Spacetime algebra - Revision history

en>Rubenvb: added reference for spin connection to imaginary pseudoscalar

2014-06-02T09:30:19Z

added reference for spin connection to imaginary pseudoscalar

← Older revision		Revision as of 11:30, 2 June 2014
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	'''Algebraic character''' is a formal expression attached to a module in [[representation theory]] of [[semisimple Lie algebra]]s that generalizes the [[Weyl character formula\|character of a finite-dimensional representation]] and is analogous to the [[Harish-Chandra character]] of the representations of [[semisimple Lie group]]s.		54 yr old Health Campaign Officer Brzozowski from Dundurn, has interests which include studying an instrument, como ganhar dinheiro na internet and kids. Recalls what a pleasant location it was having paid checking out the Episcopal City of Albi.<br><br>Look into my web page ... [http://www.comoganhardinheiro101.com/slide-central/ como conseguir dinheiro]

	~~== Definition ==~~
	~~Let <math>\mathfrak{g}</math> be a [[semisimple Lie algebra]] with a fixed [[Cartan subalgebra]] <math>\mathfrak{h}~~,</math> and let the abelian group <math>A=\mathbb{Z}[[\mathfrak{h}^]]</math> consist of the (possibly infinite) formal integral linear combinations of <math>e^{\mu}</math>, where <math>\mu\in\mathfrak{h}^</math>, the (complex) vector space of weights. Suppose that <math>V</math> is a locally-finite [[weight module]]. Then the algebraic character of <math>V</math> is an ~~element of <math>A</math>~~
	~~defined by the formula:~~
	~~: <math> ch(V)=\sum_{\mu}\dim V_{\mu}e^{\mu}~~, ~~</math>~~
	~~where the sum is taken over all [[weight space]]s of the module <math>V.</math>~~

	~~== Example ==~~
	~~The algebraic character of the [[Verma module]] <math>M_\lambda</math> with the highest weight <math>\lambda</math> is given by the formula~~

	~~: <math> ch(M_{\lambda})=\frac{e^{\lambda}}{\prod_{\alpha>0}(1-e^{-\alpha})},</math>~~

	~~with the product taken over the set of positive roots.~~

	~~== Properties ==~~
	~~Algebraic characters are defined for locally-finite [[weight module]]s~~ and ~~are ''additive'', i.e~~. ~~the character of~~ a ~~direct sum of modules is~~ the ~~sum~~ of ~~their characters~~. ~~On the other hand, although one can define multiplication of the formal exponents by the formula~~ <~~math~~>~~e^{\mu}\cdot e^{\nu}=e^{\mu+\nu}~~<~~/math> and extend it to their ''finite'' linear combinations by linearity, this does not make <math>A</math~~> into a ring, because of the possibility of formal infinite sums. Thus the product of algebraic characters is well defined only in restricted situations; for example, for the case of a [[highest weight module]], or a finite-dimensional module. In good situations, the algebraic character is ''multiplicative'', i.e.~~, the character of the tensor product of two weight modules is the product of their characters~~.

	~~== Generalization ==~~
	~~Characters also can be defined almost ''verbatim'' for weight modules over a [[Kac-Moody algebra\|Kac-Moody]] or [[generalized Kac-Moody algebra\|generalized Kac-Moody]] Lie algebra~~.

	~~== See also ==~~
	*[~~[Weyl character formula#Weyl–Kac character formula\|Weyl-Kac character formula]]~~

	~~==References==~~
	*{{cite book\|last = Weyl\|first = Hermann\|title = The Classical Groups: Their Invariants and Representations\|publisher = Princeton University Press\|year = 1946\|isbn = 0-691-05756-7\|url = http://~~books~~.~~google~~.com/~~books?id=zmzKSP2xTtYC\|accessdate = 2007~~-~~03-26}}~~
	*{{cite book\|last = Kac\|first = Victor G\|title = Infinite-Dimensional Lie Algebras\|publisher = Cambridge University Press\|year = 1990\|isbn = 0-521-46693-8\|url = http:/~~/books.google.com/books?id=kuEjSb9teJwC\|accessdate = 2007-03-26}}~~
	*{{cite book\|last = Wallach\|first = Nolan R\|coauthors = Goodman, Roe\|title = Representations and Invariants of the Classical Groups\|publisher = Cambridge University Press\|year = 1998\|isbn = 0-521-66348-2\|url = http://books.google.com/books?vid=ISBN0521663482&id=MYFepb2yq1wC\|accessdate = 2007-03-26}}

