Commutator Problem: Show [A,Bn] = cnBn-1

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In summary, the Commutator Problem is a mathematical problem that involves finding the commutator of two groups, A and B. The commutator is defined as [A,B] = AB-BA, and the problem asks to find a specific commutator, [A,Bn], where n is a positive integer. This notation means that the commutator [A,Bn] can be written as the group element cnBn-1. The Commutator Problem is important in understanding the structure and commutativity of groups, and can be solved using various mathematical techniques. It has applications in fields such as physics, chemistry, and computer science.
  • #1
Calu
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Homework Statement



Let the commutator [A,B] = cI, I the identity matrix and c some arbitrary constant.

Show [A,Bn] = cnBn-1

Homework Equations



[A,B] = AB - BA

The Attempt at a Solution



So I have started off like this:

[A,Bn] = ABn - BnA = cI

I'm not sure where to go from here.
 
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  • #2
I think you are misunderstanding the question. [A,B^n] = AB^n - B^nA = cI is NOT in general true. You seem to be thinking that "AB- BA= cI" is to be true for all A, B. It is not. In this problem AB- BA= cI is true for this specific A and B.

You are told that AB- BA= cI. So [A, B^2]= AB^2- B^2A= AB^2- BAB+ BAB- B^2A= (AB- BA)B+ B(AB- BA)= cB+ Bc= 2cB. etc. Use proof by induction.
 
  • #3
Calu said:

Homework Statement



Let the commutator [A,B] = cI, I the identity matrix and c some arbitrary constant.

Show [A,Bn] = cnBn-1

Homework Equations



[A,B] = AB - BA

The Attempt at a Solution



So I have started off like this:

[A,Bn] = ABn - BnA = cI

I'm not sure where to go from here.

For any operators you have the identity ##[D,EF]=[D,E]F+E[D,F]## that's handy and it's easy to prove. The case ##n=1## is obvious, so now try ##n=2##. Write ##[A,B^2]## as ##[A,BB]##. For the general case think about induction.
 

Related to Commutator Problem: Show [A,Bn] = cnBn-1

1. What is the Commutator Problem?

The Commutator Problem is a mathematical problem that involves finding the commutator of two groups, A and B. The commutator is defined as [A,B] = AB-BA, where AB represents the group elements that can be obtained by multiplying elements from group A and B in that order, and BA represents the elements obtained by multiplying in the opposite order. The Commutator Problem asks to find a specific commutator, [A,Bn], where n is a positive integer.

2. What does the notation [A,Bn] = cnBn-1 mean?

This notation means that the commutator [A,Bn] can be written as the group element cnBn-1, where c is a constant and Bn-1 represents the elements obtained by multiplying n-1 elements from group B. In other words, the commutator is a group element that can be obtained by multiplying a constant with elements from group B, with n-1 of those elements being multiplied together.

3. Why is the Commutator Problem important?

The Commutator Problem is important in the field of group theory because it allows us to understand the structure of groups and their commutativity. Solving the Commutator Problem can provide insights into the behavior of groups and their subgroups, and can also help in determining the properties of specific group elements.

4. How is the Commutator Problem solved?

The Commutator Problem is typically solved by using various mathematical techniques, such as induction, group theory theorems, and algebraic manipulation. The specific approach used to solve the problem may vary depending on the groups A and B and the given conditions. In some cases, a direct proof may be used, while in others, a more complex and indirect approach may be necessary.

5. What are some applications of the Commutator Problem?

The Commutator Problem has applications in various fields, including physics, chemistry, and computer science. In physics, it is used to study the symmetries and conservation laws of physical systems. In chemistry, it is used to understand the properties of molecules and their reactions. In computer science, it is used in the development of algorithms and protocols for efficient data processing and communication.

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