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Math Help - gl(3) commutation relations question

  1. #1
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    gl(3) commutation relations question

    consider the polynomial algebra C[x_1, x_2, x_3] with x_1, x_2, x_3 3 independent real variables
    show that the differential operators
    a_{ij} = x_{i}\frac{\partial}{\partial x_{j}}
    satisfy the gl(3) commutation relations

    does this mean i have to show that [a_{ij}, a_{kl}] = \delta_{kj}a_{il} - \delta_{il}a_{kj}?

    this is my working (assuming thats what i have to do):
    [a_{ij}, a_{kl}]
    = [x_i\frac{\partial}{\partial x_{j}}, x_k\frac{\partial}{\partial x_{l}}]
    = x_i\frac{\partial}{\partial x_{j}} \times x_k\frac{\partial}{\partial x_{l}} - x_k\frac{\partial}{\partial x_{l}} \times x_i\frac{\partial}{\partial x_{j}}
    = \frac{\partial x_k}{\partial x_j} \times x_i\frac{\partial}{\partial x_{l}} - \frac{\partial x_i}{\partial x_l} \times x_k\frac{\partial}{\partial x_{j}}

    but for this to equal \delta_{kj}a_{il} - \delta_{il}a_{kj}, that means that \delta_{kj} has to equal \frac{\partial x_k}{\partial x_j} and \delta_{il} has to equal \frac{\partial x_i}{\partial x_l}...is this correct?

    am i just completely on the wrong path or am i headed in the right direction?? please someone help!!!
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  2. #2
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    Re: gl(3) commutation relations question

    Quote Originally Posted by wik_chick88 View Post
    consider the polynomial algebra C[x_1, x_2, x_3] with x_1, x_2, x_3 3 independent real variables
    show that the differential operators
    a_{ij} = x_{i}\frac{\partial}{\partial x_{j}}
    satisfy the gl(3) commutation relations

    does this mean i have to show that [a_{ij}, a_{kl}] = \delta_{kj}a_{il} - \delta_{il}a_{kj}?

    this is my working (assuming thats what i have to do):
    [a_{ij}, a_{kl}]
    = [x_i\frac{\partial}{\partial x_{j}}, x_k\frac{\partial}{\partial x_{l}}]
    = x_i\frac{\partial}{\partial x_{j}} \times x_k\frac{\partial}{\partial x_{l}} - x_k\frac{\partial}{\partial x_{l}} \times x_i\frac{\partial}{\partial x_{j}}
    = \frac{\partial x_k}{\partial x_j} \times x_i\frac{\partial}{\partial x_{l}} - \frac{\partial x_i}{\partial x_l} \times x_k\frac{\partial}{\partial x_{j}}

    but for this to equal \delta_{kj}a_{il} - \delta_{il}a_{kj}, that means that \delta_{kj} has to equal \frac{\partial x_k}{\partial x_j} and \delta_{il} has to equal \frac{\partial x_i}{\partial x_l}...is this correct?
    That is correct, assuming your variables are independent.

    am i just completely on the wrong path or am i headed in the right direction?? please someone help!!!
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  3. #3
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    Re: gl(3) commutation relations question

    Quote Originally Posted by wik_chick88 View Post
    consider the polynomial algebra C[x_1, x_2, x_3] with x_1, x_2, x_3 3 independent real variables
    show that the differential operators
    a_{ij} = x_{i}\frac{\partial}{\partial x_{j}}
    satisfy the gl(3) commutation relations

    does this mean i have to show that [a_{ij}, a_{kl}] = \delta_{kj}a_{il} - \delta_{il}a_{kj}?

    this is my working (assuming thats what i have to do):
    [a_{ij}, a_{kl}]
    = [x_i\frac{\partial}{\partial x_{j}}, x_k\frac{\partial}{\partial x_{l}}]
    = x_i\frac{\partial}{\partial x_{j}} \times x_k\frac{\partial}{\partial x_{l}} - x_k\frac{\partial}{\partial x_{l}} \times x_i\frac{\partial}{\partial x_{j}}
    = \frac{\partial x_k}{\partial x_j} \times x_i\frac{\partial}{\partial x_{l}} - \frac{\partial x_i}{\partial x_l} \times x_k\frac{\partial}{\partial x_{j}}

    but for this to equal \delta_{kj}a_{il} - \delta_{il}a_{kj}, that means that \delta_{kj} has to equal \frac{\partial x_k}{\partial x_j} and \delta_{il} has to equal \frac{\partial x_i}{\partial x_l}...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: a_{ij}a_{k \ell}=x_i \frac{\partial}{\partial x_j}x_k \frac{\partial}{\partial x_{\ell}} =x_i \left( \delta_{jk} \frac{\partial}{\partial x_{\ell}} + x_k \frac{\partial^2}{\partial x_j \partial x_{\ell}} \right). thus

     [a_{ij},a_{k \ell}]=a_{ij}a_{k \ell} - a_{k \ell}a_{ij}=\delta_{jk}x_i \frac{\partial}{\partial x_{\ell}} - \delta_{j \ell}x_k \frac{\partial}{\partial x_j}= \delta_{jk}a_{i \ell} - \delta_{i \ell}a_{kj}.
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  4. #4
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    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 L_1, L_2, L_3 in terms of (*) and show that L^2 = x^2\nabla^2 - \vec x.(\vec x.\nabla + 2)\nabla

    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 \{{\alpha^i}_j = e_{ij} - e_{ji} : i, j = 1, 2, 3\}
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