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

[qtm912]

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?

 

[DimReg]

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.

 

[qtm912]

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