Generalized restricted Lie algebras and representations of the Zassenhaus algebra
The remarkable result of this paper is a description of all irreducible representations of the Zassenhaus algebra \(W(1:n)\) over an algebraically closed field \(F\) of characteristic \(p>3.\) In the standard basis \(\{ e_i, i=-1,\ldots ,p^n-2\}\) of \(L=W(1: n)\) \[ [e_i, e_j]=c_{ij}e_{i+j}, c_{ij}=\binom {i+j+1}j -\binom {i+j+1}i, \] \(\{ L_i=\langle{} e_j, j\geq i \rangle{}, i\geq -1\}\) is the standard filtration of \(L\). It is well known that \(L_0\) is a \(p\)-subalgebra of \(L\) with respect to the natural embedding \(L\) into \(\mathcal L=\text{Der }L=L+\langle{} e_{-1}^{p^k}, 0<k<n \rangle{}.\) Let \(\rho \) be an irreducible representation of \(L\) on a space \(V\). There exists a linear function \(\varTheta \in L^*\) such that \[ \rho (e_{-1})^{p^n}=\varTheta (e_{-1})^{p^n}I, \] \[ \rho (e_i)^p-\rho (e_i^{[p]})=\varTheta (e_i)^p, i\geq 0. \] The author shows that the representation \(\rho \) (and the function \(\varTheta ,\) as well) may be continued to \(\mathcal L.\) Thus, \(\rho \) may be considered as an irreducible representation of the \(p\)-algebra \(\mathcal L\) with \(p\)-character \(\varTheta\). Let \(r\) be the minimal integer \(i\) such that \(\varTheta (L_i)=0.\) All irreducible modules for \(r=-1\) (i.e. \(\varTheta =0\)) are well known. If \(0\leq r<p^n-1\) then \(V\cong \text{Ind }^{\mathcal L}_Q(Fu, \varTheta).\) Here \(Q\) is a special subalgebra in \(L_0\), \(Fu\) is a one-dimensional \(Q\)-module and induction is understood in terms of the \(\varTheta\)-reduced enveloping algebra \(U(\mathcal L)/J\) where \(J\) is an ideal generated by \(e^p-e^{[p]}-\varTheta (e)^p1, e\in \mathcal L.\) Thus, \(\dim V=p^{n+\dim L_0/Q}.\) If \(r=p^n-1\) then as \(L_0\)-module \(V\cong \text{Ind }^{L_0}_{L_s}(Fu, \varTheta), s=\frac{p^n-1}2 \) where \(Fu\) is a one-dimensional \(L_{\frac{p^n-1}2}\)-module. Thus, \(\dim V=p^{\frac{p^n-1}2}.\)
scientific article; zbMATH DE number 1218674
scientific article