gl(3) commutation relations question

consider the polynomial algebra with 3 independent real variables

show that the differential operators

satisfy the gl(3) commutation relations

does this mean i have to show that ?

this is my working (assuming thats what i have to do):

but for this to equal , that means that has to equal and has to equal ...is this correct?

am i just completely on the wrong path or am i headed in the right direction?? please someone help!!!

Re: gl(3) commutation relations question

Quote:

Originally Posted by

**wik_chick88** consider the polynomial algebra

with

3 independent real variables

show that the differential operators

satisfy the gl(3) commutation relations

does this mean i have to show that

?

this is my working (assuming thats what i have to do):

but for this to equal

, that means that

has to equal

and

has to equal

...is this correct?

That is correct, assuming your variables are independent.

Quote:

am i just completely on the wrong path or am i headed in the right direction?? please someone help!!!

Re: gl(3) commutation relations question

Quote:

Originally Posted by

**wik_chick88** consider the polynomial algebra

with

3 independent real variables

show that the differential operators

satisfy the gl(3) commutation relations

does this mean i have to show that

?

this is my working (assuming thats what i have to do):

but for this to equal

, that means that

has to equal

and

has to equal

...is this correct?

am i just completely on the wrong path or am i headed in the right direction?? please someone help!!!

the product rule for differentiation gives us: thus

Re: gl(3) commutation relations question

thankyou. they are specified as being independent so now i can see how that can work.

now i have to use ado's theorem to find expressions for the o(3) generators in terms of (*) and show that

any idea how im supposed to start this?

i know that ado's theorem states that every finite dimensional Lie Algebra over a fiel of characteristic 0 has a faithful finite dimensional representation, which means it can be viewed as a Lie Algebra of square matrices under the commutator bracket.

im not exactly sure what the o(3) generators are, only that they have the basis