Question on tensor notation in group theory

In summary, in the appendix of Zee's book on Group Theory, there is a discussion of commutations for SO(3). The first question asks if the commutator between two J matrices is equal to the product of the matrices minus the product in the opposite order. The second question is about an expression in the appendix that states the commutator is equal to i multiplied by the Kronecker delta with different indexes. The significance of this notation is not understood.
  • #1
BWV
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in the appendix on Group Theory in Zee's book there is a discussion of commutations for SO(3)

two questions

- does [[itex]J^{ij},J^{lk}] = J^{ij}*J^{lk}-J^{lk}*J^{ij}[/itex]?

and there is an expression in the appendix that the commutator equals i([itex]\delta^{ik}J^{jl} ...[/itex]

i don't understand the why you are multiplying the matrix by the kronecker delta with different upstairs indexes, is not it simply an identity?
 
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  • #2
bumping this - this apparently is some notation I don't understand, the Wiki entry on Special Unitary groups has this as well

304a3ac6552e3b2d08fc00e5bdb18736.png


I don't get the significance of the kronecker deltas with the different indexes
 

1. What is tensor notation in group theory?

Tensor notation in group theory is a mathematical notation used to represent the elements and operations of a group. It involves using indices and superscripts to represent the components of a tensor, which is a multi-dimensional array that transforms in a specific way under the operations of the group.

2. How is tensor notation used in group theory?

Tensor notation is used in group theory to define and manipulate the elements of a group and their transformations. It allows for a compact and concise representation of group operations and is often used in calculations and proofs in group theory.

3. What are the benefits of using tensor notation in group theory?

Tensor notation provides a more concise and efficient way to represent the elements and operations of a group compared to other notations. It also allows for easier manipulation and calculation of group properties and transformations.

4. Are there any drawbacks to using tensor notation in group theory?

One potential drawback of tensor notation is that it can be more abstract and difficult to understand for those unfamiliar with it. It also requires a certain level of mathematical knowledge and skill to use effectively.

5. Can tensor notation be used in other areas of science besides group theory?

Yes, tensor notation is a common notation used in many areas of science, including physics, engineering, and computer science. It is particularly useful in fields that involve multidimensional data and transformations, such as fluid dynamics and general relativity.

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