Understanding Lie Groups: SO(1,1) and Dimensionality

In summary, SO(2) is a group that has an interpretation as rotations that preserve the dot product in two dimensions. SO(1,1) is a group that also has an interpretation as rotations but it is not the usual dot product you are familiar with.
  • #1
qtm912
38
1
I am familiar with what SO(2) means for example but am unclear what SO(1,1) refers to. This came up in a classical physics video lecture when lie groups were discussed and the significance of the notation was glossed over.
Second question: is the dimensionality of such a group the same as the number of generators?
 
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  • #2
SO(2) has an interpretation as rotations that preserve the dot product in two dimensions. So a^2 + b^2 = a'^2 + b'^2 where a' and b' are the result of applying an element of SO(2) to the two component vector with components a and b.

SO(1,1) is also preserves such a product, but it's not the usual dot product you are familiar with. Rather, it says that a^2 - b^2 = a'^2 - b'^2 (notice the minus sign). So extending this notation, SO(m,n) would be m minus signs and n plus signs, and acts of vectors of dimension m+n. This is most useful in Special Relativity, where the lorentz group is SO(1,3).

I think the dimension of a group is the number of generators, but I don't remember for sure.
 
  • #3
DimReg said:
SO(2) has an interpretation as rotations that preserve the dot product in two dimensions. So a^2 + b^2 = a'^2 + b'^2 where a' and b' are the result of applying an element of SO(2) to the two component vector with components a and b.

SO(1,1) is also preserves such a product, but it's not the usual dot product you are familiar with. Rather, it says that a^2 - b^2 = a'^2 - b'^2 (notice the minus sign). So extending this notation, SO(m,n) would be m minus signs and n plus signs, and acts of vectors of dimension m+n. This is most useful in Special Relativity, where the lorentz group is SO(1,3).

I think the dimension of a group is the number of generators, but I don't remember for sure.

Thank you very much, it is clear now
 

1. What is a Lie group?

A Lie group is a type of mathematical group that is also a smooth manifold, meaning it locally resembles Euclidean space. It is named after the mathematician Sophus Lie and is used to study continuous symmetries in mathematics, physics, and other fields.

2. What is SO(1,1) and why is it important?

SO(1,1) is the special orthogonal group of dimension 1,1. It is important because it is a fundamental example of a Lie group that is not compact, meaning it does not have finite size or volume. It is also used in physics to study the symmetries of spacetime in special relativity.

3. What does "dimensionality" mean in the context of Lie groups?

Dimensionality refers to the number of real parameters needed to describe the elements of a Lie group. In the case of SO(1,1), it has dimension 2, meaning it can be described by two real numbers. This is important for understanding the structure and properties of the group.

4. How are Lie groups related to physics?

Lie groups are widely used in physics to study symmetries and transformations in physical systems. For example, the special orthogonal group SO(3) is used to study the symmetries of three-dimensional space in classical mechanics, and the special unitary group SU(2) is used to study the symmetries of quantum systems.

5. What is the significance of studying SO(1,1) in particular?

SO(1,1) is a non-compact Lie group, meaning it has infinite size or volume. This makes it a valuable example for understanding the properties and behavior of non-compact groups, which can be more difficult to study than compact groups. It also has applications in physics, as mentioned in the answer to question 2.

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