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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