# Understanding Weights & Roots: Why Does T Act Trivially?

• jdstokes
In summary, a weight space is a 1-dimensional irreducible representation of the maximal torus in a Lie algebra. When this representation is restricted to the torus, it becomes a direct sum of weights. In the adjoint representation, the roots are defined as the nontrivial weights. The trivial weights occur when the torus acts trivially on its own tangent space. This is because the adjoint action of the torus on itself is identity, while the action on the rest of the Lie algebra may be non-trivial, leading to the presence of nontrivial weights or roots.
jdstokes
I'm having trouble understanding the idea of a weight space.

Suppose $\mathfrak{g}$ is the Lie alebra of G with maximal torus T and Cartan subalgebra $\mathfrak{t}$. The weights are the (1-dimensional) irreducible represenations of T. If we restrict any representation $\rho : G \to GL(V)$ to T ($\rho|_T : T \to GL(V)$) then we get a direct sum of weights $\alpha_i$. If $\rho$ is taken to be the adjoint representation, then the roots are defined to be the nontrivial weights of this rep.

My question concerns the trivial weights. Why exactly is it that T acts trivially on its own tangent space $\mathfrak{t}$?

I realized that this is pretty obvious since the adjoint action of T on itself is just identity because T is abelian. Thus for any $t \in T$, the linear transformation $Ad|_T (t) : \mathfrak{g} \to \mathfrak{g}$ acts trivially on the Cartan subalgebra $\mathfrak{h}$. The T-action on the remainder of the Lie algebra, however may be non-trivial, which is where the nontrivial weights (ie roots) enter.

## 1. What are weights and roots?

Weights and roots are mathematical concepts used in algebra and number theory. Weights refer to the coefficients of variables in an equation, while roots refer to the solutions or values that make the equation true.

## 2. How are weights and roots related?

Weights and roots are related through equations and polynomials. The weights in an equation determine the roots or solution to that equation. Conversely, the roots can be used to find the weights in an equation.

## 3. What is meant by "T acts trivially"?

When we say "T acts trivially", it means that the variable or unknown quantity T has no effect on the equation or solution. This can happen when T has a weight of zero, making it essentially disappear from the equation.

## 4. Why is it important to understand weights and roots?

Understanding weights and roots is important in solving equations and problems in mathematics. It allows us to determine the solutions to equations and to manipulate equations to find the desired outcome. It also helps us understand the relationship between different variables in an equation.

## 5. How can I improve my understanding of weights and roots?

To improve your understanding of weights and roots, it is important to practice solving equations and working with polynomials. You can also read up on the concepts and their applications in algebra and number theory. Seeking help from a tutor or teacher can also greatly improve your understanding.

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