}</ref>~~
	* In a ~~splittable Lie algebra, there ''may'' exist Cartan subalgebras~~ that ~~are~~ not ~~splitting~~.<~~ref~~>~~{{Harv\|Bourbaki\|2005\|loc=Chapter VIII, Section 2: Root System of a Split Semi~~-~~Simple Lie Algebra, Exercise 2 ''b'' [http://books~~.~~google.com/books?id=Yh1RHnYCDNsC&pg=PA77 p. 77]}}</ref>~~
	* Direct sums of ~~splittable Lie algebras~~ and ~~ideals in splittable Lie algebras are splittable~~.

	~~=== Split real Lie algebras ===~~
	~~{{see also\|Real form}}~~
	~~For a real Lie algebra~~, ~~splittable is equivalent~~ to ~~either of these conditions:<ref>{{Harv\|Onishchik\|Vinberg\|1994\|loc=p~~. ~~157}}</ref>~~
	* The real rank equals the complex rank.
	* The [[Satake diagram]] has neither black vertices nor arrows.

	~~Every complex semisimple Lie algebra has a unique (up~~ to ~~isomorphism) split real Lie algebra~~, ~~which is also semisimple~~, and ~~is simple if~~ and ~~only if the complex Lie algebra is~~.<~~ref~~>~~{{Harv\|Onishchik\|Vinberg\|1994\|loc=Theorem 4.4, p. 158}}~~<~~/ref~~>

	~~For real semisimple Lie algebras, split Lie algebras~~ are ~~opposite~~ to ~~[[compact Lie algebra]]~~s – the ~~corresponding Lie group~~ is ~~"as far as possible" from being compact~~.

	~~== Examples ==~~
	~~The split real forms for~~ the ~~complex semisimple Lie algebras are:<ref>{{Harv\|Onishchik\|Vinberg\|1994\|loc=p~~. ~~158}}</ref>~~
	* <math>A_n, ~~\mathfrak{sl}_{n+1}(\mathbf{C}): \mathfrak{sl}_{n+1}(\mathbf{R})</math>~~
	* <math>B_n, ~~\mathfrak{so}_{2n+1}(\mathbf{C}): \mathfrak{so}_{n~~,~~n+1}(\mathbf{R})</math>~~
	* <math>C_n, ~~\mathfrak{sp}_n(\mathbf{C}): \mathfrak{sp}_n(\mathbf{R})~~<~~/math~~>
	* <~~math~~>~~D_n~~, ~~\mathfrak{so}_{2n}(\mathbf{C}): \mathfrak{so}_{n~~,~~n}(\mathbf{R})~~<~~/math~~>
	* Exceptional Lie algebras: <~~math~~>~~E_6~~, ~~E_7, E_8, F_4~~, ~~G_2</math> have split real forms~~ ''~~E''I~~, ~~''E''V~~, ~~''E''VIII~~, '~~'F''~~I~~, ''G'~~'.
	~~These are~~ the ~~Lie algebras of the split real groups of the complex Lie groups~~.

	~~Note that for ''sl'' and ''sp~~,'~~' the real form is the real points of (the Lie algebra of) the same [[algebraic group]]~~, ~~while~~ for ~~''so'' one must use the split forms (of maximally indefinite index), as SO is compact~~.

	~~== See also ==~~
	* [[Compact Lie algebra]]
	* [[Real form]]
	* [[Split-complex number]]
	* [[Split orthogonal group]]

	~~== References ==~~
	~~{{reflist}}~~
	~~{{refbegin}}~~
	* {{Citation \| title=Elements of Mathematics: Lie Groups and Lie Algebras: Chapters 7–9 \| authorlink=Nicolas Bourbaki \| first=Nicolas \| last=Bourbaki \| chapter = VIII: Split Semi-simple Lie Algebras \| year = 2005 \| chapterurl = http://books.google.com/books?id=Yh1RHnYCDNsC&pg=PA69 }}
	* {{Citation \| title=Lie groups and Lie algebras III: structure of Lie groups and Lie algebras\|first1 = A. L. \| last1 = Onishchik \| first2 = Ėrnest Borisovich \| last2 = Vinberg \| year = 1994 \| chapter = 4.4: Split Real Semisimple Lie Algebras \| chapterurl = http://books.google.com/books?id=l8nJCNiIQAAC&pg=PA157 \| pages = 157–158 }}
	~~{{refend}}~~

	~~{{DEFAULTSORT:Split Lie Algebra}}~~
	~~[[Category:Properties of Lie algebras]]~~

en>Cyberbot II: Removing {{Blacklisted-links}}. No blacklisted links were found. (Peachy 2.0 (alpha 5))

2013-10-04T01:31:39Z

Removing {{Blacklisted-links}}. No blacklisted links were found. (Peachy 2.0 (alpha 5))

