Let $g$ a finite-dimensional complex simple Lie algebra and $\sigma$ a finite order Dynikin diagram automorphism of $g$. 


Consider $\tilde g$ as the loop algebra associated to $g$, and $\tilde g^\sigma$ as the twisted affine Lie algebra (associated to $g$) in the spirit of the theory developed in the book ''Infinite dimensional Lie algebra" by Kac, or ''Lie algebras of Finite and Affine Type'' by Roger Carter.


It is easy to see that given an action of $\tilde g$ we have an action of $\tilde g^\sigma$ which is given by the restriction of the action of $\tilde g$. On the opposite way, my question is: Given an action of $\tilde g^\sigma$ on a module $M$, is it possible to extend this action to $\tilde g$?


Moreover, if we restrict to the context of universal highest-weight modules for $\tilde g^\sigma$ can we produce a highest-weight module for $\tilde g$ by extending the action of $\tilde g^\sigma$?


What should be a reference for this subject concerning about these extensions?


Thanks,



twisted affine algebras
To add to what Carnahan has posted. There is a difference between the representation theories of untwisted and twisted affine algebras. But they seem to be unified through vertex algebra theory.


Vertex operator algebras themselves are "untwisted" but admit twisted modules. Haisheng Li has some results that make this statement more concrete, but essentially, while there are twisted VOA modules, there are no twisted VOAs. 


So though Affine VOAs are associated with untwisted affine algebras, they do include representations of the corresponding twisted affine. 


You may want to take a look at some of Haisheng Li's work in this area. For example:


Li, Haisheng
A new construction of vertex algebras and quasi-modules for vertex algebras.
Adv. Math. 202 (2006), no. 1, 232–286. 17B69


Take a look at section 7, specifically, Proposition 7.4


