Algebraic group actions in the cohomology theory of Lie algebras of Cartan type
In contrast to classical simple Lie algebras, Lie algebras of Cartan type \({\mathfrak g}\) and their representations cannot be characterized in terms of algebraic groups and their infinitesimal subgroups, which makes the application of cohomological methods developed within the representation theory of modular classical Lie algebras more difficult. Nevertheless, important classes of restricted \({\mathfrak g}\)-modules have the natural structure of \(G\)-modules where \(G\) is a subgroup in \(\Aut {\mathfrak g}\). Of particular interest is the case when \(G=T\) is a maximal torus of \(\Aut {\mathfrak g}\). In order to make use of both structures the authors construct a special infinite-dimensional Hopf algebra \(\text{Dist} (G)u ({\mathfrak g})\) where \(\text{Dist} (G)\) is the distribution algebra of an algebraic group \(G\) and \(u({\mathfrak g})\) is the \(u\)-algebra of \({\mathfrak g}\). In the first part of the paper, for a Hopf algebra \(H\) and an \(H\)-module algebra \(A\) containing a common Hopf subalgebra \(D\), normal in \(H\), the Hopf algebra \(P=A \# H/I\) where \(I\) is an ideal of \(A \# H\), generated by \(d \otimes 1-1 \otimes d\), \(d \in D\) is constructed. Making use of the representation theory of Hopf algebras developed by \textit{Z. Lin} [J. Algebra 154, 152-187 (1993; Zbl 0766.17014) and J. Algebra 166, 100-129 (1994; Zbl 0810.17005)] the connection of homological functors for admissible categories of modules over \(P,H\) and \(A\) is investigated. In the second part of the paper this technique is used for the case \(A= u({\mathfrak g})\), \(H= \text{Dist} (G)\), \(D= u({\mathfrak g}_G)\) where \( {\mathfrak g}\) is a Lie \(p\)-algebra of Cartan type, \(G\subset \Aut {\mathfrak g}\), \({\mathfrak g} = \text{Lie} G\) is a Lie algebra of \(G\). The corresponding Hopf algebra \(P\) is denoted by \(\text{Dist} (G) u ({\mathfrak g})\). The description of projective \(u({\mathfrak g})\)-modules in the category of finite-dimensional \(\text{Dist} (G)u ({\mathfrak g})\)-modules is given. The complexity and support varieties of \(\text{Dist} (G)u ({\mathfrak g})\)-modules are studied.
scientific article; zbMATH DE number 868024