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14 votes

What is the convex hull of the quaternionic symmetries of the 3 dimensional cube?

This is the disphenoidal 288-cell, which is the dual of the bitruncated 24-cell. This is also mentioned in the "Geometry" section of the Wikipedia article on the 288-cell. It has 48 vertices, and 336 ...
M. Winter's user avatar
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11 votes

The Hypercomplex Structure of $SU(3)$

I'm assuming that you have a copy of Dominic Joyce's 1992 JDG article, "Compact hypercomplex and quaternionic manifolds" handy. Write $\frak{g} = \frak{su}(3)$ as a direct sum $$ \frak{su}(3) = \...
Robert Bryant's user avatar
8 votes

Compact quaternionic Kahler manifolds of negative curvature: examples

Any (Riemannian) symmetric space admits a cocompact lattice. This is due to A. Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2, 1963, pp.111-122. The quaternionic hyperbolic space ...
Igor Belegradek's user avatar
8 votes

Hyper-Kaehler Strucutre for Compact Lie Groups?

The answer is already 'no' for the simplest case: $$ S^1\times \mathrm{SU}(2) = (\mathbb{H}{\setminus}\{0\})/\mathbb{Z}, $$ which is clearly hypercomplex, but cannot even be Kähler, much less ...
Robert Bryant's user avatar
8 votes

Do these definitions of integrable quaternionic structure agree?

These two 'definitions' do not agree. Also, you should be careful about your choice of sources. Most differential geometers use the terminology 'almost quaternionic' to mean that the structure group ...
Robert Bryant's user avatar
6 votes

compact manifold as a hyperkahler quotient of an infinite-dimensional affine space

Wow, my thesis; it's been a while! Perhaps I was/am confused but I'll add a few remarks in case it helps. I don't know of any examples of a compact hyperkahler manifold obtained as an infinite-...
Oliver Nash's user avatar
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4 votes

Rozansky-Witten invariants of hyperkahler manifolds and independence of complex structure

There is an alternative approach which was explained to me 2 days ago, I believe it provides a proof of "independence of choice of complex structure compatible with the hyperkahler structure"...
BS math's user avatar
  • 91
3 votes

What do the Pauli matrices say about the Threefold Way?

Q: What do the Pauli matrices say about the Threefold Way? In the context of Dyson's threefold way, the Pauli matrices produce two of the three ensembles of random Hamiltonians. A Hermitian matrix $H$ ...
Carlo Beenakker's user avatar
2 votes

How does the concept of a hermitian metric generalize to a hyperkahler manifold?

Here are some small facts about Quaternionic Geometry. First let us leave metrics aside, for the moment. There are 2 analogues of complex manifolds in the quaternionic world, namely triholomorphic ...
Malkoun's user avatar
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