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I know the following definitions (or notions) of a Lie algebra root:

  1. Lie algebra roots are the eigenvalues of a Cartan subalgebra in the adjoint representation. In other words, to find the roots of a Lie algebra, find a Cartan subalgebra $\{H_i\}$ and then find the eigenvalues $a_{ij}$ of their adjoint representation, i.e. solve $[\rho_{\mathrm{adj}}(H_i), E_j] = a_{ij} E_j$.
  2. Lie algebra roots are the weights of the adjoint representation.
  3. Lie algebra roots are the vectors connecting the weights of the fundamental representation. In other words, to find the roots of a Lie algebra, find the weights $\{w_i\}$ of the fundamental representation. Then the roots $a_{ij}$ can be found by $a_{ij} = w_i-w_j$.

I'd like to know why these three notions are equivalent.

  • Am I correct that 1. is just another way of saying 2.? Are 1. and 2. by definition equal, or is there anything to prove to show their equivalence?
  • How can I show that 1. and 3. are equivalent?
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I still think 1. and 2. are the same thing.

As for 3. it's easy to show the equivalence to 1.:

Let $H$ be in a Cartan subalgebra, $\{E_\alpha\}$ a positive generator corresponding to the root $\alpha$ (i.e. $[H,E_\alpha]=\alpha E_\alpha$) and $\mathbb V_\lambda$ a weight space of an irrep $\rho$, i.e. $\rho(H)v = \lambda v$ for all $v\in\mathbb V_\lambda$. Then $\rho(E_\alpha)\mathbb V_\lambda \subset\mathbb V_{\alpha+\lambda}$, because

\begin{align} \rho(H)\rho(E_\alpha)v &= \rho(HE_\alpha)v \\ &= \rho(E_\alpha H)v + \rho([H,E_\alpha])v \\ &= \rho(E_\alpha)\rho(H)v + \alpha \rho(E_\alpha)v \\ &= (\lambda + \alpha)\rho(E_\alpha)v \end{align}

so $\rho(E_\alpha)v\in\mathbb V_{\lambda + \alpha}$.

This means that the weights of an irrep $\rho$ are positioned in a lattice and the roots of the Lie Algebra are the lattice spacings, i.e. $w_i - w_j = \alpha$.

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