/pdf/automorphism-groups-of-unipotent-groups-of-chevalley-type-4orw8i6hwt.pdf

Automorphisms of affine unipotent groups in positive characteristic

Automorphism groups of unipotent groups of Chevalley type

Let G be a connected unipotent algebraic group defined over the perfect field k. We show that polynomial generators x1, , x for the ring k[G] can be chosen so that if N is any connected normal kclosed subgroup of G, then I(N) can be generated by codim N p-polynomials in x1 X * xn where p = char k. Moreover k[G/N] cani also be generated as a polynomial algebra over k by p-polynomials. Introduction. These results are essentially an extension of a theorem of Rosenlicht [4, Theorem 1]. We use the notation and conventions of [1] throughout this paper. Recall that a p-polynomial in k[T] is a linear form if p = 0 and a polynomial all of whose exponents are powers of p if p > 0. A p-polynomial in k[xl, *. , xn] is a sum of p-polynomials in each of the single variables xl, x . A function f E k[G] will be called additive if /(ab) = /(a) + /(b) for all closed points a, b in G. 1. Frattini coordinates. Let G be a unipotent algebraic group. The Frattini subgroup of G is the intersection of all closed subgroups of codimension one. We shall denote this group by Fr(G). Proposition 1. If G is a unipotent algebraic group then Fr(G) is a closed characteristic subgroup of G. If G is connected and defined over the perfect field k, then Fr(G) is connected and defined over k. Moreover in the connected case G/Fr(G) has the structure of a vector group (over k if G is defined over k) and is the maximal such quotient. Proof. The first assertion is immediate. Let H C G be a closed subgroup of G of codimension one. Since G/H Ga7 H contains the commutator subgroup of G and the group generated by the pth powers of the elements of G. It follows that Fr(G) also contains these subgroups. Thus G/Fr(G) is connected, commutative and of exponent p hence by Received by the editors December 28, 1973. AMS(MOS) subject classifications (1970). Primary 14L10, 20G15.

/pdf/defining-normal-subgroups-of-unipotent-algebraic-groups-1zic301hph.pdf

Defining normal subgroups of unipotent algebraic groups

The subject of algebraic groups has had a rapid development in recent years. Leaving aside the late research by many people on the Albanese and Picard variety, it has received much substance and impetus from the work of Severi on commutative algebraic groups over the complex number field, that of Kolchin, Chevalley, and Borel on algebraic groups of matrices, and especially Weil's research on abelian varieties and algebraic transformation spaces. The main purpose of the present paper is to give a more or less systematic account of a large part of what is now known about general algebraic groups, which may be abelian varieties, algebraic groups of matrices, or actually of neither of these types.

Automorphisms of affine unipotent groups in positive characteristic

Citations

Automorphism groups of unipotent groups of Chevalley type

Defining normal subgroups of unipotent algebraic groups

References

Some Basic Theorems on Algebraic Groups

Some rationality questions on algebraic groups

Introduction to Affine Algebraic Groups

Automorphisms of affine algebraic groups

Related Papers (5)

Automorphisms of certain unipotent groups

Conjugacy classes of affine automorphisms of K n and linear automorphisms of P n in the

Outer unipotent classes in automorphism groups of simple algebraic groups

Automorphisms of Orthogonal Groups in Characteristic 2, II

On the Automorphisms of the Classical Groups