Group of unit quaternions

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Definition

This group is denoted $S^{3}, S U (2), S^{0} (H), S^{1} (C)$ and is defined in a number of equivalent ways.

As the group of unit quaternions

Denote by $H$ the division ring of Hamiltonian quaternions. The group we are interested in is the multiplicative subgroup of $H^{*}$ comprising those unit quaternions $a + b i + c j + d k$ satisfying $a^{2} + b^{2} + c^{2} + d^{2} = 1$ . Note that $a, b, c, d \in R$ (and are allowed to be equal). Explicitly, the multiplication is given by:

$(a_{1} + b_{1} i + c_{1} j + d_{1} k) (a_{2} + b_{2} i + c_{2} j + d_{2} k) = (a_{1} a_{2} - b_{1} b_{2} - c_{1} c_{2} - d_{1} d_{2}) + (a_{1} b_{2} + a_{2} b_{1} + c_{1} d_{2} - c_{2} d_{1}) i + (a_{1} c_{2} + a_{2} c_{1} + d_{1} b_{2} - d_{2} b_{1}) j + (a_{1} d_{2} + a_{2} d_{1} + b_{1} c_{2} - b_{2} c_{1}) k$

The identity element is:

$1 + 0 i + 0 j + 0 k$

The inverse is given by:

$(a + b i + c j + d k)^{- 1} = a - b i - c j - d k$

As the special unitary group

The group can also be defined as the special unitary group of degree two over the field of complex numbers. It is denoted $S U (2)$ or $S U (2, m a t h b b C)$ .