Groups of Multiplicative Type
Separable algebras, besides describing connected components, are related to a familiar kind of matrix and can lead us to another class of group schemes. One calls an n × n matrix g separable if the subalgebra k[g] of End(k n ) is separable. We have of course k[g] ≃ k[X]/p(X) where p(X) is the minimal polynomial of g Separability then holds iff k[g]⊗\( bar k \) = \( bar k \)[g] ⋍ \( bar k \)[X]/p(X) is separable over k. This means that p has no repeated roots over k, which is the familiar criterion for g to be diagonalizable over \( bar k \). (We will extend this result in the next section.) Then p is separable in the usual Galois theory sense, its roots are in k s , and g is diagonalizable over k s .
KeywordsAbelian Group Hopf Algebra Group Scheme Closed Subgroup Invertible Element
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