4.4. Symmetric Novikov algebras Therefore, [[x,y]+,[x,x]+]+ = [x,[y,[x,x]+]+]+ if and only if x 2 (xy) + x 2 (yx) = x(yx 2 ) + x(x 2 y). Remark that we have following identities: x 2 (xy) = x 3 y − (x 2 ,x,y) = x 3 y − (x,x 2 ,y), x 2 (yx) = (x 2 y)x − (x 2 ,y,x) = x 3 y − (y,x 2 ,x), x(yx 2 ) = (xy)x 2 − (x,y,x 2 ) = x 3 y − (y,x,x 2 ), x(x 2 y) = x 3 y − (x,x 2 ,y). It means that we have only to check the formula (y,x 2 ,x) = (y,x,x 2 ). It is clear by the identities (III) and (V). Then we can conclude that J(N) is a Jordan algebra. Corollary 4.4.18. If (N,B) is a symmetric Novikov algebra satisfying (V) then (J(N),B) is a pseudo-Euclidean Jordan algebra. Proof. It is obvious since B([x,y]+,z) = B(xy + yx,z) = B(x,yz + zy) = B(x,[y,z]+), for all x,y,z ∈ J(N). Remark 4.4.19. Obviously, Jordan-Novikov algebras are power-associative but in general this is not true for Novikov algebras. Indeed, if Novikov algebras were power-associative then they would satisfy (V). That would imply they were Jordan-admissible. But, that is a contradiction as shown in Example 4.4.16. Lemma 4.4.20. Let N be a Novikov algebra then [x,yz]+ = [y,xz]+, for all x,y,z ∈ N. Proof. By (III), for all x,y,z ∈ N one has (xy)z + y(xz) = x(yz) + (yx)z. Combined with (IV), we obtain: (xz)y + y(xz) = x(yz) + (yz)x. That means [x,yz]+ = [y,xz]+, for all x,y,z ∈ N. Proposition 4.4.21. Let (N,B) be a symmetric Novikov algebra then following identities: (1) x[y,z] = [y,z]x = 0, and therefore [x,yz]+ = [x,zy]+, (2) [x,y]+z = [x,z]+y, (3) [x,yz]+ = [xy,z]+ = x[y,z]+ = [x,y]+z, (4) x[y,z]+ = [y,z]+x, hold for all x,y,z ∈ N. Proof. Let x,y,z and t be elements in N. (1) By Proposition 4.4.6 and Lemma 4.4.8, L [y,z] = 0 so one has (1). (2) We have B([x,y]+z,t) = B(y,[x,zt]+) = B(y,[z,xt]+) = B([z,y]+x,t). Therefore, [x,y]+z = [z,y]+x. Since the product [ , ]+ is commutative then [y,x]+z = [y,z]+x. 122
4.4. Symmetric Novikov algebras (3) By (1) and Lemma 4.4.20, [x,yz]+ = [x,zy]+ = [z,xy]+ = [xy,z]+. Since B is associative with respect to the product in N and in J(N) then B(t,[xy,z]+) = B([t,xy]+,z) = B([t,yx]+,z) = B([y,tx]+,z) = B(tx,[y,z]+) = B(t,x[y,z]+). It implies that [xy,z]+ = x[y,z]+. Similarly, one has: B([x,y]+z,t) = B(x,[y,zt]+) = B(x,[y,tz]+) = B(x,[t,yz]+) = B([x,yz]+,t). So [x,y]+z = [x,yz]+. (4) By (2) and (3), x[y,z]+ = [x,y]+z = [y,x]+z = [y,z]+x. Corollary 4.4.22. Let (N,B) be a symmetric Novikov algebra then (J(N),B) is a symmetric Jordan-Novikov algebra. Proof. We will show that [[x,y]+,z]+ = [x,[y,z]+]+, for all x,y,z ∈ N. Indeed, By Proposition 4.4.21 one has [[x,y]+,z]+ = [2xy,z]+ = 2[z,xy]+ = 2[x,yz]+ = [x,[y,z]+]+. Hence, the product [ , ]+ are both commutative and associative. That means J(N) be a Jordan- Novikov algebra. It results that if a Novikov algebra is symmetric then it is Jordan-admissible. In fact, we have the much stronger result as follows: Proposition 4.4.23. Let N be a symmetric Novikov algebra then the product on N is associative, that is x(yz) = (xy)z, for all x,y,z ∈ N. Proof. First, we need the lemma: Lemma 4.4.24. Let N be a symmetric Novikov algebra then N 2 ⊂ Z(N). Proof. By Lemma 4.4.8, one has [x,y] = xy−yx ∈ Ann(N) ⊂ Z(N), for all x,y ∈ N. Also, by (4) of Proposition 4.4.21, x[y,z]+ = [y,z]+x, for all x,y,z ∈ N, that means [x,y]+ = xy + yx ∈ Z(N), for all x,y ∈ N. Hence, xy ∈ Z(N), for all x,y ∈ N, i.e. N 2 ⊂ Z(N). Let x,y,z ∈ N. By above Lemma, one has (yz)x = x(yz). Combined with (IV), (yx)z = x(yz). On the other hand, [x,y] ∈ Ann(N) implies (yx)z = (xy)z. Therefore, (xy)z = x(yz). A general proof of the above proposition can be found in [AB10], Lemma II.4 which holds for all symmetric left-symmetric superalgebras. By Corollary 4.4.9, if N is a symmetric Novikov algebra then g(N) is 2-step nilpotent. However, J(N) is not necessarily 2-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 [[x,y]+,z]+ = 0, for all x,y,z ∈ N. That implies J(N) is also a 2-step nilpotent Jordan algebra. The converse is also true. 123
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Université de Bourgogne - Dijon UF
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Remerciements Je remercie l’Ambas
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Contents 3.4.2 Quadratic dimension
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Introduction structures can be dete
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Introduction and Di red (n + 2) th
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Introduction and obtain the followi
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y Introduction We realize that with
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Introduction by I. Bajo, S. Benayad
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Introduction (2) there exists an i-
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xix
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Notations 0 Ik where J = ∈ gl(2
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1.1. Definitions and gε(V ) = {C
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1.2. Nilpotent orbits Lie algebra g
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1.2. Nilpotent orbits where µi = i
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1.2. Nilpotent orbits The point of
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1.3 Semisimple orbits We recall the
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1.4. Invertible orbits Let us now c
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1.5. Adjoint orbits in the general
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2.1. Preliminaries (3) Set the map
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2.1. Preliminaries Proof. Let I be
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2.2. Singular quadratic Lie algebra
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If l = k then we stand at (k − 1)
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2.2. Singular quadratic Lie algebra
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Consider Then dim(g) = 6 ∑ i=3 2.
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2.2. Singular quadratic Lie algebra
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2.2. Singular quadratic Lie algebra
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2.2. Singular quadratic Lie algebra
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2.2. Singular quadratic Lie algebra
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2.2. Singular quadratic Lie algebra
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2.2. Singular quadratic Lie algebra
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2.2. Singular quadratic Lie algebra
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2.2. Singular quadratic Lie algebra
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(2) g g ′ if and only if g i g
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2.4. 2-step nilpotent quadratic Lie
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2.4. 2-step nilpotent quadratic Lie
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2.4. 2-step nilpotent quadratic Lie
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Chapter 3 Singular quadratic Lie su
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