Point symmetry generators

In summary, the conversation discusses the problem of calculating the product of two operators in Lie algebras. The final expression is explained as a result of the difference between the two operators and the functions that multiply the derivatives. It is suggested to do the computation and subtract the second derivative terms to get the correct first derivative terms. The speaker expresses their dislike for tedious calculations but acknowledges their importance in understanding mathematics.
  • #1
2
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I am taking a first course in Lie algebras and currently working with this problem (see attached file). I understand that the product of the two operators should be regarded as composition. How to explain the final expression?
Regards
Staffan
 

Attachments

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  • #2
Haven't you tried just doing the computation? That's how you learn mathematics- not just by looking at formula and expecting to understand them, but by actually doing the calculations!

You are given that
[tex]Z_j= \xi_j(x,u)\frac{\partial}{\partial x}+ \chi_j(x,u)\frac{\partial}{\partial u}[/tex]
for j= 1 and 2. In other words, the difference between Z1 and Z2 is the functions multiplying the derivatives.

If f is any function of x and u (any reason for using x and u instead of x and y?) then
[tex]Z_1 Z_2(f)= \xi_1(x,u)\frac{\partial}{\partial x}+ \chi_1(x,u)\frac{\partial}{\partial u}[ \xi_2(x,u)\frac{\partial f}{\partial x}+ \chi_2(x,u)\frac{\partial f}{\partial u}][/tex]
[tex]= \xi_1(x,u)\frac{\partial}{\partial x}[\xi_2(x,u)\frac{\partial f}{\partial x}+ \chi_2(x,u)\frac{\partial f}{\partial u}]+ \chi_j(x,u)\frac{\partial}{\partial u}[\xi_2(x,u)\frac{\partial f}{\partial x}+ \chi_2(x,u)\frac{\partial f}{\partial u}][/tex]
[tex]= \xi_1\xi_2\frac{\partial^2 f}+ \xi_1 \frac{\partial \xi_2}{\partial x}\frac{\partial f}{\partial x}+ \cdot\cdot\cdot[/tex]

Finish that, then do the same for [itex]Z_2Z_1[/itex] and subtract. All the second derivative terms (those not involving derivatives of [itex]\xi_1[/itex], [itex]\xi_2[/itex], [itex]\chi_1[/itex], or [itex]\chi_2[/itex]) will cancel leaving only first derivative terms.
 
  • #3
Yes, I *have* calculated but I didn't manage to get the 1st derivatives right. Thanks a lot!
 
  • #4
Yeah, I hate tedious calculations like that!
 

What is the definition of a point symmetry generator?

A point symmetry generator is a point in a geometric figure that, when rotated 180 degrees around it, produces the same figure.

How many point symmetry generators can a geometric figure have?

A geometric figure can have multiple point symmetry generators, but it must have at least one.

What is the relationship between point symmetry generators and rotational symmetry?

Point symmetry generators are the points of a figure that have rotational symmetry of order 2, meaning they can be rotated 180 degrees and still maintain the same appearance.

What is the difference between a point symmetry generator and a line symmetry?

A point symmetry generator is a single point in a figure that creates symmetry, while a line symmetry is a line that divides the figure into two equal halves that are mirror images of each other.

How are point symmetry generators used in real-life applications?

Point symmetry generators are used in a variety of fields, such as art, architecture, and engineering, to create aesthetically pleasing designs and structures. They are also used in mathematics to study the properties of geometric figures and in physics to understand the symmetries of physical laws and systems.

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