# Lie Generators - PowerPoint PPT Presentation

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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.

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Lie Generators

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## 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 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

### 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.

### One Parameter

• A one-parameter subgroup can always be written in a standard form.

• 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

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