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SL(2,H)
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Group Theory
group theory
Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Spin geometry
Contents
Definition
By one denotes the special linear group of matrices with coefficients in the quaternions, where “special” refers to their Dieudonné determinant being unity:
(here is the norm(-square) on quaternions).
Properties
Relation to
Every quaternionic unitary matrix (hence in Sp(2)) happens to have unit Dieudonné determinant (Cohen-De Leo 99, Cor. 6.4). Therefore we have a subgroup inclusion
Relation to
Under the conjugation action on Hermitian matrices with coefficients in the quaternions, is identified with Spin(5,1) and its canonical action on Minkowski spacetime .
For more on this see at spin representation, supersymmetry and division algebras and geometry of physics – supersymmetry.
exceptional spinors and real normed division algebras
Lorentzian spacetime dimension | spin group | normed division algebra | brane scan entry |
---|
| | the real numbers | super 1-brane in 3d |
| | the complex numbers | super 2-brane in 4d |
| SL(2,H) | the quaternions | little string |
| Spin(9,1) “SL(2,O)” | the octonions | heterotic/type II string |
References
Last revised on September 25, 2021 at 06:32:01.
See the history of this page for a list of all contributions to it.