Minimal affine varieties of superalgebras with superinvolution: a characterization (Q6592917)

This paper provides a complete characterization of the affine minimal varieties of algebras with superinvolution over an algebraically closed field of characteristic zero.\N\NThe authors first review the basic background on superalgebras with superinvolution, including the notions of homogeneous elements, superinvolution, and polynomial identities. They then classify the finite dimensional simple superalgebras with superinvolution, which play a crucial role in the construction of the main objects of study.\N\NThe main result is that any affine minimal variety of algebras with superinvolution is generated by a suitable algebra \(UT_\mathbf{g}^{\diamond}\big(A_1,\ldots,A_m\big)\), where \(A_1, \dots, A_m\) are finite dimensional simple superalgebras and \(\mathbf{g}\) is a translation vector in \(\mathbb{Z}_2^m\). These algebras are introduced as a ``universal enveloping target'' for all possible finite dimensional superalgebras with superinvolution having the same semisimple part as \(A_1\oplus \cdots\oplus A_m\).\N\NThe authors first prove that any subvariety \(\mathcal{U}\) of the variety generated by \(UT_\mathbf{g}^{\diamond}\big(A_1,\ldots,A_m\big)\) must satisfy the exponential inequality, using a detailed analysis of certain multilinear polynomials encoding the dimensions of the simple components. Then, the main result is established by showing that \(UT_\mathbf{g}^{\diamond}\big(A_1,\ldots,A_m\big)\) itself generates a minimal variety, by proving that any proper subvariety has strictly smaller superexponent.\N\NThis provides a complete classification of the affine minimal varieties of superalgebras with superinvolution, extending previous results on the structure of such varieties. The paper introduces several technical tools, such as Capelli-type polynomials and controlled basic evaluations, that may find applications in the study of polynomial identities of superalgebras.