SU(2)

SU(2) is the group of two-by-two unitary complex matrices with determinant one. It is a compact, simply connected Lie group that double-covers the rotation group SO(3).

Unitary matrix: A complex square matrix whose conjugate transpose is its inverse. Every SU(2) matrix preserves the complex inner product because it is unitary.
Quaternion: An extension of the real numbers with three imaginary units and noncommutative multiplication. Unit quaternions give a concrete parametrization of SU(2).
Quantum spin: Intrinsic angular momentum in quantum mechanics, characterized by discrete spin values and corresponding state spaces. Spin transformations are represented by SU(2), including the spin-one-half case.
U(2): The group of two-by-two unitary complex matrices, with determinants allowed anywhere on the unit circle. U(2) differs by dropping SU(2)'s determinant-one condition.
Double cover: A covering map in which each point of the target has exactly two preimages. The map from SU(2) to SO(3) assigns two SU(2) elements to each spatial rotation.
Special linear group: The group of invertible matrices over a field with determinant one. SU(2) imposes unitarity on the determinant-one matrices of dimension two.
Exponential map: A map from a Lie algebra to its Lie group defined by exponentiating algebra elements. Exponentiating elements of su(2) generates continuous families of SU(2) transformations.
Bloch sphere: A unit sphere representation of the pure states of a two-level quantum system, with antipodal points corresponding to orthogonal states. SU(2) acts on qubit states and induces rotations of the Bloch sphere.
SO(3): The group of real three-dimensional orthogonal matrices with determinant one. SO(3) describes ordinary spatial rotations, while SU(2) is its double cover.
Fundamental group: A group that records equivalence classes of closed loops in a topological space. SU(2) is simply connected, whereas SO(3) has a nontrivial fundamental group.

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