Special unitary group

In mathematics, the special unitary group of degree n, denoted SU(n), is the Lie group of n×n unitary matrices with determinant 1 (i.e., real-valued determinant, not complex as for general unitary matrices). The group operation is that of matrix multiplication. The special unitary group is a subgroup of the unitary group U(n), consisting of all n×n unitary matrices. As a compact classical group, U(n) is the group that preserves the standard inner product on Cⁿ. It is itself a subgroup of the general linear group, SU(n) ⊂ U(n) ⊂ GL(n, C).

The SU(n) groups find wide application in the Standard Model of particle physics, especially SU(2) in the electroweak interaction and SU(3) in quantum chromodynamics.

The simplest case, SU(1), is the trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+I, −I}.SU(2) is also identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations.

Sukhoi Su-2

The Sukhoi Su-2 (Russian: Сухой Су-2) was a Soviet scout and light bomber aircraft used in the early stages of World War II. It was the first airplane designed by Pavel Sukhoi. The basic design received an engine and armament upgrade (Su-4) and was modified for the ground attack role (ShB).

Development

In 1936, Joseph Stalin released a requirement for a multipurpose combat aircraft. Codenamed Ivanov, the airplane had to be capable of performing reconnaissance and then attacking the targets it located. P.O. Sukhoi was working in the Tupolev OKB at the time and designed the "Ivanov" aircraft under the tutelage of Andrei Tupolev. The resulting ANT-51 flew on 25 August 1937 with M.M. Gromov at the controls. Powered by an 610 kW (820 hp) Shvetsov M-62 air-cooled radial engine,the ANT-51 reached 403 km/h (220 kn, 250 mph) at 4,700 m (15,420 ft). This was considered insufficient but since the basic design was sound, it was decided to re-test with a more powerful engine. Equipped with a 746 kW (1,000 hp) Tumansky M-87 engine, the ANT-51 reached 468 km/h (255 kn, 290 mph) at 5,600 m (18,370 ft) and was accepted into production as BB-1 (Blizhniy Bombardirovschik; Russian: Ближний Бомбардировщик — Short-range Bomber). In 1940, the aircraft was renamed Su-2 and the unreliable M-87 engine was replaced with a Tumansky M-88. This lightened version with a M-88B engine reached 512 km/h (275 kn, 320 mph) in testing.