On restricted Leibniz algebras.

*(English)*Zbl 1162.17001A Leibniz algebra over a field \(k\) is a \(k\)-module which admits a bilinear bracket map that satisfies the Leibniz identity: \([x,[y,z]] = [[x,y],z] - [[x,z],y]\). Any Lie algebra over \(k\) is also a Leibniz algebra. But in general, the bracket in the Leibniz case need not be anti-commutative as for a Lie algebra. Of interest here are Leibniz algebras over a field of prime characteristic \(p\). For Lie algebras over such a field, one can consider a \(p\)-restricted Lie algebra and the notion of a restricted module. Associated to such a Lie algebra is the restricted enveloping algebra which is a quotient of the universal enveloping algebra. Restricted modules are precisely modules for the restricted enveloping algebra. For Leibniz algebras, the notion of a \(p\)-restricted Leibniz algebra (and restricted Leibniz module) was introduced by A. S. Dzhumadil’daev and S. A. Abdykassymova [C. R. Acad. Sci., Paris, Sér. I Math. 332, 1047–1052 (2001; Zbl 1037.17003)]. The main goal of this paper is to develop an appropriate analogue of the restricted enveloping algebra.

The authors first show that any diassociative algebra admits the structure of a \(p\)-restricted Leibniz algebra. More precisely, there is a functor from the category of diassociative algebras to the category of \(p\)-restricted Leibniz algebras. They show that this functor admits a left adjoint by constructing for any \(p\)-restricted Leibniz algebra, a restricted universal diassociative algebra. However, an alternate structure is needed to classify the restricted Leibniz modules. Given a restricted Leibniz algebra, the authors construct a restricted universal enveloping algebra and show that the category of right modules over this algebra is equivalent to the category of restricted Leibniz modules (over that Leibniz algebra).

Finally, the authors consider the tensor product of a Leibniz algebra with a Zinbiel algebra (which is dual in a Koszul sense to the notion of a Leibniz algebra). Such a tensor product is known to be a Lie algebra. Here the authors show that (over arbitrary characteristic) such a tensor product in fact admits the structure of a pre-Lie algebra. Further, in prime characteristic, they show that such a tensor product admits the structure of a restricted pre-Lie algebra, from which it follows that it admits the structure of a restricted Lie algebra.

The authors first show that any diassociative algebra admits the structure of a \(p\)-restricted Leibniz algebra. More precisely, there is a functor from the category of diassociative algebras to the category of \(p\)-restricted Leibniz algebras. They show that this functor admits a left adjoint by constructing for any \(p\)-restricted Leibniz algebra, a restricted universal diassociative algebra. However, an alternate structure is needed to classify the restricted Leibniz modules. Given a restricted Leibniz algebra, the authors construct a restricted universal enveloping algebra and show that the category of right modules over this algebra is equivalent to the category of restricted Leibniz modules (over that Leibniz algebra).

Finally, the authors consider the tensor product of a Leibniz algebra with a Zinbiel algebra (which is dual in a Koszul sense to the notion of a Leibniz algebra). Such a tensor product is known to be a Lie algebra. Here the authors show that (over arbitrary characteristic) such a tensor product in fact admits the structure of a pre-Lie algebra. Further, in prime characteristic, they show that such a tensor product admits the structure of a restricted pre-Lie algebra, from which it follows that it admits the structure of a restricted Lie algebra.

Reviewer: Christopher P. Bendel (Menomonie)

##### MSC:

17A32 | Leibniz algebras |

17B35 | Universal enveloping (super)algebras |

17B50 | Modular Lie (super)algebras |

17B55 | Homological methods in Lie (super)algebras |

##### Keywords:

diassociative algebra; Leibniz algebra; pre-Lie algebra; restricted Leibniz algebra; restricted Lie algebra; Zinbiel algebra
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\textit{I. Dokas} and \textit{J.-L. Loday}, Commun. Algebra 34, No. 12, 4467--4478 (2006; Zbl 1162.17001)

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