special unitary group

English

Noun

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 , 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, , 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:

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

Usage notes

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.

Translations