10
$\begingroup$

Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$, or its compact form over $\mathbb{R}$. Recall that the automorphism group $\operatorname{Aut}(\mathfrak{g})$ is of the form $G^{\mathrm{adj}}\rtimes \operatorname{Out}(\mathfrak{g})$, where $G^{\mathrm{adj}}$ is the adjoint form of $G$ and $\operatorname{Out}(\mathfrak{g})$ is the group of Dynkin diagram automorphisms.

A theorem of Kac parameterizes the conjugacy classes of finite-order elements of $\operatorname{Aut}(\mathfrak{g})$. It is quoted for example in Theorem 8.3.3.1 of these lecture notes.

Briefly, you write down the affine Dynkin diagram of $\mathfrak{g}$, with its numbering (meaning numbers $m_i$ attached to each root, so that the affine root is given the number $m_0 = 1$, and the vector $\vec m$ is in the kernel of the Cartan matrix); then a finite-order element is, up to conjugacy, a pair $\rho \in \operatorname{Out}(\mathfrak{g})$ of order $r$ and some nonnegative rational numbers $a_i$ assigned to the roots of the $\rho$-folded Dynkin diagram (the twisted affine Dynkin diagram of the $\rho$-folded diagram of $\mathfrak{g}$), so that $\sum a_i m_i = 1/r$. (The order is the common denominator of the $a_i$.) These data are considered up to Dynkin diagram automorphism.

It is pretty easy to see why these data parameterize elements of $\operatorname{Aut}(\mathfrak{g})$. Indeed, the idea is to describe the maximal torus therein as $(\mathbb Q/\mathbb Z) \otimes (\text{weight lattice})$, and then to track any further Weyl group actions.

Is there a similarly simple description of the power operations $x \mapsto x^n$ on this set, for $n \in \mathbb Z$?

The easiest case is when $\rho = \mathrm{id}$, in which case the power operation is "$a_i \mapsto na_i$," more or less. But that map quickly destroys any bounds on $\sum a_i m_i$, so you need to subtract off some integers and then act by the Weyl group.

Improve this question
$\endgroup$
6
  • $\begingroup$ Do you mean "some nonnegative rational numbers $a_i$" (rather than $n_i$)? $\endgroup$
    – Mikhail Borovoi
    Jan 16, 2020 at 18:53
  • $\begingroup$ Do you mean "the power operations $x\mapsto x^n$ on this set, for $n\in\mathbb Z$" rather than "for $a\in \mathbb Z$"? $\endgroup$
    – Mikhail Borovoi
    Jan 16, 2020 at 18:56
  • $\begingroup$ @MikhailBorovoi Oops, yes, I will correct it. $\endgroup$
    – Theo Johnson-Freyd
    Jan 17, 2020 at 5:52
  • 1
    $\begingroup$ @MikhailBorovoi I think all I need is that $\mathfrak{g}$ be semisimple. If it is not simple, then the Dynkin diagram may be disconnected, and there may be outer automorphisms that permute the components of the diagram. I think that what I'm calling "the twisted affine Dynkin diagram of the $\rho$-folded diagram of $\mathfrak{g}$" is what you are calling the "$\rho$-twisted affine Dynkin diagram", but perhaps I am using the terminology wrong? $\endgroup$
    – Theo Johnson-Freyd
    Jan 18, 2020 at 8:28
  • 1
    $\begingroup$ I think that there is no standard terminology, and I find your terminology to be better than mine. Anyway, to vertices of what do you attach the numbers $m_i$, and how do you define them? $\endgroup$
    – Mikhail Borovoi
    Jan 18, 2020 at 9:56

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.