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### Lie groups

The **Lie groups** are frequently used in quantum mechanics.

##### SO(2)

The "S" stands for "special", which refers to the fact that reflections are excluded.

The "O" stands for "orthogonal", which means a group of rotations where orthogonality (right angles) are preserved.

The "2" stands for 2 dimensions.

##### SU(2)

The "S" stands for "special", which refers to the fact that reflections are excluded.

The "U" stands for "unitary" (preserving of the unit-norm nature of complex vectors), which refers to rotations in the complex number space.

The "2" stands for 2 dimensions.

##### SU(3)

The "S" stands for "special", which refers to the fact that reflections are excluded.

The "U" stands for "unitary" (maintaining the character of the unit standard of complex vectors), which refers to rotations in the complex plane.

The "3" stands for 3 dimensions, the "eightfold way" and is needed for multiplets.

##### SU(5)

The "S" stands for "special", which refers to the fact that reflections are excluded.

The "U" stands for "unitary" (maintaining the character of the unit standard of complex vectors), which refers to rotations in the complex plane.

The "5"stands for 24 dimensions.

##### U(1)

The "U" stands for "unitary" (preserving of the unit-norm nature of complex vectors), which refers to rotations in the complex number space.

The "1" stands for 1 dimension.

## HistoryThe Norwegian mathematician Sophus Lie (1842 - 1899) has developed these groups. |