Nonfinitely Based Varieties of Right Alternative Metabelian Algebras

Abstract

Since 1976, it is known from the paper by V. P. Belkin that the variety RA₂ of right alternative metabelian (solvable of index 2) algebras over an arbitrary field is not Spechtian (contains nonfinitely based subvarieties). In 2005, S. V. Pchelintsev proved that the variety generated by the Grassmann RA₂-algebra of finite rank r over a field ℱ, for char(ℱ) ≠ 2, is Spechtian iff r = 1. We construct a nonfinitely based variety 𝔐 generated by the Grassmann 𝒱-algebra of rank 2 of certain finitely based subvariety 𝒱 ⊂ RA₂ over a field ℱ, for char(ℱ) ≠ 2, 3, such that 𝔐 can also be generated by the Grassmann envelope of a five-dimensional superalgebra with one-dimensional even part.

Key Words:

• Grassmann algebra
• Nonfinitely based variety of algebras
• Right alternative metabelian algebra
• Spechtian variety of algebras
• Superalgebra

2010 Mathematics Subject Classification:

• 17D15
• 17A50
• 17A70

ACKNOWLEDGMENTS

The author is very grateful to his supervisor Prof. I. P. Shestakov and to the IME-USP for the kind hospitality and the creative atmosphere. The author is also very thankful to Prof. S. V. Pchelintsev for his suggesting the problem and for the useful discussions on the obtained results.

Notes

Communicated by A. Elduque.