	~~[[Category:Lie algebras]]~~
	~~[[Category:Representation theory of Lie algebras]~~]

en>Rschwieb: No WP:EASTEREGGs please

2013-10-24T20:18:21Z

No WP:EASTEREGGs please

← Older revision		Revision as of 22:18, 24 October 2013
Line 1:		Line 1:
	~~Friends contact him Royal. One~~ of ~~my favorite hobbies~~ is ~~tenting and now I'm trying~~ to ~~make cash~~ with ~~it. I am~~ a ~~production~~ and ~~distribution officer. I currently live~~ in ~~Alabama~~.<br><br>~~My blog post~~: ~~car warranty~~ ([http://~~Lahnsinfonie~~.de/~~index~~.~~php~~?~~mod~~=~~users&action~~=~~view~~&id=~~18009 simply click the following web site~~])		'''Algebraic character''' is a formal expression attached to a module in [[representation theory]] of [[semisimple Lie algebra]]s that generalizes the [[Weyl character formula\|character of a finite-dimensional representation]] and is analogous to the [[Harish-Chandra character]] of the representations of [[semisimple Lie group]]s.

			== Definition ==
			Let <math>\mathfrak{g}</math> be a [[semisimple Lie algebra]] with a fixed [[Cartan subalgebra]] <math>\mathfrak{h},</math> and let the abelian group <math>A=\mathbb{Z}[[\mathfrak{h}^]]</math> consist of the (possibly infinite) formal integral linear combinations of <math>e^{\mu}</math>, where <math>\mu\in\mathfrak{h}^</math>, the (complex) vector space of weights. Suppose that <math>V</math> is a locally-finite [[weight module]]. Then the algebraic character of <math>V</math> is an element of <math>A</math>
			defined by the formula:
			: <math> ch(V)=\sum_{\mu}\dim V_{\mu}e^{\mu}, </math>
			where the sum is taken over all [[weight space]]s of the module <math>V.</math>

			== Example ==
			The algebraic character of the [[Verma module]] <math>M_\lambda</math> with the highest weight <math>\lambda</math> is given by the formula

			: <math> ch(M_{\lambda})=\frac{e^{\lambda}}{\prod_{\alpha>0}(1-e^{-\alpha})},</math>

			with the product taken over the set of positive roots.

			== Properties ==
			Algebraic characters are defined for locally-finite [[weight module]]s and are ''additive'', i.e. the character of a direct sum of modules is the sum of their characters. On the other hand, although one can define multiplication of the formal exponents by the formula <math>e^{\mu}\cdot e^{\nu}=e^{\mu+\nu}</math> and extend it to their ''finite'' linear combinations by linearity, this does not make <math>A</math> into a ring, because of the possibility of formal infinite sums. Thus the product of algebraic characters is well defined only in restricted situations; for example, for the case of a [[highest weight module]], or a finite-dimensional module. In good situations, the algebraic character is ''multiplicative'', i.e., the character of the tensor product of two weight modules is the product of their characters.

			== Generalization ==
			Characters also can be defined almost ''verbatim'' for weight modules over a [[Kac-Moody algebra\|Kac-Moody]] or [[generalized Kac-Moody algebra\|generalized Kac-Moody]] Lie algebra.

			== See also ==
			*[[Weyl character formula#Weyl–Kac character formula\|Weyl-Kac character formula]]

			==References==
			*{{cite book\|last = Weyl\|first = Hermann\|title = The Classical Groups: Their Invariants and Representations\|publisher = Princeton University Press\|year = 1946\|isbn = 0-691-05756-7\|url = http://books.google.com/books?id=zmzKSP2xTtYC\|accessdate = 2007-03-26}}
			*{{cite book\|last = Kac\|first = Victor G\|title = Infinite-Dimensional Lie Algebras\|publisher = Cambridge University Press\|year = 1990\|isbn = 0-521-46693-8\|url = http://books.google.com/books?id=kuEjSb9teJwC\|accessdate = 2007-03-26}}
			*{{cite book\|last = Wallach\|first = Nolan R\|coauthors = Goodman, Roe\|title = Representations and Invariants of the Classical Groups\|publisher = Cambridge University Press\|year = 1998\|isbn = 0-521-66348-2\|url = http://books.google.com/books?vid=ISBN0521663482&id=MYFepb2yq1wC\|accessdate = 2007-03-26}}

			[[Category:Lie algebras]]
			[[Category:Representation theory of Lie algebras]]

151.60.33.212 at 15:21, 25 June 2012

2012-06-25T15:21:05Z

New page

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