Representations of finite Lie algebras
This paper is devoted to the study of the representation theory of finite Lie algebras, realised as fixed point subalgebras under a Frobenius morphism. Similar ideas have been developed for associative algebras in [\textit{B. Deng} and \textit{J. Du}, Trans. Am. Math. Soc. 358, No. 8, 3591--3622 (2006; Zbl 1095.16007)]. The main setting is as follows: let \(\mathbb{F}_q\) be the field with \(q\) elements, \(k\) its algebraic closure, \(\mathfrak{g}\) a finite-dimensional Lie algebra over \(k\) and \(F\) a (\(q\)-semilinear) Frobenius morphism on \(\mathfrak{g}\). Then the object of study is the fixed point subalgebra \(\mathfrak{g}^F\), and the relationship between representations of \(\mathfrak{g}^F\) and \(F\)-stable representations of \(\mathfrak{g}\). In the first parts general features of Lie algebras with a Frobenius morphism are investigated. Using the Frobenius twist functor, the authors provide connections between simple (or indecomposable) modules over \(\mathfrak{g}\) and \(\mathfrak{g}^F\). As a first application they determine all Frobenius maps over classical simple Lie algebras over \(k\), with \(\mathrm{char}(k) > 3\). As a corollary one retrieves the well-known classification of all \(\mathbb{F}_q\)-forms of \(\mathfrak{g}\). The last sections are devoted to the case when \(\mathfrak{g}^F\) is restricted, or equivalently when \(\mathfrak{g}\) is restricted and \(F\) is compatible with the \(p\)-mapping. The authors define the \(F\)-orbital reduced enveloping algebra \(U_{\underline{\chi}}(\mathfrak{g})\), which is associated with the \(F\)-orbit \(\underline{\chi}\) of \(\chi\). Various equivalences of categories are described, between appropriate full subcategories of \(\mathfrak{g}\)-modules with \(p\)-character \(\chi\) and module categories over \(U_{\underline{\chi}}(\mathfrak{g})\). The behaviour of these constructions when \(\chi\) is twisted by a \(k\)-algebra automorphism of \(\mathfrak{g}\) is also investigated. Finally the whole theory is illustrated with the example of \(\mathfrak{sl}_2(\mathbb{F}_q)\).
scientific article; zbMATH DE number 5599085
scientific article