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A characterization of \((-1,-1)\)-Freudenthal-Kantor triple systems. (English) Zbl 1253.17005
An \((\epsilon,\delta)\) Freudenthal-Kantor triple system is a triple system \((U,xyz)\) such that \([L(u,v),L(x,y)]=L(uvx,y)+\epsilon L(x,vuy)\) and \(K(K(u,v)x,y)=L(y,x)K(u,v)-\epsilon K(u,v)L(x,y)\) for any \(u,v,x,y\in U\), where \(L(u,v)x=uvx\), \(K(u,v)x=uxv-\delta vxu\). These systems are building blocks for \(5\)-graded Lie algebras and superalgebras. The more traditional Kantor triple systems (or generalized Jordan systems of second order) are precisely the \((-1,1)\) Freudenthal-Kantor triple systems.
The structurable algebras of Allison are strongly connected to Kantor triple systems see [J. R. Faulkner, “Structurable triples, Lie triples, and symmetric spaces”, Forum Math. 6, No. 5, 637–650 (1994; Zbl 0813.17001)]. In this paper a class of anti-structurable algebras is considered. The main result relates these algebras with the \((-1,-1)\) Freudenthal-Kantor triple systems. Also, the \((-1,-1)\) Freudenthal-Kantor triple systems with a left unit (\(eex=x\) for any \(x\)) are shown to be described in terms of the bilinear multiplication \(x\cdot y=exy\).
MSC:
17A40 Ternary compositions
17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.)
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