The Freiheitssatz and the automorphisms of free right-symmetric algebras. (English) Zbl 1168.17002
An algebra \(A\) over a field \(k\) is said to be right-symmetric in case \((x,y,z)=(x,z,y)\) holds for any \(x,y,z\in A\), where \((x,y,z)=(xy)z-x(yz)\) is the associator of the elements \(x,y,z\). Right-symmetric algebras are Lie-admissible and are strongly related to locally affine manifolds.
A number of interesting results for these algebras is proved in the paper under review:
1) The word problem for right-symmetric algebras with a single defining relation is decidable.
2) If a right symmetric algebra is generated by elements \(x_1,\ldots,x_n\) subject to just one defining relation involving \(x_n\), then the subalgebra generated by \(x_1,\ldots,x_{n-1}\) is free on these generators (Freiheitssatz).
3) Any two-generated subalgebra of a free right-symmetric algebra is free.
4) Any automorphism of the free right-symmetric algebra on two generators is tame.
Several open problems on subalgebras and automorphisms of free right-symmetric algebras are posed too.
A number of interesting results for these algebras is proved in the paper under review:
1) The word problem for right-symmetric algebras with a single defining relation is decidable.
2) If a right symmetric algebra is generated by elements \(x_1,\ldots,x_n\) subject to just one defining relation involving \(x_n\), then the subalgebra generated by \(x_1,\ldots,x_{n-1}\) is free on these generators (Freiheitssatz).
3) Any two-generated subalgebra of a free right-symmetric algebra is free.
4) Any automorphism of the free right-symmetric algebra on two generators is tame.
Several open problems on subalgebras and automorphisms of free right-symmetric algebras are posed too.
Reviewer: Alberto Elduque (Zaragoza)
MSC:
| 17A30 | Nonassociative algebras satisfying other identities |
| 17A36 | Automorphisms, derivations, other operators (nonassociative rings and algebras) |
| 17A50 | Free nonassociative algebras |
| 17D25 | Lie-admissible algebras |
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