Definition Lie group Lie group is a smooth manifold with group structure and it plays a big role in modern geometry because its structure is very convenient to use to describe relations between different transformations in spaces. Besides groups Lie and Lie algebras connected to them are useful to be considered as a symmetry of a physical system. Let’s consider two basic examples of Lie groups and the general definition. Denote 𝑆 1 a boundary of a circle on a plane that has a center in the origin point and its radius is equal to 1. Now let’s perceive that every point of the plain, that is uniquely defined by two numbers x and y, represents a complex number 𝑥 + 𝑖𝑦. It is easy to see that we do the right thing by giving every two numbers 𝑥1 + 𝑖𝑦1 and 𝑥2 + 𝑖𝑦2 two different points, because 𝑥1 + 𝑖𝑦1 = 𝑥2 + 𝑖𝑦2 means 𝑥1 − 𝑥2 = 𝑖(𝑦2 − 𝑦1 ), which entails (𝑥1 − 𝑥2 )2 = −(𝑦2 − 𝑦1 )2 , now we see that 𝑥1 = 𝑥2 and 𝑦1 = 𝑦2 . Now we can observe an amazing thing: we can multiply points on a plane and moreover by multiplying two points on our 𝑆 1 we get a point, that also(!) lies on 𝑆 1 : 𝑥12 + 𝑦12 = 1 and 𝑥22 + 𝑦22 = 1 brings us to |(𝑥1 + 𝑖𝑦1 )(𝑥2 + 𝑖𝑦2 )|2 = |𝑥1 𝑥2 − 𝑦1 𝑦2 + 𝑖(𝑥1 𝑦2 + 𝑦1 𝑥2 )|2 = (𝑥1 𝑥2 − 𝑦1 𝑦2 )2 + (𝑥1 𝑦2 + 𝑦1 𝑥2 )2 = (𝑥12 + 𝑦12 )(𝑥22 + 𝑦22 ) = 1 ∗ 1=1 Another example will transfer us to 3-dimentional space and after that we will learn how to multiply lines that go through 0 in 4-dimentional space. It is called a rotation group 𝑆𝑂3 . Let us imagine that we are placed in a fixed point and look. It is obvious that when we turn our head somehow two times in a row we arrive into a condition that could have been achieved by turning our head only once. Euler's rotation theorem states that any picture we are able to see is achievable by a single rotation of the body around some axis. Now let us map 𝜋𝑎 degree rotation around (x, y, z) axis, where −1 ≤ 𝑎 ≤ 1, 𝑥 2 + 𝑦 2 + 𝑧 2 = 1, to a point (√1 − 𝑎2 𝑥, √1 − 𝑎2 𝑦, √1 − 𝑎2 𝑧, 𝑎) and remark that points (𝑥, 𝑦, 𝑧, 𝑤) and (−𝑥, −𝑦, −𝑧, −𝑤) correspond to the equal transformations of 𝑅 3 , which means that we have mapped to the set of lines. Generally speaking, Lie groups are n-dimentional surfaces such that we know how to multiply its points with each other which means that we can represent them in varying ways, that are suitable to our aims.
