Subvarieties of the Varieties Generated by the Superalgebra M_{1, 1}(E) or M₂(𝒦)

Abstract

Let 𝒦 be a field of characteristic zero, and let us consider the matrix algebra M ₂(𝒦) endowed with the ℤ₂-grading (𝒦e ₁₁ ⊕ 𝒦e ₂₂) ⊕ (𝒦e ₁₂ ⊕ 𝒦e ₂₁). We define two superalgebras, ℛ _p and 𝒮 _q , where p and q are positive integers. We show that if 𝒰 is a proper subvariety of the variety generated by the superalgebra M ₂(𝒦), then the even-proper part of the T ₂-ideal of graded polynomial identities of 𝒰 asymptotically coincides with the even-proper part of the graded polynomial identities of the variety generated by the superalgebra ℛ _p  ⊕ 𝒮 _q . This description also affords an even-asymptotic description of the proper subvarieties of the variety generated by the superalgebra M _1,1(E) as even-asymptotically coinciding with the T ₂-ideal of the variety generated by the Grassmann envelopes G(ℛ _p ) and G(𝒮 _q ). Moreover, the following general fact is established. If two varieties of superalgebras are even-asymptotically equivalent, then they are asymptotically equivalent, and they have the same PI-exponent.

Keywords:

• ℤ₂-graded identities
• Asymptotic equivalence
• 2 × 2 Matrices

Acknowledgments

Part of this project was carried out when the second author visited the Department of Mathematics of the University of Basilicata in Potenza, Italy, in the beginning of 2000. He is very thankful for the kind hospitality and the creative atmosphere during his visit.