论文标题
在谎言代数模块上，该模块是半密象组方案的模块
On Lie algebra modules which are modules over semisimple group schemes
论文作者
论文摘要
令$ p $为素数。给定一个普通整体域$ r $上的分裂的半胶合组方案$ g $，这是一种忠实的$ \ m mathbb z _ {（p）} $ - 代数，我们对所有有限的尺寸表示$ g_k $ g_k $ g_k $ g_k $ g_k $ g $ g $ a $ k：= $ k：= freac} $ a $ $ g $ - 模块也是$ v $相对于$ r $（$ \ text {lie}（g）$ - 模块的零件的集合。我们应用此分类以获取比$ \ text {spec} k $之间的同态同构扩展的一般标准，以在$ \ text {spec} r $上的还原组方案之间的同构。我们还表明，对于简单连接的半imple组方案，在减少的$ \ mathbb Q $-代数上，其表示形式的类别等同于其Lie代数的表示类别。
Let $p$ be a prime. Given a split semisimple group scheme $G$ over a normal integral domain $R$ which is a faithfully flat $\mathbb Z_{(p)}$-algebra, we classify all finite dimensional representations $V$ of the fiber $G_K$ of $G$ over $K:=\text{Frac}(R)$ with the property that the set of lattices of $V$ with respect to $R$ which are $G$-modules is as well the set of lattices of $V$ with respect to $R$ which are $\text{Lie}(G)$-modules. We apply this classification to get a general criterion of extensions of homomorphisms between reductive group schemes over $\text{Spec} K$ to homomorphisms between reductive group schemes over $\text{Spec} R$. We also show that for a simply connected semisimple group scheme over a reduced $\mathbb Q$--algebra, the category of its representations is equivalent to the category of representations of its Lie algebra.
