Automorphic lie algebras and cohomology of root systems

This paper defines a cohomology theory of root systems which emerges naturally in the context of Automorphic Lie Algebras (ALiAs) but applies more generally to deformations of Lie algebras obtained by assigning a monomial in a finite number of variables to each weight vector. In the theory of Automorphic Lie Algebras certain problems can be formulated and partially solved in terms of cohomology, in particular one can find explicit models for an ALiA in terms of monomial deformations of the original Lie algebra. In this paper we formulate a cohomology theory of root systems and define the cup product in this context; we show that it can be restricted to symmetric forms, that it is equivariant with respect to the automorphism group of the original Lie algebra, and finally we show acyclicity at dimension two of the symmetric part, which is exactly what is needed to find models for ALiAs explicitly.