special unitary group

special unitary group (plural special unitary groups)

1. (linear algebra, group theory) For given n, the group of n×n unitary matrices with complex elements and determinant equal to one.
   • 1992 [Prentice-Hall], George H. Duffey, Applied Group Theory: For Physicists and Chemists, 2015, Dover, Unabridged Republication, page 284,

     The special unitary group in two dimensions is represented by the 2 X 2 unitary matrices whose determinants equal 1.

   • 2000, Herbert S. Green, Information Theory and Quantum Physics: Physical Foundations for Understanding the Conscious Process, Springer, page 26,

     The group is called the special unitary group in two dimensions, or SU(2), because it acts on matrices of degree 2.

   • 2004, Roger Cooke (translator), Vladimir I. Arnold, Lectures on Partial Differential Equations, [1997, Lekstii ob uravneniyakh s chastnymi proizvodnymi], Springer, page 81,

     When n = 3, the group of rotations SO(3) is isomorphic to the real three-dimensional projective space ${\displaystyle \mathbb {R} P^{3}}$ . It has a two-sheeted covering by the three-dimensional sphere (the group of unit quaternions), which in turn is isomorphic to the special unitary group ${\displaystyle SU(2)}$ , also known as the spin group of order 3, as in the following diagram:

     ${\displaystyle {\begin{aligned}S^{3}&\cong SU(2)={\text{Spin }}3\\2{\bigg \downarrow }&\\SO(3)&\cong \mathbb {R} P^{3}\end{aligned}}}$

Denoted SU(n). Each special unitary group is a Lie group and a subgroup of the unitary group U(n). SU(1) is the trivial group.