On the duality of varieties of representations of triple Lie and and triple super Lie systems. I (Q1358441)

In the paper under review the author considers the correspondence between the 2-graded Lie algebra \({\mathfrak g}={\mathfrak g}_0\oplus{\mathfrak g}_1\) over a field \(K\) of characteristic 0 and the Lie superalgebra \({\mathfrak g}^{\#}=G_0\otimes_K{\mathfrak g}_0\oplus G_1\otimes_K{\mathfrak g}_1\), where \(G=G_0\oplus G_1\) is the Grassmann algebra with its canonical \({\mathbb{Z}}_2\)-grading. The author proves that the mapping \# admits a natural interpretation in terms of polynomial identities and varieties of triple Lie algebras and superalgebras. In particular, \# defines an isomorphism between the lattices of the corresponding varieties both in the cases of algebras and representations of algebras. The main property of \# is that it sends varieties of finite dimensional and of locally nilpotent types respectively to varieties of locally nilpotent and of finite dimensional types. As a consequence, it turns out that the results for the Specht property of the varieties of representations of the \(n\)-dimensional triple Lie system and supersystem of a symmetric and of a skew-symmetric bilinear form, respectively (obtained in 1985 by I. M. Trishin and A. G. Loginov) follow from each other.