Nonfinitely Based Varieties of Right Alternative Metabelian Algebras

Abstract

Since 1976, it is known from the paper by V. P. Belkin that the variety RA 2 of right alternative metabelian (solvable of index 2) algebras over an arbitrary field is not Spechtian (contains non-finitely based subvarieties). In 2005, S. V. Pchelintsev proved that the variety generated by the Grassmann RA 2 -algebra of finite rank r over a field F , for char(F) = 2, is Spechtian iff r = 1. We construct a non-finitely based variety M generated by the Grassmann V-algebra of rank 2 of certain finitely based subvariety V ⊂ RA 2 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: non-finitely based variety of algebras, Spechtian variety of algebras, right alternative metabelian algebra, superalgebra, Grassmann algebra.

Basic superranks for varieties of algebras

“…Proof. Identities (10)- (12) imply immediately that (F V [X n ]) 2n+2 = 0. Therefore it remains to note that the element…”

“…Then by setting z := zt in (13), in view of metability, we get (11). Finally, multiplying (13) by R t R z and applying (11), we obtain (12).…”

“…The variety of right alternative algebras is a well-known source of examples of nonfinitely based varieties [2,8,12,16] over a field of characteristic zero. The similar results can be obtained for the variety of right symmetric algebras.…”