On the duality of varieties of representations of triple Lie and triple super Lie systems. II (Q1358604)

In the first part of the paper under review [\textit{Yu. P. Razmyslov}, Mosc. Univ. Math. Bull. 49, No. 6, 30-36 (1994); translation from Vestn. Mosk. Univ., Ser. I 1994, No. 6, 32-39 (1994; Zbl 0885.17022)] the author considered a 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. He showed that \(\#\) defines an isomorphism between the lattices of varieties of triple Lie systems and supersystems. In the present, second part of the paper the author applies the correspondence \(\#\) in the study of the triple Lie system of a symmetric bilinear form and the triple Lie supersystem of a skew-symmetric bilinear form; in both cases the form is nondegenerate and on an infinite dimensional vector space. The main result is that the variety of triple Lie systems related with the symmetric form is dual with respect to \(\#\) to the variety of triple Lie supersystems related with the skew-symmetric form. A similar result holds also for the varieties of representations of the triple systems of the forms. The main result allows to translate in the language of supersystems the result of S. Yu. Vasilovskij giving the basis of identities of the triple system of the symmetric form and to obtain a basis of the identities of the triple supersystem of the skew-symmetric form. In the case of identities of representations of triple Lie (super)systems the author shows that all multilinear identities of the forms over a field of characteristic different from 2 follow from those of degree \(\leq 7\).