Spin groups

The spin group $S p i n (n, m)$ is the double cover of the group $S O (n, m)$ (the special orthogonal group). The latter is the group of special orthogonal group of signature $n, m$ .

They can be approached by means of Clifford algebras.

Spin Groups:

The double cover of $S O (2)$ is $Spin (2)$ , which is isomorphic to $U (1) = S U (1)$ . They are the unitary complex numbers.
The double cover of $S O (3)$ is given by $S p i n (3)$ . This is isomorphic to $S U (2)$ and can also be represented by unit quaternions.
The double cover of the Lorentz group $S O^{+} (1, 3)$ is $S p i n (1, 3)$ , which is equivalent to $S L (2, C)$ , the special linear group of 2x2 complex matrices. The complex Lie algebra $sl (2, C)$ associated with the Special Linear group of 2x2 complex matrices with determinant 1 can be decomposed into a direct sum of two real algebras isomorphic to $su (2)$ . This decomposition is expressed as

sl (2, C) = su (2) \oplus (\pm i) su (2),

where $su (2)$ represents the algebra of rotation generators in three-dimensional space, and $(\pm i) su (2)$ corresponds to the algebra of boost generators (Lorentz boosts) which in physical terms relate to transformations altering an object's velocity without rotation. The coefficients $- i$ and $+ i$ have to do with chirality...

The group $S O^{+} (1, 3)$ contains six "basis" matrices:
Rotations:

Λ_{y z} (θ) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & \cos θ & - \sin θ \\ 0 & 0 & \sin θ & \cos θ \end{matrix}]

Λ_{z x} (θ) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos θ & 0 & \sin θ \\ 0 & 0 & 1 & 0 \\ 0 & - \sin θ & 0 & \cos θ \end{matrix}]

Λ_{x y} (θ) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos θ & - \sin θ & 0 \\ 0 & \sin θ & \cos θ & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

Boosts:

Λ_{t x} (ϕ) = [\begin{matrix} \cosh ϕ & - \sinh ϕ & 0 & 0 \\ - \sinh ϕ & \cosh ϕ & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

Λ_{t y} (ϕ) = [\begin{matrix} \cosh ϕ & 0 & - \sinh ϕ & 0 \\ 0 & 1 & 0 & 0 \\ - \sinh ϕ & 0 & \cosh ϕ & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

Λ_{t z} (ϕ) = [\begin{matrix} \cosh ϕ & 0 & 0 & - \sinh ϕ \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ - \sinh ϕ & 0 & 0 & \cosh ϕ \end{matrix}]

They can be seen like associated to the following matrices (or their opposites) in $S p i n (1, 3) = S L (2, C)$ :

[\begin{matrix} \cos \frac{θ}{2} & - i \sin \frac{θ}{2} \\ - i \sin \frac{θ}{2} & \cos \frac{θ}{2} \end{matrix}]

[\begin{matrix} \cos \frac{θ}{2} & - \sin \frac{θ}{2} \\ \sin \frac{θ}{2} & \cos \frac{θ}{2} \end{matrix}]

[\begin{matrix} e^{- i θ / 2} & 0 \\ 0 & e^{i θ / 2} \end{matrix}]

[\begin{matrix} \cosh \frac{ϕ}{2} & - \sinh \frac{ϕ}{2} \\ - \sinh \frac{ϕ}{2} & \cosh \frac{ϕ}{2} \end{matrix}]

[\begin{matrix} \cosh \frac{ϕ}{2} & i \sinh \frac{ϕ}{2} \\ - i \sinh \frac{ϕ}{2} & \cosh \frac{ϕ}{2} \end{matrix}]

[\begin{matrix} e^{- ϕ / 2} & 0 \\ 0 & e^{ϕ / 2} \end{matrix}]

Related: They have a representation in the corresponding Clifford algebra and, also, they can be seen like elements of the Clifford algebra. See Clifford algebra#The spin group and the spinors.
Related: spin representation