Irreducible graded modules over graded Lie algebras (Q790920)

Finite-dimensional irreducible graded modules over finite-dimensional graded Lie algebras over a field of characteristic \(p>0\) are investigated in this article. Let \(L=L_{-q}+\ldots+L_ r\) be a graded Lie algebra and \(M=M_{-q}+\ldots+M_ R\) a graded \(L\)-module. The author determines a number of criteria for \(M\) to be isomorphic to \[ \mathrm{Ind}(L,M_ R)=\overline{U(L)} \otimes_{\iota(U({\mathcal L}_ 0))} M_ R, \] where \(\overline{U(L)}=U(L)/J\), \(J\) being the two-sided ideal of \(U(L)\) generated by the elements \(y^{p^{m(y)}}\), \(y\in L_{-q}+\ldots+L_{-1}\), where \(m(y)\) is the least nonnegative integer such that \(\mathrm{ad}_ L\!^{p^{m(y)}}y=0\). These criteria are used in calculating the minimum possible dimension of an irreducible graded \(L\)-module, where \(L\) is a simple Lie algebra of Cartan type \(H_{2n}(F)\) or \(K_{2n+1}(F)\).