Extensions of modules over Hopf algebras arising from Lie algebras of Cartan type
Let \(\mathfrak g\) be a Lie \(p\)-algebra of Cartan type with the automorphism group \(G\), \(G=G_0\ltimes N^+\), where \(N^+\) is the unipotent subgroup, \(G_0\) is the subgroup of linear automorphisms. For a group scheme \(G\) the distribution algebra of \(G\) is denoted by \(D(G)\). Let \(G_r=\ker F^r\) where \(F^r: G\rightarrow G\) is the \(r\)th iteration of the Frobenius morphism. According to the authors' paper [J. Algebra 179, 852-888 (1996; Zbl 0862.17015)] a cocommutative Hopf algebra \(A_r=D(G_r)\mathfrak u(\mathfrak g)\), as well as \(A=D(G)\mathfrak u(\mathfrak g)\), where \(\mathfrak u(\mathfrak g)\) is the \(u\)-algebra of \(\mathfrak g\) can be built. In the paper the structure of the groups \(\text{Ext} _{A_r}^1(\mathcal L(\lambda),\mathcal L(\lambda))\) and \(\text{Ext} _A^1(\mathcal L(\lambda),\mathcal L(\lambda))\) is studied for an irreducible \(A_r\)-module \(\mathcal L(\lambda)\) (correspondingly, an irreducible \(A\)-module) with an anti-dominant weight \(\lambda \in -\Lambda _r\) where \(\Lambda _r\) is the character group of Frobenius kernel \(T_r\) of maximal torus \(T\) in \(G\). \(\Lambda _r\) is treated as a subset in \(X(T)_+\) of dominant weights of \(T\). The major results are obtained for Lie \(p\)-algebras of types \(W(m:\mathbf 1),~CS(m:\mathbf 1)\): {Theorem 5.1} Suppose \(p>2\) and \(p^{r-1}>m\). For \(\lambda \in -\Lambda _r\) \[ \text{Ext} _{A_r}^1(\mathcal L(\lambda),\mathcal L(\lambda))=0. \] Theorem 5.8 If \(p\geq m+1\) then \[ \text{Ext} _{\mathfrak u(\mathfrak g)}^1(\mathcal L(\lambda),\mathcal L(\lambda))=0 \] for all \(\lambda \in -\Lambda _1\). In the first part of the work the general conditions for all Lie \(p\)-algebras of Cartan type sufficient for vanishing \(\text{Ext} _{A_r}^1(\mathcal L(\lambda),\mathcal L(\lambda))\) are given. They reduce the problem to calculation of \(\text{Ext} _{D(B_r^+)}^1(\mathcal L(\lambda),\lambda)\) and \(\text{Ext} _{D(N_r^+)}^1(\mathcal L(\lambda),\lambda)\) where \(B^+\) is a standard Borel subgroup of \(G\). For \(A\)-modules it is almost evident that \(\text{Ext} _A^1(\mathcal L(\lambda),\mathcal L(\lambda))=0\) (Proposition 6.1). In the conclusion it is shown how \(\text{Ext} _{A_r}^1(\mathcal L(\lambda),\mathcal L(\lambda))\) can be recursively calculated if \(\text{Ext} _{A_1}^1(\mathcal L(\lambda),\mathcal L(\lambda))\) and \(\text{Ext} _{D(G_1)}^1(\mathcal L(\lambda),\mathcal L(\lambda))\) are known. The relation between extensions of \(A\)-modules and \(A_r\) cohomology is established.
scientific article; zbMATH DE number 1475286