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On the number of abelian left symmetric algebras. (English) Zbl 0953.17001

The authors give a proof for the fact that, for any \(n\geq 6\), there are infinitely many conjugacy classes of abelian Lie subgroups of the affine group \(\text{Aff}(\mathbb R^n)\) acting simply transitively on \(\mathbb R^n\). This result has been observed by A.T. Vasquez in the seventies. However, a proof did not appear in the literature, as far as I know. In the present article an easy algebraic proof is given. It is shown that up to isomorphism there are infinitely many commutative associative algebras of dimension \(n\) for any \(n\geq 6\). Since these algebras correspond to left-symmetric algebras with underlying abelian Lie algebra the result follows from this.

MSC:

17A30 Nonassociative algebras satisfying other identities
17B30 Solvable, nilpotent (super)algebras
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