Notes for AKT-091006/0:11:24: Difference between revisions

Revision as of 11:02, 30 October 2011

Choose a basis $(X_{a})_{a=1}^{\dim {\mathcal {G}}}$ for ${\mathcal {G}}$ and a basis $(e_{a})_{a=1}^{\dim(R)}$ for $R$ .

Notation:

[X_{a},X_{b}]=f_{a,b}^{c}X_{c}

, where

f_{ab}^{c}\in \mathbb {Q}

are the structure constants

\left\langle X_{a},X_{b}\right\rangle =t_{ab}

Symmetric:

t_{ab}=t_{ba}

Non-degenerate:

(t_{ab})

has an inverse,

(t^{ab})

, with

t_{ab}\cdot t^{bc}=\delta _{ac}

@@ Line 1: / Line 1: @@
-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>