Automorphic Lie algebras with dihedral symmetry

Abstract

The concept of Automorphic Lie Algebras arises in the context of reduction groups introduced in the early 1980s in the field of integrable systems. Automorphic Lie Algebras are obtained by imposing a discrete group symmetry on a current algebra of Krichever-Novikov type. Past work shows remarkable uniformity between algebras associated to different reduction groups. For example, if the base Lie algebra is sl 2 (C) and the poles of the Automorphic Lie Algebra are restricted to an exceptional orbit of the symmetry group, changing the reduction group does not affect the Lie algebra structure. In the present research we fix the reduction group to be the dihedral group and vary the orbit of poles as well as the group action on the base Lie algebra. We find a uniform description of Automorphic Lie Algebras with dihedral symmetry, valid for poles at exceptional and generic orbits.

Higher-Dimensional Automorphic Lie Algebras

“…In this paper, we focus on the exceptional cases (since they are not part of infinite families), the TOY groups. The D N -classification has been presented in [17], both for generic and exceptional G-orbits, since the D N computations can be done explicitly without the use of a computer. In addition, this is the only nonabelian group in Klein's classification whose order depends on N , which is a complication from a computational point of view, and we prefer to keep it separate.…”

“…This will be enough to construct a minimal generating set for the Automorphic Lie Algebra (see [17,19]). …”

“…We then consider (T i ⊗ k 12 [T 4 ]) T , as it is sufficient to consider entries of degree equal to the order of the group |T| (see [17,19]). …”

“…In [24], the authors study automorphic algebras associated with the dihedral group D N , starting from the finite-dimensional algebra sl 2 (C); examples of ALiAs based on sl 3 (C) were also discussed. In [17], the authors present a classification of automorphic algebras associated with the dihedral group D N , where the action is inner and no summands are trivial. A further, crucial, step towards the full classification appears in [25], where the problem is formulated in a uniform way using the theory of invariants.…”

Module structure of the center of the universal central extension of a genus zero Krichever–Novikov algebra

“…We should also mention work of S. Lombardo, A. V. Mikhailov, and J. A. Sanders dealing with the structure of genus zero multi-point Krichever-Novikov current al-gebras (and their modules) fixed under the same type of symmetries as in our work with X. Guo, R. Lu, and K. Zhao (see [LM05a], [LM05b], [LS10], and [KLS14]). They do not describe how the center of the universal central extension decomposes into a direct sum of irreducible modules for the automorphism group of the coordinate ring.…”