Notes for AKT-091006/0:11:24 - Revision history

Iva at 16:20, 30 October 2011

2011-10-30T16:20:45Z

← Older revision		Revision as of 12:20, 30 October 2011
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	Choose a basis <math>(X_a)^{\dim{\mathcal{G}}}_{a=1}</math> for <math>\mathcal{G}</math>		Choose a basis <math>(X_a)^{\dim{\mathcal{G}}}_{a=1}</math> for <math>\mathcal{G}</math>
	and a basis <math>(~~e_a~~)^{\dim(R)}_{a=1}</math> for <math>R</math>.		and a basis <math>(e_{\alpha})^{\dim(R)}_{\alpha=1}</math> for <math>R</math>.

	''Notation'':		''Notation'':

Iva at 16:02, 30 October 2011

2011-10-30T16:02:04Z

← Older revision		Revision as of 12:02, 30 October 2011
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	Choose basis <math>(X_a)~~</math>, <math>a \in \~~{~~1, 2, \cdots,~~ \dim(\mathcal{G}~~) \~~}</math> for <math>\mathcal{G}</math>		Choose a basis <math>(X_a)^{\dim{\mathcal{G}}}_{a=1}</math> for <math>\mathcal{G}</math>
	and basis <math>(e_a)~~</math>, <math>a \in \~~{~~1, 2, \cdots,~~ \dim(R) \}</math> for <math>R</math>		and a basis <math>(e_a)^{\dim(R)}_{a=1}</math> for <math>R</math>.

			''Notation'':
	~~Notation~~: ~~Let~~ <math>[X_a, X_b] = F_{a,b}^n ~~X_n~~</math>, <math>F_{ab}^n \in \mathbb{Q}</math>		:<math>[X_a, X_b] = f_{a,b}^c X_c</math>, where <math>f_{ab}^c \in \mathbb{Q}</math> are the structure constants

	<math><X_a, X_b> = t_{ab~~}</math>; <math>t^{ab}</math> is defined s.t. <math>t_{ab} \cdot t^{bc} = \delta_{ac~~}</math>		:<math>\left\langle X_a, X_b\right\rangle = t_{ab}</math>
			::Symmetric: <math>t_{ab}=t_{ba}</math>
			::Non-degenerate: <math>(t_{ab})</math> has an inverse, <math>(t^{ab})</math>, with <math>t_{ab} \cdot t^{bc} = \delta_{ac}</math>

Conan777 at 06:05, 10 October 2009

2009-10-10T06:05:51Z

← Older revision		Revision as of 02:05, 10 October 2009
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	and basis <math>(e_a)</math>, <math>a \in \{1, 2, \cdots, \dim(R) \}</math> for <math>R</math>		and basis <math>(e_a)</math>, <math>a \in \{1, 2, \cdots, \dim(R) \}</math> for <math>R</math>

			Notation: Let <math>[X_a, X_b] = F_{a,b}^n X_n</math>, <math>F_{ab}^n \in \mathbb{Q}</math>
	express all Lie-algebra operations in terms of the basis.

			<math><X_a, X_b> = t_{ab}</math>; <math>t^{ab}</math> is defined s.t. <math>t_{ab} \cdot t^{bc} = \delta_{ac}</math>

Conan777 at 05:55, 10 October 2009

2009-10-10T05:55:38Z

New page

Choose basis <math>(X_a)</math>, <math>a \in \{1, 2, \cdots, \dim(\mathcal{G}) \}</math> for <math>\mathcal{G}</math>
and basis <math>(e_a)</math>, <math>a \in \{1, 2, \cdots, \dim(R) \}</math> for <math>R</math>

express all Lie-algebra operations in terms of the basis.