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Left-symmetric structures on simple modular Lie algebras. (English) Zbl 0824.17021

Left-symmetric structures on Lie algebras arise in the theory of affine manifolds. It was known that a finite-dimensional simple Lie algebra \({\mathfrak g}\) over a field \(k\) does not admit any left-symmetric structure if \(\text{char} (k)=0\). The author shows that when \(\text{char} (k) =p>0\) and \({\mathfrak g}\) is classical, \({\mathfrak g}\) admits a left-symmetric structure only in case \(p\mid\dim {\mathfrak g}\). On the other hand, some nonrestricted simple Lie algebras of Cartan type, e.g., the Block algebras \({\mathcal L} (G, \delta, f)\) of dimension \(p^ n-1\), admit left- symmetric structures for every \(p>0\).

MSC:

17B50 Modular Lie (super)algebras
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