Algebras with superautomorphism: simple algebras and codimension growth

Let A be an associative algebra endowed with a superautomorphism φ. In this paper we completely classify the finite-dimensional simple algebras with superautomorphism of order ≤ 2. Moreover, after generalizing the Wedderburn–Malcev Theorem in this setting, we prove that the sequence of φ-codimensions of A is polynomially bounded if and only if the variety generated by A does not contain the group algebra of ℤ2 and the algebra of 2 × 2 upper triangular matrices with suitable superautomorphisms.