Group gradings on restricted Cartan-type Lie algebras (Q764073)

Gradings are an important key to understand the structure of Lie algebras, including the root space decomposition as one of the most relevant examples. The classification of the gradings on (finite-dimensional) simple Lie \(F\)-algebras is almost complete in the classical case when the ground field \(F\) is algebraically closed. The paper under review begins the study of some simple non classical restricted Lie algebras. It classifies up to isomorphisms, over algebraically closed fields of characteristic \(p>3\), the gradings over an abelian group \(G\) on the following simple restricted Lie algebras of Cartan type: the Witt algebra \(W=W(m,\underline{1})\), the special algebra \(S^{(1)}\) for \(S=S(m, \underline{{1}})\) if \(m\geq3\) and the Hamiltonian algebra \(H^{(2)}\) for \(H=H(2,\underline{1})\). It also classifies the fine gradings on such Cartan type Lie algebras up to equivalence. All this material has just been compiled in the monograph [\textit{A. Elduque} and \textit{M. Kochetov}, Gradings on simple Lie algebras. Providence, RI: AMS (2013; Zbl 1281.17001)]. The main tool used to obtain the classification for one of the above algebras is to find another algebra which shares the same automorphism group scheme and whose gradings are easier to study. There is a bijective correspondence between the classifications of gradings on these two algebras, because the \(G\)-gradings on an algebra \(U\) are in correspondence with the morphisms \(G^D\to {\mathbf{Aut}}(U)\), being \(G^D\) the group scheme represented by the Hopf algebra \(FG\). This approach had been used before, for instance for \(\mathfrak{g}_2\) and the octonion algebra. The role of \textit{helper} is played in this paper by the algebra \(\mathcal{O}=\mathcal{O}(m,\underline{1})\), that is, the truncated polynomial algebra \(F[x_1,\dots,x_m]/(x_1^p,\dots,x_m^p)\), whose gradings are classified without difficulties at the end of the work. Then the authors take into account (and prove when necessary) the following isomorphisms of group schemes in the case \(p>3\): \( {\mathbf{Aut}}(\mathcal{O})\cong {\mathbf{Aut}}(W)\), \( {\mathbf{Aut}}_S(\mathcal{O})\cong {\mathbf{Aut}}(S^{(1)})\) where \( {\mathbf{Aut}}_S(\mathcal{O})=\mathrm{Stab}_{ {\mathbf{Aut}}(\mathcal{O})}(\langle\omega_S\rangle)\) for \(\omega_S\) the differential form given by \(dx_1\wedge \dots\wedge dx_m\), and \( {\mathbf{Aut}}_H(\mathcal{O})\cong {\mathbf{Aut}}(H^{(2)})\) where \( {\mathbf{Aut}}_H(\mathcal{O})=\mathrm{Stab}_{ {\mathbf{Aut}}(\mathcal{O})}(\langle\omega_H\rangle)\) for \(\omega_H\) the Hamiltonian form (this isomorphism is proved for \(m=2r\), although the gradings are only computed for \(r=1\)). Thus there is a bijective correspondence between \(G\)-gradings on \(\mathcal{O}\) and \(G\)-gradings on \(W\) (and between their isomorphism classes), as well as between \(G\)-gradings on \(S^{(1)}\) (respectively \(H^{(2)}\)) and \(S\)-admissible gradings on \(\mathcal{O}\) (respectively \(H\)-admissible gradings).