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The special unitary group is the subgroup of the unitary group on the elements with determinant equal to 1.
For a natural number, the special unitary group is the group of isometries of the -dimensional complex Hilbert space which preserve the volume form on this space. It is the subgroup of the unitary group consisting of the unitary matrices with determinant .
More generally, for any complex vector space equipped with a nondegenerate Hermitian form , is the group of isometries of which preserve the volume form derived from . One may write if is obvious, so that is the same as . By , we mean , where has positive eigenvalues and negative ones.
ADE classification and McKay correspondence
See at representation theory of the special unitary group.
We discuss aspects of SU(2), hence
As a matrix group is equivalent to the subgroup of the general linear group on those of the form
where are complex numbers and denotes complex conjugation.
The underlying manifold of is diffeomorphic to the 3-sphere .
See at spin group – Exceptional isomorphisms.
The Lie algebra as a matrix Lie algebra is the sub Lie algebra on those matrices of the form
The standard basis elements of given by the above presentation are
These are called the Pauli matrices.
The Pauli matrices satisfy the commutator relations
The maximal torus of is the circle group . In the above matrix group presentation this is naturally identified with the subgroup of matrices of the form
The coadjoint orbits of the coadjoint action of on are equivalent to the subset of the above matrices with for some .
These are regular coadjoint orbits for .
See at spin group – Exceptional isomorphisms.
Howard Georgi, §13 in: Lie Algebras In Particle Physics, Westview Press (1999), CRC Press (2019) [doi:10.1201/9780429499210]
with an eye towards application to (the standard model of) particle physics
Last revised on August 24, 2024 at 11:44:33. See the history of this page for a list of all contributions to it.