On automorphisms of block-triangular polynomial translation groups

Abstract

Let GA n be the group of all polynomial transformations of affine space A n (affine Cremona group) over field K of zero characteristic. For a given decomposition A n = A n 1 ⊕ … ⊕ A n q we define the block-unitriangular polynomial translations of A n , as transformations of the kind x i ↦ x i + a i ( x 1, …, x i − 1 ), where X i = ( x i, j ) ∈ A n i , j = 1, 2 …, n i , i = 1, 2, …, q and a i are vectorpolynomials. Such transformations form the subgroup U n of GA n which can be considered as iterated algebraic wreath product of groups of ordinary translations of affine spaces A n i . The normalizer B n = N GA n ( U n ) may be decomposed into semidirect product: B n = ( GL n 1 ( K) × … × GL n q ( K)) · U n . There are two opposite examples of groups B n : AGL n ( K) — affine group, q = 1, n = n 1; — Jonq'ear group — the group of all triangular transformations, q = n, n i = 1. The groups U n , B n have a structure of algebraic groups of infinite dimension. Main purpose of the article is to describe algebraic automorphisms of groups U n , B n . The principal results are 1. (1) the endomorphisms of a polynomial ring in several variables as an infinite-dimension translation module form the ring which is isomorphic to a formal exponential power series ring; 2. (2) the structure of Aut U n is ascertained, action of automorphisms is written in the explicit form; 3. (3) every regular automorphism of B n is an interior one.