add Heisenberg group

[mateuszbaran]

Do we have it documented somewhere what orthogonality means on a Lie group?

[kellertuer]

Can you elaborate what orthogonality you mean?

[mateuszbaran]

I mean, we need a metric to call a basis orthonormal or orthogonal. When the exp/log of that metric don't match exp/log we have for that group, it might be a bit weird.

[kellertuer]

I mean, we need a metric to call a basis orthonormal or orthogonal. When the exp/log of that metric don't match exp/log we have for that group, it might be a bit weird.

On the Lie group there is no metric, that is why I was confused; but you refer to the Lie algebra? That does come with an inner product anyways since it is a tangent space.
What is the actual point where this is currently a problem? If it is the DefaultLieAlgebraOrthogonalBasis I am fine renaming that to a name without Orthogonal

[mateuszbaran]

Hm, it looks like I forgot again how orthogonality works in LieGroups.jl.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 99.20%. Comparing base (932a647) to head (a1b60ff).
Report is 1 commits behind head on main.

Additional details and impacted files

@@            Coverage Diff             @@
##             main      #19      +/-   ##
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+ Coverage   99.12%   99.20%   +0.08%     
==========================================
  Files          12       13       +1     
  Lines         683      756      +73     
==========================================
+ Hits          677      750      +73     
  Misses          6        6              

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[kellertuer]

Up to one small remark this looks good. Thanks!