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su(2)
Contents
Context
Lie theory
∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Cohomology
Homotopy
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
Contents
Idea
The Lie algebra is the special case of special unitary Lie algebras for , underlying the Lie group SU(2) (the special unitary group for ).
Idea
The Lie algebra is equivalently given as follows:
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the Lie algebra on 3 generators subject to the following relations on their Lie bracket:
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the Lie algebra spanned by ( times) the three Pauli matrices with Lie bracket their commutator in their matrix algebra.
Properties
Pauli matrix presentation
Proposition
The Lie algebra 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
Another common basis in use is the Cartan-Weyl basis
Complexification
The complexification of is the special linear Lie algebra (see at sl(2)) (…)
References
Textbook accounts:
See also
Last revised on March 18, 2020 at 14:21:19.
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