Lie Generators. Lie groups are continuous. Continuous coordinate system Finite dimension Origin is identity. The multiplication law is by analytic functions. Two elements x , y Consider z = xy There are N analytic functions that define the coordinates. Based on 2 N coordinates.
Dimension is n2
All Lie groups are isomorphic to subgroups of GL(n, R).
Let x, y GL(n, R).
Coordinates are matrix elements minus dab
Find the coordinates of z=xy.
Analytic in coordinatesGL as Lie Group
Consider rotations about the Euclidean x-axis.
May use either angle or sine
The choice gives different rules for multiplication.Single-axis Rotation
Solve the differential equation.
The matrix a is an infinitessimal generator of g(e)Transformation Generator
Using standard form