On the Lie algebra associated to S-unitary matrices

A natural generalization of unitary groups arising from sesquilinear forms which are assumed neither Hermitian nor skew-Hermitian is considered. Let S∈Mn(C). An S-unitary matrix A is a matrix A∈GLn(C) such that ASA⁎=S. The set US of all S-unitary matrices is a matrix Lie group. A formula for the real dimension of the associated Lie algebra uS when S is nonsingular and normal is derived. When S is invertible and unitary, it is shown that uS is the direct sum of some Lie algebras associated to the indefinite unitary groups. Finally, the dimension formula is applied to a class of permutation matrices.