Automorphic Lie Algebras with Dihedral Symmetry

Automorphic Lie Algebras are interesting because of their fundamental nature and their role in our understanding of symmetry. Particularly crucial is their description and classification as it allows us to understand and apply them in different contexts, from mathematics to physical sciences. While the problem of classification of Automorphic Lie Algebras with dihedral symmetry was already considered in the past (see for instance [5], [6], and [7]), it was never addressed in full generality, where one takes any two representations of the dihedral group. Indeed, all results so far have been obtained under the simplifying assumption that one can use the same dihedral representation to define an action on either the space of vectors or matrices, and over the polynomials. In this paper we present a complete classification of Automorphic Lie Algebras with dihedral symmetry, starting from any two representations. In the light of this result we show that Automorphic Lie Algebras with dihedral symmetry are isomorphic, thereby extending the uniformity result of Lombardo and Sanders [7] and Bury [1]. Furthermore, we consider also the case of invariant vectors and compute the corresponding Molien functions; their knowledge allows us to compute symmetric rational maps related to the energy of Skyrmions with dihedral symmetry.