Leibniz algebras in characteristic \(p\). (Q2747420)

The paper under review presents a definition of a restricted Leibniz algebra \(Q\) in characteristic \(p\), and then presents a condition for the non-vanishing of the Leibniz cohomology of \(Q\). In particular, let \(k\) be an algebraically closed field of characteristic \(p > 0\), and let \(Q\) be a (left) Leibniz algebra over \(k\). By definition [\textit{J.-L. Loday} and \textit{T. Pirashvili}, Math. Ann. 296, 139--158 (1993; Zbl 0821.17022)], NEWLINE\[NEWLINE [[x, \, y], \, z] = [x, \, [y, \, z]] - [y, \, [x, \, z]] NEWLINE\]NEWLINE for all \(x, \;y, \;z \in Q\), although the bracket is not necessarily skew-symmetric. For \(x \in Q\), define an operator \(\ell_x : Q \to Q\) by \(\ell_x (y) = [x, \, y]\). The authors define \(Q\) to be weakly restricted if for any \(x \in Q\), there is a non-negative integer \(\ell(x)\) and some element of \(Q\) denoted \(x^{[p^{\ell(x)}]}\) with NEWLINE\[NEWLINE \ell^{p^{\ell(x)}}_x = \ell_{x^{[p^{\ell(x)}]}}. NEWLINE\]NEWLINE If \(\ell(x) = 1\), then \(Q\) is referred to as a restricted or a \(p\)-Leibniz algebra. If \(Q\) is a weakly restricted Leibniz algebra, then a \(Q\)-module \(M\) is called weakly restricted if NEWLINE\[NEWLINE \ell^{p^{\ell(x)}}_x \, m = \ell_{x^{[p^{\ell(x)}]}} \, m NEWLINE\]NEWLINE for all \(x \in Q\) and \(m \in M\). Proven is that for a weakly restricted Leibniz algebra \(Q\) and \(M\) an irreducible \(Q\)-module with \(HL^*(Q, \, M) \neq 0\), then \(M\) must be weakly restricted. Here \(HL^*\) denotes Leibniz cohomology. Compare with [\textit{N. Jacobson}, Trans. Am. Math. Soc. 50, 15--25 (1941; Zbl 0025.30301)]. For a prime \(p\) and a natural number \(m\), particular study of the Zassenhaus algebra \(L = W_1 (m)\) is offered, where \(L\) is the Lie algebra with basis \(\{ e_i \}\), \(i = -1\), 0, 1, 2,\dots, \(p^m - 2\), and bracket NEWLINE\[NEWLINE [e_i, \, e_j] = \;\Bigg( \binom{i+j+1}{j} - \binom{i+j+1}{i} \Bigg) e_{i+j}. NEWLINE\]NEWLINE In particular \(W_1(m)\) is simple and weakly restricted with (strong) restrictedness holding only for \(m = 1\). Let \(O_1(m)\) be the divided power algebra on \(x^{(i)}\), \(0 \leq i \leq p^m - 1\), and product NEWLINE\[NEWLINE x^{(i)} x^{(j)} = \binom{i+j}{i} x^{(i+j)} . NEWLINE\]NEWLINE For \(t \in k\), define a \(W_1(m)\) module structure on \(O_1(m)\) by NEWLINE\[NEWLINE (e_i)_t \, x^{(j)} = \Bigg( \binom{i+j}{i+1} + t\binom{i+j}{i} \Bigg) x^{(i+j)}. NEWLINE\]NEWLINE If \(p > 7\), \(M\) an irreducible antisymmetric \(W_1(m)\)-module, and NEWLINE\[NEWLINEHL^2(W_1(m), \, M) \neq 0,NEWLINE\]NEWLINE then \(M\) is isomorphic to one of six possible modules, describable in terms of the choices \(t = -2\), \(-1\), \(1\), \(2\), \(3\), and the module \(M=k\).