Color Involutions in Graded Algebras: Representability Theorems and Specht Property
<jats:p> Let <jats:italic>G</jats:italic> be a finite abelian group and let <jats:italic>A</jats:italic> be a <jats:italic>G</jats:italic> -graded algebra with color involution <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$*.$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> We first establish a Wedderburn–Malcev decomposition in case <jats:italic>A</jats:italic> 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 <jats:italic>A</jats:italic> satisfies the same <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$*$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -identities as the Grassmann envelope of a finite-dimensional <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$(G\times \mathbb {Z}_2)$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>×</mml:mo> <mml:msub> <mml:mi>Z</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> -graded algebra <jats:italic>B</jats:italic> with a suitable color involution <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$\sharp $$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>♯</mml:mo> </mml:math> </jats:alternatives> </jats:inline-formula> related to <jats:inline-formula> <jats:alternatives> <jats:tex-math>$$*$$</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . </jats:p>