# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  L. V. Sabinin and P. O. Mikheev, The Theory of Smooth Bol Loops. Lectures [in Russian], Univ. Druzhby Narodov, Moscow (1985). · Zbl 0584.53001  P. O. Mikheev, ”Commutator algebras of right-alternative algebras,” in: Quasigroups and Their Systems [in Russian], Shtiintsa, KishinËv, 1990,113, pp. 62–65. · Zbl 0723.17028  A. A. Sagle, ”On anti-commutative algebras and general Lie triple systems,” Pacific J. Math.,15, No. 1, 281–291 (1965). · Zbl 0142.27502  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.  P. O. Mikheev and L. V. Sabinin, Smooth Quasigroups and Geometries [in Russian], VINITI, Moscow, 1988,20, pp. 75–110 (Itogi Nauki i Tekhniki).  A. T. Gainov, ”Identical relations for binary-Lie algebras,” Uspekhi Mat. Nauk,12, No. 3, 141–146 (1957).  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  V. T. Filippov, ”AT-ideal of a Mal’tsev algebra,” Sibirsk. Mat. Zh.,29, No. 3, 148–155 (1988). · Zbl 0661.17029  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  V. T. Filippov, ”To the theory of Mal’tsev algebras,” Algebra i Logika,16, No. 1, 101–108 (1977).  V. T. Filippov, ”Central simple Mal’tsev algebras,” Algebra i Logika,15, No. 2, 235–242 (1976).  V. T. Filippov, ”Prime Mal’tsev algebras,” Mat. Zametki,31, No. 4, 669–678 (1982). · Zbl 0494.17012
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.