Lie Algebra differentiable manifold

In summary, the conversation discusses the relationship between a Lie group, its corresponding Lie algebra, and the representation space for the group. The Lie algebra is a vector space with a defined bracket operation, which can be used in the exponential mapping to generate the Lie group. In the example given, su(2) is the Lie algebra for the Lie group SO(2). The conversation also mentions that a representation space for a Lie group can be a set of nxn matrices, with the elements of the group being represented by a matrix in this space.
  • #1
Hymne
89
1
Okey, I have problem with the foundation of lie algebra. This is my understanding:

We have a lie group which is a differentiable manifold. This lie group can for example be SO(2), etc.

Then we have the Lie algebra which is a vectorspace with the lie bracket defined on it: [. , .].
This lie algebra will, when we put it in the exponential mapping, give the Lie group.

For example: su(2) = R(0, 1; -1, 0).

I hope this is correct.

The we come to representation space.. Well in the example above our elements of the Lie group, G, will be represented by a matrix: A = exp(t x) where x belongs to the lie algebra, and t is our parameter. Does this mean that the vector space of n,n matrices is our representation space?
 
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  • #2
Apart of minor inexactitudes: when you have a group consisting of some set of nxn matrices, then, yes, the set of all nxn matrices can be considered as a particular representation space - usually a reducible one.
 

1. What is a Lie algebra?

A Lie algebra is a mathematical structure that is used to study the properties and transformations of differentiable manifolds. It consists of a vector space together with a binary operation called the Lie bracket, which is used to measure the failure of the product rule on the manifold.

2. What is a differentiable manifold?

A differentiable manifold is a mathematical object that is used to describe smooth, curved spaces. It is a generalization of the concept of a plane or a space to ones that have more complex shapes, such as spheres or tori. Differentiable manifolds are important in many areas of mathematics, physics, and engineering.

3. How are Lie algebras and differentiable manifolds related?

Lie algebras and differentiable manifolds are closely related because the Lie algebra of a differentiable manifold captures the infinitesimal symmetries of the manifold. In other words, it describes the transformations that preserve the structure of the manifold locally. This connection allows for the use of Lie algebra techniques in studying differentiable manifolds.

4. What are some applications of Lie algebras and differentiable manifolds?

Lie algebras and differentiable manifolds have numerous applications in mathematics, physics, and engineering. They are used to study and understand symmetries in physical systems, such as in quantum mechanics and general relativity. They also play a crucial role in the development of geometric control theory, which has applications in robotics, aerospace engineering, and other fields.

5. How can one learn more about Lie algebras and differentiable manifolds?

There are many resources available for learning about Lie algebras and differentiable manifolds. One can start by studying advanced calculus, linear algebra, and abstract algebra, which are all foundational topics for understanding these concepts. There are also many textbooks, online courses, and research articles that provide a comprehensive introduction to these subjects. Additionally, attending seminars and conferences related to Lie algebras and differentiable manifolds can also be a great way to learn more and stay updated on current research in the field.

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