What does the weights of Lie group mean?

Question

Let $\Delta=\{\alpha_1,\alpha_2\}$ be the simple root system

of the exceptional Lie group $G_2$ with $\alpha_1$ is short and $\alpha_2$ is long, so $\lambda_1=2\alpha_1+\alpha_2,\lambda_2=3\alpha_1+2\alpha_2$ are the fundamental dominant weights. Let $T$ be the maximal torus of $G_2,$ then $H^*(BT;Z)$ is a polynomial algebra on two generators. Can we see $\alpha_1$ and $\alpha_2$ as the generators? What about $\lambda_1$ and $\lambda_2$?

Answer 1

Let T be an arbitrary compact torus. The second cohomology group of BT (with arbitrary coefficients, call that ring k) generates the full cohomology freely as an algebra. In other words, if you pick a k-basis x₁, x₂,... of H²(BT), then you get an isomorphism of H*(BT) with k[x₁, x₂,...].

Now let's specialise to the case k=ℤ. In that case, the second cohomology group of BT is canonically isomorphic to the group of characters of T, i.e., to the group of homomorphisms from T → S¹. Given a character χ :T → S¹, the corresponding element of H²(BT) is represented by (the first Chern class of) the complex line bundle ET ×_T ℂ_χ → BT = ET/T.

Now back to your question. The elements α₁ and α₂ form a basis of BT, where T now refers to the maximal torus of G₂. So you get an isomorphism H*(BT;ℤ) $\xrightarrow{\sim}$ ℤ[α₁, α₂]. But λ₁ and λ₂ also form a basis of BT. So you get another isomoprhism H*(BT;ℤ) $\xrightarrow{\sim}$ ℤ[λ₁, λ₂].

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Let me also answer the question in the title of your question:

By "weight of a Lie group", one means a homomorphism from its maximal torus to S¹.