Lie generators
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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 Generators


Lie group operation

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


Gl as lie group

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

Transformed Curves

  • All Lie groups have coordinate systems.

    • May define differentiable curves

  • The set x(e) may also form a group.

    • Subgroup g(e)


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.

Single-axis Rotation


One parameter

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


Transformation generator

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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