Toral arrangements and hyperplane arrangements

Abstract

We consider pairs, (T, A), where T is a torus and A is a finite set of characters of T. Then dA = (ker(dx) | X ∈A) is a finite set of hyperplanes in the Lie algebra of T. Let 풪Tbe the coordinate ring of T, and 풪T, e the local ring of the identity in T. In analogy with hyperplane arrangements, put y = Πi(xi − 1), and consider the set, D(A), of derivations, θ, of 풪T that satisfy θ(y) ∈ y풪t. The main results are that the localization of D(A) at the identity of T is a free 풪T, e module if and only if dA is a free hyperplane arrangement, and that if this is the case, then the exponents of dA can be recovered from A. © 1998 Rocky Mountain Mathematics Consortium.