# zbMATH — the first resource for mathematics

On restricted Leibniz algebras. (English) Zbl 1162.17001
A 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.

##### MSC:
 17A32 Leibniz algebras 17B35 Universal enveloping (super)algebras 17B50 Modular Lie (super)algebras 17B55 Homological methods in Lie (super)algebras
Full Text:
##### References:
 [1] DOI: 10.1016/S0022-4049(03)00120-8 · Zbl 1040.18011 [2] DOI: 10.1081/AGB-100105972 · Zbl 1006.17002 [3] Dzhumadil’daev A. S., C. R. Acad. Sci. Paris Sér. I Math. 332 pp 1047– (2001) [4] DOI: 10.1016/S0022-4049(97)00066-2 · Zbl 0936.17003 [5] DOI: 10.1007/3-540-45328-8_5 [6] DOI: 10.2307/2372701 · Zbl 0055.26505 [7] Jacobson N., Lie Algebras (1979) [8] Loday J.-L., Enseign. Math. (2) 39 pp 269– (1993) [9] Loday J.-L., Math. Scand. 77 pp 189– (1995) [10] DOI: 10.1007/3-540-45328-8_2 [11] DOI: 10.1007/BF01445099 · Zbl 0821.17022 [12] Strade H., Modular Lie Algebras and Their Representations (1998) · Zbl 0923.17020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.