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Thread: 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

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

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

    $\displaystyle (\alpha,\alpha)=1,\ \ (\beta,\beta)=\frac{-1}{2}\frac{-1}{2}+\frac{\sqrt{3}}{2}\frac{\sqrt{3}}{2}=1$ and $\displaystyle (\alpha,\beta)=-\frac{1}{2}$ 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

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

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

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

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

    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

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

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

    $\displaystyle (\alpha,\alpha)=1,\ \ (\beta,\beta)=\frac{-1}{2}\frac{-1}{2}+\frac{\sqrt{3}}{2}\frac{\sqrt{3}}{2}=1$ and $\displaystyle (\alpha,\beta)=-\frac{1}{2}$ 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

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

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

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

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

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

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

    Then, the kth fundamental weight is


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

    I didn't study your book, but I think p38 shows the relevant example. I think $\displaystyle G_{ij}$ 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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