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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

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

- 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.

- 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)

Using standard form

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