Trace((ad_x)(ad_y)) = 2n(trace(xy))

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E_{mj}, E_{mi}] - [E_{mi}, E_{mj}]= \sum_{i=1}^n \sum_{j=1}^n \sum_{m=1}^n x_{mi} y_{mj} [E_{mj}, E_{mi
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antiemptyv
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Homework Statement



Let L be the Lie algebra [tex]sl(n, F)[/tex] and [tex]X = (x_{ij}, Y = (y_{ij}) \in L[/tex].

Prove

[tex]\kappa(X,Y) = 2n Tr(XY)[/tex],

where [tex]\kappa(,)[/tex] is the Killing form and [tex]Tr()[/tex] is the trace form.

Homework Equations



For any unit matrix [tex]E_{ij}[/tex] and any [tex]X \in L[/tex],

[tex]XE_{ij} = \sum_{m=1}^n x_{mi} E_{mj}[/tex] and
[tex]E_{ij}X = \sum_{m=1}^n x_{jm}E_{im}.[/tex]

The Attempt at a Solution



I have reduced this to the following:

[tex]Tr(XY) = \sum_{k=1}^n \sum_{m=1}^n x_{mk}y_{km}[/tex]

[tex]\kappa(X,Y) = Tr(ad_X ad_Y) = \sum_{k=1}^n (x_{ik}y_{ki} + y_{kj}x_{jk} ) - 2 \sum_{i=1}^n \sum_{j=1}^n x_{ii}y_{jj}[/tex]

Perhaps I have been at this for too long, but I don't see why exactly 2n times the first expression is equivalent to the second expression. Any guidance would be appreciated.
 
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  • #2


To prove that \kappa(X,Y) = 2n Tr(XY), we can use the fact that the Killing form is defined as \kappa(X,Y) = Tr(ad_X ad_Y), where ad_X and ad_Y are the adjoint actions of X and Y, respectively.

Using the given equations, we can rewrite the adjoint actions as:

ad_X(E_{ij}) = [X, E_{ij}] = \sum_{m=1}^n x_{mi} E_{mj} - \sum_{m=1}^n x_{mj} E_{im}

ad_Y(E_{ij}) = [Y, E_{ij}] = \sum_{m=1}^n y_{mi} E_{mj} - \sum_{m=1}^n y_{mj} E_{im}

Substituting these into the definition of the Killing form, we get:

\kappa(X,Y) = Tr(ad_X ad_Y) = \sum_{i=1}^n \sum_{j=1}^n (ad_X(E_{ij}))_{ij} (ad_Y(E_{ij}))_{ji}

= \sum_{i=1}^n \sum_{j=1}^n (\sum_{m=1}^n x_{mi} E_{mj} - \sum_{m=1}^n x_{mj} E_{im})_{ij} (\sum_{m=1}^n y_{mj} E_{mi} - \sum_{m=1}^n y_{mi} E_{mj})_{ji}

= \sum_{i=1}^n \sum_{j=1}^n (\sum_{m=1}^n x_{mi} y_{mj} - \sum_{m=1}^n x_{mj} y_{mi}) (\sum_{m=1}^n y_{mi} x_{mj} - \sum_{m=1}^n y_{mj} x_{mi})

= \sum_{i=1}^n \sum_{j=1}^n (\sum_{m=1}^n x_{mi} y_{mj} y_{mi} x_{mj}) - (\sum_{m=1}^n x_{mi} y_{mj} y_{mj} x_{mi})

= \sum_{
 

FAQ: Trace((ad_x)(ad_y)) = 2n(trace(xy))

What is the meaning of the equation "Trace((ad_x)(ad_y)) = 2n(trace(xy))"?

The equation represents a relationship between two operators, ad_x and ad_y, which are used in mathematical physics to describe the motion and behavior of particles in a system. The left side of the equation represents the trace of the composition of these operators, while the right side represents the trace of their product. This equation is important in understanding the dynamics of systems with multiple particles.

What is the significance of the number 2n in the equation?

The number 2n is a scaling factor that accounts for the dimensionality of the system. In mathematical physics, it is often necessary to account for the number of dimensions in a system, and this number is represented by n. In this equation, the presence of 2n indicates that the trace of the composition of the operators is twice the trace of their product, due to the dimensionality of the system.

How is this equation used in scientific research?

This equation is used in a variety of fields, including quantum mechanics, statistical mechanics, and differential geometry. It allows scientists to calculate and analyze the behavior of particles in complex systems, such as atoms and molecules. It is also used in theoretical physics to understand the dynamics of particle interactions and the properties of physical systems.

What are some real-world applications of this equation?

One application of this equation is in the study of molecular dynamics, which is used to simulate the behavior of molecules in a system. This equation can also be applied to the study of quantum systems, such as atoms and subatomic particles. Additionally, it is used in the development of mathematical models for physical systems and in the analysis of data from experiments and observations.

Are there any limitations to this equation?

Like any mathematical model, this equation has limitations and may not accurately describe all systems. It is based on certain assumptions and simplifications, which may not always hold true in real-world scenarios. Additionally, the equation may not be applicable to all types of systems or particles, and further research and experimentation may be needed to fully understand the dynamics of a particular system.

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