A derangement is a permutation with no fixed points, and various combinatorial formulas exist for the number of derangements of n objects. I will describe a new algebra which has a basis indexed by the derangements of k things. This algebra occurs as the algebra of all linear operators on the kth tensor power of the Lie algebra sl(n) commuting with the adjoint action of sl(n). Motivation for this work (which is joint work with Georgia Benkart) comes from the classic situation of Schur-Weyl duality between the action of general linear groups and symmetric groups on tensor powers of a vector space. The work can be regarded as an an extension of that classic duality to a new context.