← Older revision		Revision as of 03:31, 4 October 2013
Line 1:		Line 1:
	~~Today~~, ~~there are several other types of web development and blogging software available to design and host your website blogs online~~ and that ~~too~~ in ~~minutes~~, ~~if not hours~~. ~~What I advise you do next~~ is ~~save the backup data file to~~ a ~~remote place like~~ a CD-~~ROM~~, ~~external disk drive if you have one or a provider such as Dropbox. If you enjoyed this information and you would such as to obtain even more facts concerning~~ [http://~~mmservice~~.dk/~~backup_plugin_327971 wordpress dropbox backup~~] ~~kindly browse through~~ the ~~internet site~~. ~~A pinch~~ of ~~tablet centric strategy can get your Word~~ - ~~Press site miles ahead~~ of ~~your competitors~~, ~~so here are few strategies that will give your Wordpress websites~~ and ~~blogs an edge~~ over ~~your competitors:. If you're using Wordpress and want to make your blog a "dofollow" blog~~, ~~meaning that links from your blog pass on~~ the ~~benefits of Google pagerank, you can install one of~~ the ~~many dofollow plugins available~~. This ~~particular wordpress plugin~~ is ~~essential for not only having~~ the ~~capability where you improve your position~~, ~~but to enhance your organic searches~~ for ~~your website~~. ~~<br><br>Luckily~~, ~~for Word~~ - ~~Press users~~, ~~WP Touch plugin transforms your site into an IPhone style theme. After~~ all, ~~Word - Press is free~~, ~~many of the enhancements for Word~~ - ~~Press like themes and plugins are also free~~, ~~and~~ there ~~is plenty of free information online about how to use Word~~ - ~~Press~~. ~~There are number of web services that offer Word press development across the world. Furthermore~~, ~~with the launch~~ of ~~Windows 7 Phone is the smart phone market nascent App. W3C compliant HTML and~~ a ~~good open source powered by Word~~ - ~~Press CMS site is regarded as the prime minister~~. <br><br>~~Digital photography is~~ a ~~innovative effort~~, ~~if you removethe stress to catch every position~~ and ~~viewpoint of~~ a ~~place~~, ~~you free yourself up~~ to ~~be more innovative and your outcomes will be much better~~. The ~~nominee in each category with~~ the ~~most votes was crowned the 2010 Parents Picks Awards WINNER and~~ has ~~been established as the best product, tip or place in that category~~. ~~Whether or not it's an viewers on your web page, your social media pages, or your web page, those who have~~ a ~~present and effective viewers of "fans" are best best for provide provides~~, ~~reductions~~, and ~~deals to help re-invigorate their viewers~~ and ~~add to their main point here~~. ~~Storing write-ups in advance would have~~ to ~~be neccessary with~~ the ~~auto blogs~~. ~~Websites using this content based strategy~~ are ~~always given top scores by Google~~. <br><br>It has become a more prevalent cause of infertility and the fertility clinic are having more and more couples with infertility problems. Find more information about Design To Wordpress here. To do this, ~~you should link your posts to other relevant posts that you've created. There are many advantages of hiring Wordpress developers for Wordpress project development~~:~~. Word - Press offers constant updated services and products~~, ~~that too~~, ~~absolutely free of cost.~~ <br><br>~~This advice is critical because you don~~'~~t want to waste too expensive time establishing your Word - Press blog~~ the ~~exact method~~. ~~It can run as plugin~~ and ~~you can still get to that whole database just in circumstance your webhost does not have a c - Panel area. While deciding couple should consider~~ the ~~expertise~~ of the ~~doctor,clinics success rate~~,the ~~costs~~ of ~~fertility treatment~~,~~including fertility tests~~ and ~~IVF costs~~ and ~~overall ones own financial budget~~. ~~with posts or testimonials updated as they are uploaded to a particular section of the website~~. ~~However, if you're just starting out your blog site or business site, you can still search for an ideal theme for it without breaking your bank account~~.		{{Lie groups \|Semi-simple}}

			In the [[mathematical]] field of [[Lie theory]], a '''split Lie algebra''' is a pair <math>(\mathfrak{g}, \mathfrak{h})</math> where <math>\mathfrak{g}</math> is a [[Lie algebra]] and <math>\mathfrak{h} < \mathfrak{g}</math> is a '''splitting [[Cartan subalgebra]]''', where "splitting" means that for all <math>x \in \mathfrak{h},</math> <math>\operatorname{ad}_{\mathfrak{g}} x</math> is [[triangularizable]]. If a Lie algebra admits a splitting, it is called a '''splittable Lie algebra'''.<ref>{{Harv \|Bourbaki \|2005 \|loc=Chapter VIII, Section 2: Root System of a Split Semi-Simple Lie Algebra, [http://books.google.com/books?id=Yh1RHnYCDNsC&pg=PA77 p. 77]}}</ref> Note that for reductive Lie algebras, the Cartan subalgebra is required to contain the center.

			Over an [[algebraically closed field]] such as the [[complex numbers]], all [[semisimple Lie algebra]]s are splittable (indeed, the Cartan subalgebra acts not only by triangularizable matrices but a fortiori by diagonalizable ones) and all splittings are conjugate; thus split Lie algebras are of most interest for non-algebraically closed fields.

