Color Involutions in Graded Algebras: Representability Theorems and Specht Property

Let G be a finite abelian group and let A be a G-graded algebra with color involution ∗. We first establish a Wedderburn–Malcev decomposition in case A is finite-dimensional. Then, we prove several results for algebras with polynomial identities in this setting, including the Hook theorem, the Specht property, and, most notably, the Representability theorems. The latter assert that A satisfies the same ∗-identities as the Grassmann envelope of a finite-dimensional (G×Z2)-graded algebra B with a suitable color involution ♯ related to ∗.