6533b823fe1ef96bd127dfe6

RESEARCH PRODUCT

Subvarieties of the Varieties Generated by the SuperalgebraM1, 1(E) orM2(𝒦)

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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-expo...