https://github.com/olivierverdier/multiaffine.jl

Implementation of the multi-affine group and its actions
https://github.com/olivierverdier/multiaffine.jl

Last synced: about 5 hours ago
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Implementation of the multi-affine group and its actions

• Host: GitHub
• URL: https://github.com/olivierverdier/multiaffine.jl
• Owner: olivierverdier
• License: mit
• Created: 2024-06-20T16:33:47.000Z (5 months ago)
• Default Branch: main
• Last Pushed: 2024-10-22T10:23:40.000Z (28 days ago)
• Last Synced: 2024-10-22T11:04:38.871Z (28 days ago)
• Language: Julia
• Homepage: https://olivierverdier.github.io/MultiAffine.jl/
• Size: 408 KB
• Stars: 0
• Watchers: 1
• Forks: 0
• Open Issues: 0
• Metadata Files:
  • Readme: README.md
  • License: LICENSE

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README

        
# MultiAffine

[![Build Status](https://github.com/olivierverdier/MultiAffine.jl/actions/workflows/CI.yml/badge.svg?branch=main)](https://github.com/olivierverdier/MultiAffine.jl/actions/workflows/CI.yml?query=branch%3Amain)

[![codecov](https://codecov.io/gh/olivierverdier/MultiAffine.jl/graph/badge.svg?token=aTe2GSxvIw)](https://codecov.io/gh/olivierverdier/MultiAffine.jl)

[![](https://img.shields.io/badge/docs-dev-blue.svg)](https://olivierverdier.github.io/MultiAffine.jl/)

Given a group $`H`$, represented in dimension $`n`$ over some field $`\mathbf{F}`$

this package implements the group

```math

(\mathbf{F}^n)^k \rtimes H

```

The group consists of matrices of the form

```math

[X;h] \equiv \begin{bmatrix}

\mathbf{1} & \mathbf{0} \\

X & h

\end{bmatrix}

```

where $`h`$ is a matrix belonging to the group $`H`$,

and $`X`$ is a $`n × k`$ matrix.

If we denote such an element by $`[X;h]`$,

the multiplication law is

```math

[X;h] [X';h'] = [X+hX';hh']

```

For instance, to implement the group

```math

(\mathbf{C}^2)^2 \rtimes U(2)

```

you can use

```julia

using MultiAffine

using Manifolds

G = MultiAffineGroup(Unitary(2), 2)

identity_element(G) # ([0.0 0.0; 0.0 0.0], ComplexF64[1.0 + 0.0im 0.0 + 0.0im; 0.0 + 0.0im 1.0 + 0.0im])

```

When the group $`H`$ is the [special orthogonal group](https://en.wikipedia.org/wiki/Orthogonal_group) $`SO(n)`$, one can use the alias `MultiDisplacementGroup(n,k)` to implement the group

```math

(\mathbf{R}^n)^k \rtimes SO(n)

```

This has the same effect as calling `MultiAffineGroup(SpecialOrthogonal(n), k)`. 

```julia

G = MultiDisplacementGroup(3,2)

identity_element(G) # ([0.0 0.0; 0.0 0.0; 0.0 0.0], [1.0 0.0 0.0; 0.0 1.0 0.0; 0.0 0.0 1.0])

```