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Homogeneous Bol algebras. (English. Russian original) Zbl 0854.17001
Sib. Math. J. 35, No. 4, 818-825 (1994); translation from Sib. Mat. Zh. 35, No. 4, 919-926 (1994).
A Bol algebra \(B\) is a vector space over a field \(\Phi\) with a bilinear operation \(xy\) and a trilinear operation \([x, y, z]\) that satisfy the following identities: \[ x^2= 0, \quad [x, x, y]= 0, \quad [x, y, z]+ [y, z, x]+ [z, x, y] =0, \] \[ [x, y, z]t- [x, y, t]z+ [z, t, xy]- [x, y, zt]+ (xy) (zt)=0, \qquad \text{and} \] \[ [x, y, [z, t, v ]]= [[x, y, z], t, v]+ [z, [x, y, t], v]+ [z, t, [x, y, v]]. \] Also, if the trilinear operation is of the form \([x, y, z]= \alpha (xy) z+\beta (yz) x+ \gamma (zx) y\) for some fixed \(\alpha, \beta, \gamma\in \Phi\), then the Bol algebra \(B\) is said to be homogeneous.
For \(B\) a homogeneous Bol algebra over a field \(\Phi\) of characteristic \(\neq 2, 3\), the author proves the following:
(1) \(B\) is either a metabelian algebra, or a \(\delta\)-algebra with \(\delta\neq 3\). (An anti-commutative algebra is metabelian if it satisfies the identity \((xy) (zt)= 0\); and it is a \(\delta\)-algebra if it satisfies the identity \(J(x, y, z)x= J(x, y, xz)+ \delta (xy) (xz)\), where \(J(x, y, z)= (xy)z+ (yz)x+ (zx)y\) and \(\delta\) is a fixed element in \(\Phi\).) Also, when \(B\) is a \(\delta\)-algebra, if \(\delta=0\) then \(B\) is a Mal’tsev algebra; and if \(\delta\neq 0\) then \(B\) satisfies the identity \(J(vt, xy, z)= 2((vt) (xy))z\).
(2) If \(B\) is not a Mal’tsev algebra, then \((B^2 )^3 =0\).
(3) If \(B\) is semiprime, then \(B\) is a Mal’tsev algebra and is isomorphically embeddable in the commutator algebra \(A^{(-)}\) of some alternative algebra \(A\) over \(\Phi\).

MSC:
17A30 Nonassociative algebras satisfying other identities
17A40 Ternary compositions
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[1] L. V. Sabinin and P. O. Mikheev, The Theory of Smooth Bol Loops. Lectures [in Russian], Univ. Druzhby Narodov, Moscow (1985). · Zbl 0584.53001
[2] P. O. Mikheev, ”Commutator algebras of right-alternative algebras,” in: Quasigroups and Their Systems [in Russian], Shtiintsa, KishinĂ‹v, 1990,113, pp. 62–65. · Zbl 0723.17028
[3] A. A. Sagle, ”On anti-commutative algebras and general Lie triple systems,” Pacific J. Math.,15, No. 1, 281–291 (1965). · Zbl 0142.27502
[4] V. T. Filippov, ”Homogeneous triple systems,” in: Studies on the Theory of Rings and Algebras [in Russian], Inst. Mat. (Novosibirsk), Novosibirsk, 1989,16, pp. 164–184.
[5] P. O. Mikheev and L. V. Sabinin, Smooth Quasigroups and Geometries [in Russian], VINITI, Moscow, 1988,20, pp. 75–110 (Itogi Nauki i Tekhniki).
[6] A. T. Gainov, ”Identical relations for binary-Lie algebras,” Uspekhi Mat. Nauk,12, No. 3, 141–146 (1957).
[7] A. Sagle, ”A Malcev algebra,” Trans. Amer. Math. Soc.,101, No. 3, 426–458 (1961). · Zbl 0101.02302 · doi:10.1090/S0002-9947-1961-0143791-X
[8] V. T. Filippov, ”AT-ideal of a Mal’tsev algebra,” Sibirsk. Mat. Zh.,29, No. 3, 148–155 (1988). · Zbl 0661.17029
[9] A. A. Sagle, ”On simple extended Lie algebras over fields of characteristic zero,” Pacific J. Math.,15, No. 2, 621–642 (1965). · Zbl 0134.26904
[10] V. T. Filippov, ”To the theory of Mal’tsev algebras,” Algebra i Logika,16, No. 1, 101–108 (1977).
[11] V. T. Filippov, ”Central simple Mal’tsev algebras,” Algebra i Logika,15, No. 2, 235–242 (1976).
[12] V. T. Filippov, ”Prime Mal’tsev algebras,” Mat. Zametki,31, No. 4, 669–678 (1982). · Zbl 0494.17012
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