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## Lie Generators

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**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 2N coordinates Lie Group Operation**The general linear groups GL(n, R) are Lie groups.**Represent transformations Dimension is n2 All Lie groups are isomorphic to subgroups of GL(n, R). Example Let x, y GL(n, R). Coordinates are matrix elements minus dab Find the coordinates of z=xy. Analytic in coordinates GL as Lie Group**Transformed Curves**• All Lie groups have coordinate systems. • May define differentiable curves • The set x(e) may also form a group. • Subgroup g(e)**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. Single-axis Rotation**One Parameter**• A one-parameter subgroup can always be written in a standard form. • Start with arbitrary represenatation • Differentiable function m • Assume that there is a parameter • The differential equation will have a solution. • Invert to get parameter S1**The standard form can be used to find a parameter a**independent of e. Solve the differential equation. The matrix a is an infinitessimal generator of g(e) Transformation Generator Using standard form next