On the factorability of T2-ideals of minimal supervarieties and a special class of minimal superalgebras

Abstract

Let F be an algebraically closed field of characteristic zero. In this thesis, we describe necessary and suficient conditions for the factorability of T2-ideals of minimal supervarieties of a fixed superexponent. In light of the characterization of the minimal supervarieties of a fixed superexponent given by Di Vincenzo, da Silva and Spinelli, our main strategy is describing the factorability of the T2-ideals of the upper block triangular matrix algebras (...) equipped with elementary Z2-gradings, where A1, . An are finite-dimensional simple superalgebras. In addition, we classify, up to Z2-graded isomorphism, the minimal superalgebras (...). We also state that the concept of Z2-regularity establishes a nice connection between the factorability of the T2-ideal of (...) and the number of isomorphism classes of (...). As another approach, we define an r-special minimal superalgebra as a minimal superalgebra A with semisimple part (...) such that, for one index (...), whereas for all remaining indices (...) We obtain necessary and suficient conditions for the supervariety generated by A to be minimal. Moreover, we classify, up to Z2-graded isomorphism, the r-special minimal superalgebras. In particular, we prove that any r-special minimal superalgebra is a quotient of an appropriate subalgebra of an upper block triangular matrix algebra, with respect to an elementary Z2-grading. As a consequence of our results, we state that the r-special minimal superalgebras are determined, up to Z2-graded isomorphism, by the Z2-graded polynomial identities satisfied by them.