# SU(2) and su(2) have different dimensions?

• I

## Main Question or Discussion Point

The Lie group SU(2) is the set of unitary 2x2 matrices with determinant 1.
These matrices can be written

a b
-b* a*

Thus, as a manifold, we can think of a coordinate chart consisting of the four real numbers making up the two complex numbers a and b. It is a manifold of dimension 4.

The Lie algebra su(2) is the set of 2x2 matrices of the form

ic d+if
-d+if -ic

where here c, d, f are real numbers. Thus the dimension of su(2) is 3.

But su(2) is also supposed to be the tangent space of the identity element of SU(2). Shouldn't the tangent space of a manifold at any point have the same dimension as the manifold?

Either my identification of the dimension of SU(2) or su(2) is wrong, or my claim that the tangent space dimension is always equal to the manifold dimension is wrong. Which is it?

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Mentor
The Lie group SU(2) is the set of unitary 2x2 matrices with determinant 1.
These matrices can be written

a b
-b* a*

Thus, as a manifold, we can think of a coordinate chart consisting of the four real numbers making up the two complex numbers a and b. It is a manifold of dimension 4.
No, the condition on the determinant reduces it by another dimension:
$|a|^2 + |b|^2 = 1$
The Lie algebra su(2) is the set of 2x2 matrices of the form

ic d+if
-d+if -ic

where here c, d, f are real numbers. Thus the dimension of su(2) is 3.
But su(2) is also supposed to be the tangent space of the identity element of SU(2). Shouldn't the tangent space of a manifold at any point have the same dimension as the manifold?

Either my identification of the dimension of SU(2) or su(2) is wrong,
Yes.
or my claim that the tangent space dimension is always equal to the manifold dimension is wrong. Which is it?
Manifold and tangent space are of the same dimension.

No, the condition on the determinant reduces it by another dimension:
$|a|^2 + |b|^2 = 1$
D'oh! I knew that. Thanks!