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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.

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