M. =. ?1 and M. *. , The formulas in Theorem 3

. Hence, the problem of classification of singular quadratic Lie superalgebras of type S 1 (up to i-isomophisms) can be reduced to the classification of O(q 0 ) × Sp(q 1 )-orbits of o(q 0 ) ? sp(q 1 ), where O(q 0 ) × Sp(q 1 ) denotes the direct product of two groups O(q 0 ) and Sp

. Proof, By Lemma 3.2.4 we can assume that g is reduced By Proposition 3.2.3, g is also reduced. Since g g then we can identify g = g as a Lie superalgebra equipped with the bilinear forms B, B and we have two dup-numbers: dup B (g)

. Proof, First, g can be realized as the double extension g = (CX 0 ? CY 0 )

C. Let and . C|, Assume that D is an invertible centromorphism The condition (1) of Lemma 3.4.20 implies that D ? ad(X) = ad(X) ? D, for all X ? g and then DC = CD. Using formula (1) of Corollary 3.5.4 and

. Hence, That means J(N) be a Jordan- Novikov algebra. It results that if a Novikov algebra is symmetric then it is Jordan-admissible, fact, we have the much stronger result as follows

?. Let, By above Lemma, one has (yz)x = x(yz) Combined with (IV), (yx)z = x(yz)

J. However, -step nilpotent, for example the one-dimensional Novikov algebra Cc with c 2 = c and B(c, c) = 1. If N is a symmetric 2-step nilpotent Novikov algebra then (xy)z = 0, for all x, y, z ? N. So, That implies J(N) is also a 2-step

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