Irreducible holomorphic symplectic manifolds, also called hyperkähler manifolds, are one of the building blocks of complex manifolds with trivial first Chern class. We will classify their symmetries. Thanks to breakthroughs in the theory of hyperkähler manifolds, we can translate this geometric problem into an arithmetic one, namely, into a classification of conjugacy classes of isometries in the orthogonal group of an (indefinite) lattice. We propose to construct the isometries using hermitian forms over number fields. This yields an algorithmic approach to the classification.