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 [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 $ℝP^3$. 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 $SU(2)$, also known as the spin group of order 3, as in the following diagram:

     $\begin{array}{cc}{S}^3& \cong SU(2)=\text{Spin }3\\ 2\downarrow & \\ SO(3)& \cong ℝP^3\end{array}$

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

group of n×n matrices with complex elements and determinant 1

• Finnish: erityinen unitaarinen ryhmä
• German: spezielle unitäre Gruppe f
• Polish: specjalna symetria unitarna f
• Russian: специальная унитарная группа f (specialʹnaja unitarnaja gruppa)