Isomorphism problems and groups of automorphisms for Ore extensions K[x][y; f(d)/(dx) ] (prime characteristic)

2021/07/21 by Bavula, V. V.
#13N10 #16D60 #16P90 #16S32 #16U20 #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2107.09977

Abstract

Let Λ(f) = K[x][y; f(d)/(dx) ] be an Ore extension of a polynomial algebra K[x] over an arbitrary field K of characteristic p>0 where f∈ K[x]. For each polynomial f, the automorphism group of the algebras Λ(f) is explicitly described. The automorphism group \rm Aut_K(Ł(f))=S\rtimes G_f is a semidirect product of two explicit groups where G_f is the \em eigengroup of the polynomial f (the set of all automorphisms of K[x] such that f is their common eigenvector). For each polynomial f, the eigengroup G_f is explicitly described. It is proven that every subgroup of \rm Aut_K(K[x]) is the eigengroup of a polynomial. It is proven that the Krull and global dimensions of the algebra Λ(f) are 2. The prime, completely prime, primitive and maximal ideals of the algebra Λ(f) are classified.

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