Understanding Lie Algebras: Structure Constants and Commutation Relations

In summary, the commutator in the Baker-Campbell-Hausdorff formula is proportional to a linear combination of the group's generators due to closure. These constants of proportionality are known as the Structure Constants and can be used to determine the commutation relations between all the generators and thus the entire group in any representation. The exponential map and Lie algebra are important concepts in understanding Lie groups, and recommended resources for further understanding are Greiner & Muller's "QM - Symmetries" and Ballentine's "QM - A Modern Development".
  • #1
welatiger
85
0
the commutator in the Baker-Campbell-Hausdorff formula must be proportional to some linear
combination of the generators of the group (because of closure)
The constants of proportionality are called the Structure Constants
of the group, and if they are completely known, the commutation relations between
all the generators are known, and so the entire group can be determined in any
representation you want.

I want to understand this?
 
Physics news on Phys.org
  • #2
welatiger said:
I want to understand this?
The exponential map is an elementary part of the theory of Lie groups. A Lie group depends continuously on some parameters, and differentiating a group element wrt these parameters near the group identity yields the Lie algebra -- which can then exponentiated to recover the group.

Try Greiner & Muller "QM -- Symmetries" for a physicist-friendly introduction. Or maybe even Ballentine's "QM -- A Modern Development".

[Maybe this thread belongs in the group theory forum?]
 

Related to Understanding Lie Algebras: Structure Constants and Commutation Relations

1. What are Lie algebras?

Lie algebras are mathematical structures that are used to study continuous symmetries, such as those found in physical systems. They are a type of algebraic structure that is closely related to group theory.

2. How are Lie algebras different from other algebraic structures?

Lie algebras are different from other algebraic structures because they are specifically designed to study continuous symmetries, whereas other structures may focus on discrete symmetries. Lie algebras also have a specific set of properties, such as closure under a specific operation called the Lie bracket, that distinguish them from other structures.

3. What are some applications of Lie algebras?

Lie algebras have a wide range of applications in mathematics, physics, and engineering. They are used to study symmetries in differential equations, group representations, and quantum mechanics. They also have applications in areas such as control theory, robotics, and computer vision.

4. How do Lie algebras relate to Lie groups?

Lie algebras and Lie groups are closely related, with Lie groups being the continuous version of Lie algebras. Every Lie group has a corresponding Lie algebra, and the Lie algebra captures the local properties of the Lie group. This allows for the use of Lie algebras in the study of Lie groups.

5. Are there any open problems in the study of Lie algebras?

Yes, there are still many open problems in the study of Lie algebras. Some current research topics include the classification of simple Lie algebras, the study of infinite dimensional Lie algebras, and the connection between Lie algebras and other areas of mathematics, such as algebraic geometry and combinatorics.

Similar threads

I Jacobi identity of Lie algebra intuition
    • Differential Geometry
Replies
5
Views
2K
I What is the rank of the SU(2)xSU(2) algebra?
    • High Energy, Nuclear, Particle Physics
Replies
10
Views
2K
Deriving the commutation relations of the Lie algebra of Lorentz group
    • Advanced Physics Homework Help
Replies
3
Views
2K
A Differential structure of the group of automorphism of a Lie group
    • Differential Geometry
Replies
0
Views
323
A Topology on a space of Lie algebras
    • Linear and Abstract Algebra
Replies
2
Views
3K
I Lie Algebras & Relativity: Usefulness & Purpose
    • Special and General Relativity
Replies
10
Views
1K
I Proving Commutator Identity for Baker-Campbell-Hausdorff Formula
    • Linear and Abstract Algebra
Replies
4
Views
2K
Structure constants in Lie algebra
    • High Energy, Nuclear, Particle Physics
Replies
3
Views
6K
I Non-Commutation Property and its Relation to the Real World
    • Quantum Physics
Replies
19
Views
1K
I Fundamental representation and adjoint representation
    • Quantum Physics
Replies
27
Views
972

Hot Threads

Recent Insights

Back
Top