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Math Help - Lie Algebra-Roots and Dynkin basis

  1. #1
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    Lie Algebra-Roots and Dynkin basis

    Given a Lie algebra I understand all the stuff about the Cartan-Weyl basis, the roots and stuff, so for example, for A2, the roots are

    \alpha=(1,0), \ \ \beta=\frac{1}{2}(-1,\sqrt{3}),\ \ \gamma=\frac{1}{2}(1,\sqrt{3})

    with simple roots +\alpha and +\beta. The scalar products are

    (\alpha,\alpha)=1,\ \ (\beta,\beta)=\frac{-1}{2}\frac{-1}{2}+\frac{\sqrt{3}}{2}\frac{\sqrt{3}}{2}=1 and (\alpha,\beta)=-\frac{1}{2} using the definition of the scalar product in this basis from which one calculates the Cartan matrix and angles between roots blah blah blah...

    I am reading 'Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists' By Jürgen Fuchs, Christoph Schweigert and I do not understand the Dynkin basis. It says the quadratic form matrix is given by

    G_{ij}=(\Lambda_{(i)},\Lambda_{(j)}) and G^{ij}=(\check{\alpha}^{(i)},\check{\alpha}^{(j)})  =\frac{2}{(\alpha^{(i)},\alpha^{(i)})}A_{ij}

    where A^{ij} is the cartan matrix, \check{\alpha} is the co-root \check{\alpha}=\frac{2\alpha}{(\alpha,\alpha)} and \Lambda is the weight in the dual space with \Lambda_{(i)}(\check{\alpha}^{(j)})=\delta_i^j. From this, they claim (\check{\alpha}^{(j)})_i=(\Lambda_{(i)})^j=\delta_  i^j and the Dynkin basis is

    (\alpha^{(i)})^j=A^{ij}.

    How does one calculate the matrix elements G_{ij}? It seems a recursive definition of G_{ij} because it is written in terms of the inner product but to define the inner product in this basis one needs to already know G_{ij}

    Does my confusion make sense to anyone and could someone please clear this up for me?
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  2. #2
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    Quote Originally Posted by ppyvabw View Post
    Given a Lie algebra I understand all the stuff about the Cartan-Weyl basis, the roots and stuff, so for example, for A2, the roots are

    \alpha=(1,0), \ \ \beta=\frac{1}{2}(-1,\sqrt{3}),\ \ \gamma=\frac{1}{2}(1,\sqrt{3})

    with simple roots +\alpha and +\beta. The scalar products are

    (\alpha,\alpha)=1,\ \ (\beta,\beta)=\frac{-1}{2}\frac{-1}{2}+\frac{\sqrt{3}}{2}\frac{\sqrt{3}}{2}=1 and (\alpha,\beta)=-\frac{1}{2} using the definition of the scalar product in this basis from which one calculates the Cartan matrix and angles between roots blah blah blah...

    I am reading 'Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists' By Jürgen Fuchs, Christoph Schweigert and I do not understand the Dynkin basis. It says the quadratic form matrix is given by

    G_{ij}=(\Lambda_{(i)},\Lambda_{(j)}) and G^{ij}=(\check{\alpha}^{(i)},\check{\alpha}^{(j)})  =\frac{2}{(\alpha^{(i)},\alpha^{(i)})}A_{ij}

    where A^{ij} is the cartan matrix, \check{\alpha} is the co-root \check{\alpha}=\frac{2\alpha}{(\alpha,\alpha)} and \Lambda is the weight in the dual space with \Lambda_{(i)}(\check{\alpha}^{(j)})=\delta_i^j. From this, they claim (\check{\alpha}^{(j)})_i=(\Lambda_{(i)})^j=\delta_  i^j and the Dynkin basis is

    (\alpha^{(i)})^j=A^{ij}.

    How does one calculate the matrix elements G_{ij}? It seems a recursive definition of G_{ij} because it is written in terms of the inner product but to define the inner product in this basis one needs to already know G_{ij}

    Does my confusion make sense to anyone and could someone please clear this up for me?
    Let \Delta={\alpha_1, \ldots, \alpha_r} be a base consisting of positive simple roots. The fundamental weights with respect to \Delta are the elements \mu_1, \ldots, \mu_r satisfying

    2\dfrac{<\mu_k, \alpha_m>}{<\alpha_m, \alpha_m>}=\delta_{km}, k, m=1,\ldots, r.

    Then, the kth fundamental weight is


    \mu_k=\dfrac{1}{2}\mu\dfrac{<\alpha_k, \alpha_k>}{<\mu, \alpha_k>}, where \mu is an integral element.

    I didn't study your book, but I think p38 shows the relevant example. I think G_{ij} needs fundamental weights which can be obtained from the above formula. My knowledge of this area is limited, so hope this is a right direction
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