×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] N. Jacobson, ”Lie and Jordan triple systems,” Am. J. Math.,71, 149–170 (1949). · Zbl 0034.16903
[2] W. G. Lister, ”A structure theory of Lie triple systems,” Trans. Am. Math. Soc.,72, 217–242 (1952). · Zbl 0046.03404
[3] K. Yamaguti, ”On the Lie triple system and its generalization,” J. Sci. Hiroshima Univ.,21, No. 3, 155–160 (1958). · Zbl 0084.18404
[4] M. Kikkawa, ”On Killing-Ricci forms of Lie triple algebras,” Pac. J. Math.,96, No. 1, 153–161 (1981). · Zbl 0475.17001
[5] W. G. Lister, ”Ternary rings,” Trans. Am. Math. Soc.,154, 37–55 (1971). · Zbl 0216.06901
[6] V. T. Filippov, ”Enveloping algebras of Lie triple algebras,” Preprint No. 5, Inst. Mat. Akad. Nauk SSSR, Novosibirsk (1988). · Zbl 0726.17002
[7] K. Yamaguti, ”On the theory of Malcev algebras,” Kumamoto J. Sci. Ser. A,6, No. 1, 9–45 (1963). · Zbl 0138.26203
[8] K. Yamaguti, ”On algebras of totally geodesic spaces (Lie triple systems),” J. Sci. Hiroshima Univ., Ser. A,21, No. 1, 107–113 (1957). · Zbl 0084.18405
[9] V. T. Filippov, ”Enveloping algebras of triple systems,” Pyataya Sibir. Shkola Mnogoobr. Algebr. Sistem, Tezisy Soobsch., Barnaul, 70–72 (1988).
[10] A. A. Sagle, ”Malcev algebras,” Trans. Am. Math. Soc.,101, No. 3, 426–458 (1961). · Zbl 0101.02302
[11] V. T. Filippov, ”Homogeneous triple systems,” in Research in the Theory of Rings and Algebras [in Russian], Vol. 16, Tr. Inst. Mat. Akad. Nauk SSSR (1989), pp. 164–184. · Zbl 0718.17003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.