Central extensions of the Zassenhaus algebra and their irreducible representations

It is shown that the Zassenhaus algebra $W_1(m)$ over a field of characteristic $p>3$ has, up to equivalence, a unique nontrivial central extension $\widetilde{W}_1(m)$ (the modular Virasoro algebra). For the Virasoro algebra we construct a generalized Casimir element. All the irreducible $\widetilde{W}_1(m)$-modules are described. It is shown that there is no simple graded Lie algebra with zero component $L_0\cong\widetilde{W}_1(m)$.
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