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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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##### References:
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