Special unitary group

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In abstract algebra, the special unitary group of degree n is the group of n by n unitary matrices with determinant 1 and entries from the field C of complex numbers, with the group operation that of matrix multiplication. It is written as SU(n). This is a subgroup of the unitary group U(n), itself a subgroup of the general linear group Gl(n,C).

The special unitary group SU(n) is a real Lie group of dimension n²-1.

The corresponding Lie algebra is denoted by su(n). The following matrices generate su(2) over R:

(i is the square root of -1.) This representation is often used in quantum mechanics (see Pauli matrices), to represent the spin of fundamental particles such as electrons. They also serve as unit vectors for the description of our 3 spatial dimensions in quantum relativity.

Note that the product of any two different generators is another generator, and that the generators anticommute. Together with the identity matrix,

these are also generators of the Lie algebra u(2).