Homogeneous involution on graded division algebras and their polynomial identities

In this short note, we describe the so-called homogeneous involution on finite-dimensional graded-division algebra over an algebraically closed field. We also compute their graded polynomial identities with involution. As pointed out by L. Fonseca and T. de Mello, a homogeneous involution naturally appears when dealing with graded polynomial identities and a compatible involution.