The factorability of T₂-ideals of minimal supervarieties

Abstract

Giambruno and Zaicev stated that, over a field of characteristic zero, T-ideals of minimal varieties of a fixed exponent have the factoring property. In the present article, we describe necessary and sufficient conditions for the factorability of T₂-ideals of minimal supervarieties of a fixed superexponent. In light of the characterization of minimal supervarieties of a fixed superexponent given by Di Vincenzo, da Silva, and Spinelli, the crucial point is the study of the factorability of T₂-ideals of the upper-block triangular matrix algebras $U{T}_{{\mathbb{Z}}_2}(A_1,\dots ,A_n)$ equipped with an elementary ${\mathbb{Z}}_2$-grading, where $A_1,\dots ,A_n$ are simple superalgebras. We obtain necessary and sufficient conditions for the isomorphism between two superalgebras $U{T}_{{\mathbb{Z}}_2}(A_1,\dots ,A_n)$. We also show that the concept of ${\mathbb{Z}}_2$-regularity establishes a nice connection between the factorability of the T₂-ideal of $U{T}_{{\mathbb{Z}}_2}(A_1,\dots ,A_n)$ and the number of isomorphism classes of $U{T}_{{\mathbb{Z}}_2}(A_1,\dots ,A_n)$.

Keywords:

• Factorability
• graded polynomial identities
• minimal supervarieties
• superalgebras
• superexponent

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16R10
• 16R50
• 16W55

Acknowledgements

We would like to express our gratitude to Professor Antonio Giambruno for proposing us this problem.

Additional information

Funding

The first named author was supported by CAPES. The third named author was partially supported by CNPq – Brasil grant 306534/2016-9.