Further questions on the basics of Lie Algebras

In summary, the conversation discusses questions related to Group theory, specifically the connected component of the identity, the equivalence of Lie(SO(n)) and Lie(O(n)), and the isomorphism of SU(2) Lie group and a sphere in 3D. The concepts of Lie algebra, matrix Lie groups, and the conditions for a matrix to be in the Lie algebra are mentioned. It is explained that the SU(2) Lie group is not isomorphic, but rather homeomorphic and diffeomorphic to a sphere in 3D. The process of proving this through a continuous bijection and calculating the most general complex 2x2 matrix is described. Additionally, the correct way to write matrices in Latex is explained.
  • #1
vertices
62
0
I'm reviewing my Group theory notes at the moment. I have a few questions.

1)what is the "connected component of the identity"? (How would you go about working this out?)

2)Why is Lie(SO(n))=Lie(O(n))?

3)Why is the SU(2) Lie group isomorphic to a sphere in 3D?
 
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  • #2
vertices said:
1)what is the "connected component of the identity"? (How would you go about working this out?)
The largest subgroup that's also a connected topological space. I think that for Lie groups, "connected" is equivalent to "path connected", which means that what you have to check is that for each g in the group, there's a continuous curve from the identity to g.

vertices said:
2)Why is Lie(SO(n))=Lie(O(n))?
For matrix Lie groups, we can use the simple definition of a Lie algebra: A matrix X is in the Lie algebra if exp(tX) is in the Lie group for all real numbers t. This condition implies that for both Lie groups, the Lie algebra is the real vector space of traceless hermitian n×n matrices.

Note that since det(exp(tA))=exp(tTr(A)), both the conditions ±1=det(exp(tA) and +1=det(exp(tA)) imply the same thing, that Tr(A)=0.

This book can tell you the details.

vertices said:
3)Why is the SU(2) Lie group isomorphic to a sphere in 3D?
Not Lie group isomorphic. Just homeomorpic (topological space isomorphic) and probably diffeomorphic (manifold isomorphic). What you're supposed to prove is that there's a continuous bijection from SU(2) into S3, and this is just a tedius calculation. Write down the most general complex 2×2 matrix U and find out what relationships between its components you can derive from the conditions [itex]U^\dagger U=I[/itex] and [itex]\det U=1[/itex]. You should end up with the condition that defines the unit 3-sphere.
 
  • #3
Thanks Fredrik for your help in this and other threads. Hope you had a good Christmas:)
 
  • #4
Fredrik said:
Not Lie group isomorphic. Just homeomorpic (topological space isomorphic) and probably diffeomorphic (manifold isomorphic). What you're supposed to prove is that there's a continuous bijection from SU(2) into S3, and this is just a tedius calculation. Write down the most general complex 2×2 matrix U and find out what relationships between its components you can derive from the conditions [itex]U^\dagger U=I[/itex] and [itex]\det U=1[/itex]. You should end up with the condition that defines the unit 3-sphere.

I attempted this computation as you suggested. I get matrix elements that look like this (sorry I don't know how to do matrices in Latex):

a b
-b* a

So there are two parameters, a and b. But because a=x+iy and b=w+iz, that's 4 parameter in total. So should it not belong to a 4-sphere?
 
  • #5
The fact that the determinant of the matrix must be =1 gives you [itex]w^2+x^2+y^2+z^2=1[/itex]. That's the definition of a 3-sphere. Note that a 2-sphere is a sphere and a 1-sphere is a circle. The number indicates how many dimensions the manifold has. A circle is a 1-dimensional manifold because its coordinate systems are maps into [itex]\mathbb R[/itex], not into [itex]\mathbb R^2[/itex].

Matrices start with \begin{pmatrix} and end with \end{pmatrix}. End each line except the last with \\, and put & symbols between the elements. For example, \begin{pmatrix} a & b\\ -b^* & a\end{pmatrix}

[tex]\begin{pmatrix} a & b\\ -b^* & a\end{pmatrix}[/tex]

Use vmatrix instead of pmatrix if you want a determinant.
 

Related to Further questions on the basics of Lie Algebras

1. What is a Lie algebra?

A Lie algebra is a mathematical structure that studies the properties of vector spaces and their associated operations, such as addition and multiplication. It is named after the Norwegian mathematician Sophus Lie.

2. What are the basic properties of Lie algebras?

The basic properties of Lie algebras include the concept of a Lie bracket, which is a binary operation that measures the failure of two vector fields to commute. Lie algebras also have a bilinear form, called the Killing form, which measures the symmetry of the Lie bracket.

3. What are the applications of Lie algebras?

Lie algebras have numerous applications in mathematics and physics. They are used to study the symmetries of differential equations, group theory, and quantum mechanics. They also have applications in fields such as geometry, topology, and mathematical physics.

4. How are Lie algebras related to Lie groups?

Lie algebras are the infinitesimal versions of Lie groups. A Lie group is a continuous group of symmetries, while a Lie algebra is the tangent space at the identity element of the Lie group. This means that the Lie algebra captures the local structure of the Lie group.

5. What are some important Lie algebras?

Some important Lie algebras include the special linear algebra, orthogonal algebra, and the symplectic algebra, which are all related to classical groups. Other significant Lie algebras include the affine, Virasoro, and Kac-Moody algebras, which have applications in geometry, string theory, and mathematical physics.

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