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!!!
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