On minimal varieties of superalgebras with superautomorphism (Q6612163)

This paper studies \(\mathbb Z_2\)-graded algebras (superalgebras) with superinvolutions. A superinvolution on a superalgebra algebra \(A\) is a map \(\phi:A\rightarrow A\) which satisfies \(\phi^2=id\) and \(\phi(ab)=(-1)^{\alpha\beta}\phi(a) \phi(b)\), whenever \(a\) and \(b\) are homogeneous of degrees \(\alpha\) and \(\beta\), respectively. The authors generalize the notations of polynomial identities and codimensions to superalgebras with superinvolutions in the natural way. If \(A\) is a superalgebra with superinvolution, denote by \(c^\phi_n(A)\) the \(n\)-th codimension. Then, in [``Codimension growth of algebras with superautomorphism'', Preprint], \textit{A. Ioppolo} and \textit{D. La Mattina}, showed that \(\lim_{n\rightarrow\infty} c_n^\phi(A)^{1/n}\) always exists and is an integer, denoted \(\exp^\phi(A)\). \(A\) is called minimal if for every \(B\) satisfying more identities than \(A\), \(\exp^\phi(B)<\exp^\phi (A)\). The classification of minimal (ordinary) p.i. algebras was accomplished by \textit{A. Giambruno} and \textit{M. Zaicev} in [Adv. Math. 174, No. 2, 310--323 (2003; Zbl 1035.16013); Trans. Am. Math. Soc. 355, No. 12, 5091--5117 (2003; Zbl 1031.16015)]. In the finite dimensional case they showed that every minimal p.i. algebra is p.i. equivalent to an algebra of block upper triangular matrices. The paper under review classifies all minimal finite dimensional superalgebras with superinvolution. They are also all equivalent to block upper triangular matrices with certain gradings and superinvolutions.