Question about basis of Lie algebra/tangent

[Ojisan]

Hi, I'm a newbie here and I would like to kindly ask for your collective wisdom on this forum.

I am working on a computational technique to describe a time series data on U(2), for now. Given two points x and y in U(2), I can take a matrix log of x'y to find the tangent emanating from x to y. Using the one-parameter exponential map, I can find the points along the geodesic using the so obtained tangent. My thought was to have a basis representation of the tangent space so as to come up with quantized approximate tangent which can best (in terms of geodesic distance) describe the computed tangent.

Intuitively, I want to form a grid of tangent directions which could be used at any point on U(2).

Does there exist such a basis? Where can I look for such formulations?

I apologize in advance if this a naive thought as I have no formal training in Lie group/Lie algebra. Thanks in advance for any help.

 

[jvicens]

Thank you for reaching out and sharing your project with us. Your approach to finding a basis representation of the tangent space on U(2) sounds interesting and could potentially be useful in many applications.

To answer your question, yes, there does exist a basis for the tangent space on U(2). This basis is known as the Lie algebra of U(2) and is defined as the set of all skew-Hermitian matrices with trace equal to 0. This basis is commonly denoted as su(2) and has three generators, which can be represented as the Pauli matrices.

In terms of finding formulations for your project, I would recommend looking into literature on Lie groups and Lie algebras, as well as numerical methods for computing geodesics on manifolds. Additionally, there are many resources available online, including lectures and tutorials, that can provide a more in-depth understanding of these concepts.

I hope this helps and I wish you all the best in your research. Don't hesitate to reach out for further assistance. Good luck!