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-1

The Sukhoi Su-1 or I-330 (Russian: Сухой Су-1) was a prototype Soviet high-altitude fighter aircraft built at the beginning of World War II. An improved version, designated Su-3 (I-360), was also built and tested the following year. Neither version was mass-produced.

Development

In 1939, Sukhoi was tasked with designing a high-altitude fighter, the resulting Su-1 was a conventional monoplane with a streamlined wooden semi-monocoque fuselage, low-set all-metal wing made of duralumin and un-pressurized cockpit. The key feature of the aircraft was a pair of TK-2 turbochargers driven by exhaust gases from the Klimov M-105P engine. The prototype was completed at the Sukhoi plant in Kharkov in May 1940, flying for the first time, on 15 June 1940, with A.P. Chernyavsky at the controls. Testing continued until 3 August, when Chernavskii mistakenly landed with the landing gear retracted. After the completion of repairs in mid-September, testing resumed, but an in-flight engine failure on 2 October resulted in a dead-stick landing. Limited flight testing continued until April 1941, with the Su-1 reaching a top speed of 641 km/h (345 kn, 400 mph) at 10,000 m (32,810 ft). However, the turbochargers proved unreliable and without them the aircraft was inferior in performance to the Yakovlev Yak-1.