Representations of Cartan type Lie algebras in characteristic \(p\).
Let \(K\) be an algebraically closed field of characteristic \(p>2\) and let \(L\) denote the derivation algebra of the truncated polynomial algebra \(\mathfrak{A}=K[x_1,\dots,x_m]/ (x_1^p,\dots,x_m^p)\). The aim of the paper under review is a description of certain representations of \(L\) similar to the representations of Lie algebras of reductive groups with \(p\)-characters in standard Levi form which was developed by \textit{E. M. Friedlander} and \textit{B. J. Parshall} [Am. J. Math. 110, No. 6, 1055--1093 (1988; Zbl 0673.17010)] as well as \textit{J. C. Jantzen} [J. Pure Appl. Algebra 152, No. 1-3, 133--185 (2000; Zbl 0976.17004)]. Let \(I\) be a subset of \(S=\{1,\dots,m\}\) and denote by \(\mathfrak{A}(I)\) the subalgebra of \(\mathfrak{A}\) generated by \(\{x_i\mid i\in I\}\), by \(L(I)\) the derivation algebra of \(\mathfrak{A}(I)\), and by \(\mathfrak{h}(I)\) the standard maximal torus \(\bigoplus_{i=1}^m Kx_i\frac{\partial}{\partial x_i}\) of \(L(I)\). Each subset \(I\) of \(S\) determines a grading of \(\mathfrak{A}\) by considering \(x_i\) for \(i\in I\) as a homogeneous element of degree one and for \(i\not\in I\) as a homogeneous element of degree zero. This grading of \(\mathfrak{A}\) induces a \(\mathbb{Z}^m\)-grading \(L=\bigoplus_{\alpha\in\mathbb{Z}^m} L_{\alpha,I}\) and a \(\mathbb{Z}\)-grading \(L=\bigoplus_{q\geq -1}L_{[q],I}\) on \(L\). Here \(L_I=L_{0,I}\) is generated as a module over \(\mathfrak{A}(S\setminus I)\) by \(\{x_i \frac{\partial}{\partial x_i}\mid i\in I\}\cup\{\frac{\partial}{\partial x_i}\mid i\in S\setminus I\}\), i.e., \(L_I\) is a semidirect product of \(L(S\setminus I)\) and an abelian ideal isomorphic to \(\mathfrak{A}(S\setminus I)\otimes\mathfrak{h}(I)\), and \(L_{[0],I}\) is a semidirect product of \(L(S\setminus I)\) and an ideal isomorphic to \(\mathfrak{A}(S \setminus I)\otimes\mathfrak{gl}(I)\). Let \(\chi\) be a linear form on \(L\) and denote by \(u_\chi(\mathfrak{g})\) the \(\chi\)-reduced universal enveloping algebra of any restricted subalgebra \(\mathfrak{g}\) of \(L\). It is shown that for every linear form \(\chi\) on \(L\) there exists a subset \(I\) of \(S\) such that \(\chi(L_{\alpha,I})=0\) for every \(0\neq\alpha\in\mathbb{Z}^m\). For the rest of the paper such an index set \(I\) is fixed. Let \(\varepsilon_j\) denote the \(m\)-tuple \((\delta_{ij} )_{1\leq i\leq m}\) and set \(\pi=\sum_{j\in S\setminus I}(p-1)\varepsilon_j\). Then it is proved that the induction functor from \(\bigoplus_{q\geq 0}L_{[q],I}\) to \(L\) yields a bijection between the isomorphism classes of simple \(u_\chi(L_{[0],I})\)-modules and the isomorphism classes of simple \(u_\chi(L)\)-modules if \(\chi(x^{\pi+\varepsilon_i} \frac{\partial}{\partial x_i})\neq 0\) for some \(i\in I\). Moreover, if \(\chi\) is regular semisimple, i.e., \(\chi(x^\pi\otimes h_\alpha)\neq 0\) for every positive root \(\alpha\) of \(\mathfrak{gl}(I)\), then \(u_\chi(L_{[0],I})\) is Morita equivalent to \(u_\chi(L_I)\). In particular, the isomorphism classes of simple \(u_\chi(L)\)-modules are in one-to-one correspondence with the isomorphism classes of simple \(u_\chi(L_I)\)-modules where the latter can be described by the simple \(u_\chi(L(S\setminus I))\)-modules and the weights of the maximal torus \(\mathfrak{h}(I)\).
scientific article; zbMATH DE number 1868018
scientific article