Redirected from "Spin(3)".
Contents
Context
Group Theory
group theory
Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Related concepts
Lie theory
∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
Contents
Definition
The special unitary group for .
Proposition
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.
Properties
General
Proposition
There are isomorphisms of Lie groups
-
of with the spin group in dimension 3 and with the quaternionic unitary group in one dimension
-
of the direct product group of with itself, to Spin(4)
with respect to which the canonical inclusion is given by the diagonal map.
See at spin group – Exceptional isomorphisms.
Proposition
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
Lie algebra
Proposition
The Lie algebra (see also at su(2)) as a complex matrix Lie algebra is the sub Lie algebra on those matrices of the form
Definition
The standard basis elements of given by the above presentation are
These are called the Pauli matrices.
Proposition
The Pauli matrices satisfy the commutator relations
Coadjoint orbits
Proposition
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 .
Finite subgroups
The finite subgroup of SU(2) have an ADE classification. See this theorem.
-Structure and exceptional geometry
Representation theory
To record some aspects of the linear representation theory of SU(2).
(…)
Lemma
We have
Proof
Consider the canonical linear basis
(1)
by orthonormal quaternions
On this, the action by Sp(1)
is by left quaternion multiplication.
Consider then the following linear basis of :
with induced Lie algebra action given by
From this we find
and
and
Since everything here is invariant under cyclic permutation of the three non-zero indices it follows generally that
But this means that
Lemma
We have
Proof
Consider the following linear basis
We claim that in terms of these basis elements and those of (1) the isomorphism is given by , . This follows by direct inspection. For instance for the Lie algebra action of we find:
References
On the coadjoint orbits of :
See also