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Enveloping algebras of triple systems and algebras. (English. Russian original) Zbl 0726.17001
Algebra Logic 29, No. 1, 48-59 (1990); translation from Algebra Logika 29, No. 1, 67-81 (1990).
A triple system $$A$$ is a linear space over a field $$k$$ with a ternary multilinear multiplication $$[x,y,z]$$. Denote by $$R(A)$$ the subspace in $$\operatorname{Hom}_ k(A,A)$$ spanned by all maps $$R(y,z)$$, where $$R(y,z)x=[x,y,z]$$. Let $$N$$ be a subspace in $$\operatorname{Hom}_ k(A,A)$$ containing $$R(A)$$, $$B=A\oplus N$$. Define in $$B$$ a multiplication $$(x+n)(y+n')=xn'+R(x,y)$$. Then $$A^ 2\subseteq N$$, $$AN\subseteq A$$, $$NA=N^ 2=0$$. The algebra $$B$$ is an $$\alpha$$-envelope of $$A$$. It is standard if $$R(A)=N$$. A system $$A$$ is simple if and only if its standard $$\alpha$$-envelope is simple. Any nonassociative algebra $$C$$ is a triple system with the multiplication $$[x,y,z]=(xy)z$$. Assume that $$A$$ is (anti-) commutative. Then $$C$$ is simple if and only if its standard $$\alpha$$-envelope (as a triple system) is simple.
Let $$\mathrm{char}\,k\neq 3$$ and $$A$$ be an anticommutative triple system. Denote by $$M$$ the Lie subalgebra of a Lie algebra $$\operatorname{Hom}_ k(A,A)(-)$$ containing all operators $$R(y,z)$$. Define in $$D=A\oplus M$$ a new multiplication $(x+m)(y+m')=xm'-ym-(1/3)R(x,y)+mm'-m'm.$ $$D$$ is a $$\beta$$-envelope of $$A$$. $$D$$ is standard if $$M$$ is generated by all operators $$R(x,y)$$. $$A$$ is simple if and only if its standard $$\beta$$-envelope is simple.
Let $$A$$ be a Mal’cev algebra which is not a Lie algebra. Put in $$A$$ $$[x,y,z]=J(x,y,z)$$. Then $$A$$ is an anticommutative triple system. $$A$$ is a simple Mal’cev algebra if and only if its standard $$\beta$$-envelope is simple.

##### MSC:
 17A40 Ternary compositions 17B60 Lie (super)algebras associated with other structures (associative, Jordan, etc.) 17D10 Mal’tsev rings and algebras
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##### References:
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