Lie Generators

Lie Group Operation  Lie groups are continuous. Continuous coordinate systemContinuous coordinate system Finite dimensionFinite dimension Origin is identityOrigin 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

GL as Lie Group  The general linear groups GL( n, R ) are Lie groups. Represent transformationsRepresent transformations Dimension is n 2Dimension is n 2  All Lie groups are isomorphic to subgroups of GL( n, R ). Example  Let x, y  GL(n, R). Coordinates are matrix elements minus    Find the coordinates of z=xy. Analytic in coordinates

Transformed Curves  All Lie groups have coordinate systems. May define differentiable curves  The set x(  ) may also form a group. Subgroup g(  )

Single-axis Rotation  Parameterizations of subgroups may take different forms. Example  Consider rotations about the Euclidean x -axis. May use either angle or sine  The choice gives different rules for multiplication.

One Parameter  A one-parameter subgroup can always be written in a standard form. Start with arbitrary represenatationStart with arbitrary represenatation Differentiable function Differentiable function  Assume that there is a parameterAssume that there is a parameter  The differential equation will have a solution. Invert to get parameterInvert to get parameter S1S1

Transformation Generator  The standard form can be used to find a parameter a independent of .  Solve the differential equation.  The matrix a is an infinitessimal generator of g (  ) Using standard form next