On the Classification of Automorphic Lie Algebras

It is shown that the problem of reduction can be formulated in a uniform way using the theory of invariants. This provides a powerful tool of analysis and it opens the road to new applications of these algebras, beyond the context of integrable systems. Moreover, it is proven that sl2(C)–Automorphic Lie Algebras associated to the icosahedral group I, the octahedral group O, the tetrahedral group T, and the dihedral group Dn are isomorphic. The proof is based on techniques from classical invariant theory and makes use of Clebsch-Gordan decomposition and transvectants, Molien functions and the trace-form. This result provides a complete classification of sl2(C)–Automorphic Lie Algebras associated to finite groups when the group representations are chosen to be the same and it is a crucial step towards the complete classification of Automorphic Lie Algebras. 1 Algebraic reductions and Automorphic Lie Algebras Many integrable equations are obtained as reductions of larger systems. The fact that this is true for many equations of interest in applications makes of the reduction problem one of the central problems in the theory of integrable systems since its early days. A wide class of (algebraic) reductions can be studied in terms of reduction groups [Mik81], that is, reductions can be associated to a discrete symmetry group of the corresponding linear problem (Lax Pair), either given by the physical system or simply forced on the solutions. The simplest example of such a symmetry is the conjugation for self-adjoint operators. The requirement that a Lax pair is invariant with respect to a reduction group imposes certain algebraic constraints on the Lax operators and therefore it yields a reduction. As an illustration, consider for instance a fairly general Lax pair L = ∂x −X(x, t, λ) , M = ∂t − T (x, t, λ) , where X(x, t, λ) = X0(x, t) +X1(x, t)λ+X−1(x, t) 1 λ , T (x, t, λ) = T0(x, t) + T1(x, t)λ+ T−1(x, t) 1 λ + T2λ + T−2(x, t) 1 λ2 are n × n matrix functions of x, t and of the spectral parameter λ. The consistency condition ψtx = ψxt implies that (for all values of λ) Xt − Tx + [X , T ] = 0