Lie GeneratorsLie Group OperationGL as Lie GroupTransformed CurvesSingle-axis RotationOne ParameterTransformation GeneratorLie GeneratorsLie GeneratorsLie Group OperationLie Group OperationLie groups are continuous.Lie groups are continuous.•Continuous coordinate Continuous coordinate systemsystem•Finite dimensionFinite dimension•Origin is identityOrigin is identity),,(1 NxThe multiplication law is by The multiplication law is by analytic functions.analytic functions.•Two elements Two elements xx, , yy•Consider Consider zz = = xyxyThere are There are NN analytic analytic functions that define the functions that define the coordinates.coordinates.•Based on 2Based on 2NN coordinates coordinates),(f),,(1 NyxyzN ),,(1GL as Lie GroupGL as Lie GroupThe general linear groups The general linear groups GLGL((nn, , RR) are Lie groups.) are Lie groups.•Represent transformationsRepresent transformations•Dimension is Dimension is nn22All Lie groups are isomorphic All Lie groups are isomorphic to subgroups of to subgroups of GLGL((nn, , RR).).ExampleExampleLet x, y Let x, y  GL(n, R). GL(n, R).•Coordinates are matrix Coordinates are matrix elements minus elements minus Find the coordinates of Find the coordinates of z=xyz=xy..•Analytic in coordinatesAnalytic in coordinatesx yxz  Transformed CurvesTransformed CurvesAll Lie groups have All Lie groups have coordinate systems.coordinate systems.•May define differentiable May define differentiable curvescurvesThe set The set xx(()) may also form a may also form a group.group.•Subgroup Subgroup gg(()) Gx )()0()(ex )0(Single-axis RotationSingle-axis RotationParameterizations of Parameterizations of subgroups may take different subgroups may take different forms.forms.ExampleExampleConsider rotations about the Consider rotations about the Euclidean Euclidean xx-axis.-axis.•May use either angle or sineMay use either angle or sineThe choice gives different The choice gives different rules for multiplication.rules for multiplication.)()()(2121gggsin)]1[]1[()()(212122122121gggOne ParameterOne ParameterA one-parameter subgroup A one-parameter subgroup can always be written in a can always be written in a standard form.standard form.•Start with arbitrary Start with arbitrary represenatationrepresenatation•Differentiable function Differentiable function •Assume that there is a Assume that there is a parameterparameterThe differential equation will The differential equation will have a solution.have a solution.•Invert to get parameterInvert to get parameterS1)],([)()(2121ggg),0()0,())(),(()(2121)(0202dddd0)0( )(kfdd)()()(2121gggTransformation GeneratorTransformation GeneratorThe standard form can be used The standard form can be used to find a parameter to find a parameter aa independent of independent of . . Solve the differential equation.Solve the differential equation.The matrix The matrix aa is an infinitessimal is an infinitessimal generator of generator of gg(())egg  1011ddgggddg)()(1gggddggddg1Using standard form)()(lim1gggddgg)()()()(limggggaggg