I know the following definitions (or notions) of a Lie algebra root:
- 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 $[\rho_{\mathrm{adj}}(H_i), E_j] = a_{ij} E_j$.
- Lie algebra roots are the weights of the adjoint representation.
- 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?
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. $w_i - w_j = \alpha$.