On automorphisms of block-triangular polynomial translation groups

AbstractLet GAn be the group of all polynomial transformations of affine space An (affine Cremona group) over field K of zero characteristic. For a given decomposition An = An1 ⊕ … ⊕ Anq we define the block-unitriangular polynomial translations of An, as transformations of the kind xi ↦ xi + ai(x1, …, xi − 1), where Xi = (xi, j) ∈ Ani, j = 1, 2 …, ni, i = 1, 2, …, q and ai are vectorpolynomials. Such transformations form the subgroup Un of GAn which can be considered as iterated algebraic wreath product of groups of ordinary translations of affine spaces Ani. The normalizer Bn = NGAn(Un) may be decomposed into semidirect product: Bn = (GLn1 (K) × … × GLnq (K)) · Un. There are two opposite examples of groups Bn: AGLn(K) — affine group, q = 1, n = n1; — Jonq'ear group — the group of all triangular transformations, q = n, ni = 1. The groups Un, Bn have a structure of algebraic groups of infinite dimension. Main purpose of the article is to describe algebraic automorphisms of groups Un, Bn. 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 Un is ascertained, action of automorphisms is written in the explicit form;3.(3) every regular automorphism of Bn is an interior one