# Index Concordia

## Group theory – Week 07

Posted in Group theory by Index Guy on December 11, 2007

For our purposes, a Lie algebra is defined as a linear vector space $L$ with elements $a X + b Y$ with $a,b$ real or complex numbers. The Lie algebra has a bilinear operation $\left[ , \right]$ that satisfies:

• closure: For $X,Y \in L, [X , Y] \in L$
• antisymmetry: $\left[X, Y\right] = - \left[Y, X\right]$
• linearity of the bracket: $\left[aX + bY , Z\right] = a \left[X , Z\right] + b\left[Y , Z\right]$
• associativity: $\left[X, \left[Y, Z\right]\right] +\left[Z, \left[X, Y\right]\right] + \left[Y, \left[Z, X\right]\right]= 0$

A vector in $L$ is going to be expanded in terms of the generators $L_{a}$. The structure constants are defined by

$\left[T_{a}, T_{b}\right] = f_{ab}^{c} T_{c}.$

We will be considering real forms, where the structure constants are always real numbers.

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