			Split Lie algebras are of interest both because they formalize the [[split real form]] of a complex Lie algebra, and because split semisimple Lie algebras (more generally, split reductive Lie algebras) over any field share many properties with semisimple Lie algebras over algebraically closed fields – having essentially the same representation theory, for instance – the splitting Cartan subalgebra playing the same role as the Cartan subalgebra plays over algebraically closed fields. This is the approach followed in {{Harv\|Bourbaki\|2005}}, for instance.

			== Properties ==
			* Over an algebraically closed field, all Cartan subalgebras are conjugate. Over a non-algebraically closed fields, not all Cartan subalgebras are conjugate in general; however, in a splittable semisimple Lie algebra all ''splitting'' Cartan algebras are conjugate.
			* Over an algebraically closed field, all semisimple Lie algebras are splittable.
			* Over a non-algebraically closed field, there exist non-splittable semisimple Lie algebras.<ref>{{Harv\|Bourbaki\|2005\|loc=Chapter VIII, Section 2: Root System of a Split Semi-Simple Lie Algebra, Exercise 2 ''a'' [http://books.google.com/books?id=Yh1RHnYCDNsC&pg=PA77 p. 77]}}</ref>
			* In a splittable Lie algebra, there ''may'' exist Cartan subalgebras that are not splitting.<ref>{{Harv\|Bourbaki\|2005\|loc=Chapter VIII, Section 2: Root System of a Split Semi-Simple Lie Algebra, Exercise 2 ''b'' [http://books.google.com/books?id=Yh1RHnYCDNsC&pg=PA77 p. 77]}}</ref>
			* Direct sums of splittable Lie algebras and ideals in splittable Lie algebras are splittable.

			=== Split real Lie algebras ===
			{{see also\|Real form}}
			For a real Lie algebra, splittable is equivalent to either of these conditions:<ref>{{Harv\|Onishchik\|Vinberg\|1994\|loc=p. 157}}</ref>
			* The real rank equals the complex rank.
			* The [[Satake diagram]] has neither black vertices nor arrows.

			Every complex semisimple Lie algebra has a unique (up to isomorphism) split real Lie algebra, which is also semisimple, and is simple if and only if the complex Lie algebra is.<ref>{{Harv\|Onishchik\|Vinberg\|1994\|loc=Theorem 4.4, p. 158}}</ref>

			For real semisimple Lie algebras, split Lie algebras are opposite to [[compact Lie algebra]]s – the corresponding Lie group is "as far as possible" from being compact.

			== Examples ==
			The split real forms for the complex semisimple Lie algebras are:<ref>{{Harv\|Onishchik\|Vinberg\|1994\|loc=p. 158}}</ref>
			* <math>A_n, \mathfrak{sl}_{n+1}(\mathbf{C}): \mathfrak{sl}_{n+1}(\mathbf{R})</math>
			* <math>B_n, \mathfrak{so}_{2n+1}(\mathbf{C}): \mathfrak{so}_{n,n+1}(\mathbf{R})</math>
			* <math>C_n, \mathfrak{sp}_n(\mathbf{C}): \mathfrak{sp}_n(\mathbf{R})</math>
			* <math>D_n, \mathfrak{so}_{2n}(\mathbf{C}): \mathfrak{so}_{n,n}(\mathbf{R})</math>
			* Exceptional Lie algebras: <math>E_6, E_7, E_8, F_4, G_2</math> have split real forms ''E''I, ''E''V, ''E''VIII, ''F''I, ''G''.
			These are the Lie algebras of the split real groups of the complex Lie groups.

			Note that for ''sl'' and ''sp,'' the real form is the real points of (the Lie algebra of) the same [[algebraic group]], while for ''so'' one must use the split forms (of maximally indefinite index), as SO is compact.

			== See also ==
			* [[Compact Lie algebra]]
			* [[Real form]]
			* [[Split-complex number]]
			* [[Split orthogonal group]]

			== References ==
			{{reflist}}
			{{refbegin}}
			* {{Citation \| title=Elements of Mathematics: Lie Groups and Lie Algebras: Chapters 7–9 \| authorlink=Nicolas Bourbaki \| first=Nicolas \| last=Bourbaki \| chapter = VIII: Split Semi-simple Lie Algebras \| year = 2005 \| chapterurl = http://books.google.com/books?id=Yh1RHnYCDNsC&pg=PA69 }}
			* {{Citation \| title=Lie groups and Lie algebras III: structure of Lie groups and Lie algebras\|first1 = A. L. \| last1 = Onishchik \| first2 = Ėrnest Borisovich \| last2 = Vinberg \| year = 1994 \| chapter = 4.4: Split Real Semisimple Lie Algebras \| chapterurl = http://books.google.com/books?id=l8nJCNiIQAAC&pg=PA157 \| pages = 157–158 }}
			{{refend}}

			{{DEFAULTSORT:Split Lie Algebra}}
			[[Category:Properties of Lie algebras]]

en>Fixer88: /* Scores */

2012-05-24T20:33:06Z

Scores

New page

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