Affine Cremona Group Research Articles

On conjugacy of additive actions in the affine Cremona group

An additive action on an irreducible algebraic variety X is an effective action with an open orbit of the vector group . Any two additive actions on X are conjugate by a birational automorphism of X. We prove that, if X is the projective space, the conjugating element can be chosen in the affine Cremona group and it is given by so-called basic polynomials of the corresponding local algebra.

On new protocols of Noncommutative Cryptography in terms of homomorphism of stable multivariate transformation groups

Noncommutative cryptography is based on applications of algebraic structures like noncommutative groups, semigroups, and noncommutative rings. Its intersection with Multivariate cryptography contains studies of cryptographic applications of subsemigroups and subgroups of affine Cremona semigroups defined over finite commutative rings. Efficiently computed homomorphisms between stable subsemigroups of affine Cremona semigroups can be used in tame homomorphisms protocols schemes and their inverse versions. The implementation scheme with the sequence of subgroups of affine Cremona group that defines the projective limit was already suggested. We present the implementation of another scheme that uses two projective limits which define two different infinite groups and the homomorphism between them. The security of the corresponding algorithm is based on complexity of the decomposition problem for an element of affine Cremona semigroup into a product of given generators. These algorithms may be used in postquantum technologies.

On desynchronised multivariate algorithms of El Gamal type for stable semigroups of affine Cremona group

Families of stable cyclic groups of nonlinear polynomial transformations of affine spaces Kn over general commutative ring K of increasing with n order can be used in the key exchange protocols and related to them El Gamal multivariate cryptosystems. To use high degree of noncommutativity of affine Cremona group correspondents have to modify multivariate El Gamal algorithm via the usage of conjugations for two polynomials of kind gk and g−1 given by key holder (Alice) or giving them as elements of different transformation groups. The idea of hidden tame homomorphism and comlexity of decomposition of polynomial transwormation into word of elements of Cremona semigroup can be used. We suggest usage of new explicit constructions of infinite families of large stable subsemigroups of affine Cremona group of bounded degree as instruments of multivariate key exchange protocols. Recent results on generation of families of stable transformations of small degree and density via technique of symbolic walks on algebraic graphs are observed. Some of them used for the implementation of schemes as above with feasible computational complexity. We consider an example of a new implemented quadratic multivariate cryptosystem based on the above mentioned ideas.

On the Families of Stable Multivariate Transformations of Large Order and Their Cryptographical Applications

Abstract Families of stable cyclic groups of nonlinear polynomial transformations of affine spaces Kn over general commutative ring K of with n increasing order can be used in the key exchange protocols and El Gamal multivariate cryptosystems related to them. We suggest to use high degree of noncommutativity of affine Cremona group and modify multivariate El Gamal algorithm via conjugations of two polynomials of kind gk and g−1 given by key holder (Alice) or giving them as elements of different transformation groups. Recent results on the existence of families of stable transformations of prescribed degree and density and exponential order over finite fields can be used for the implementation of schemes as above with feasible computational complexity.

On restriction of roots on affine $$T$$ T -varieties

Let \(X\) be a normal affine algebraic variety with a regular action of a torus \(\mathbb {T}\) and \(T\subset \mathbb {T}\) be a subtorus. We prove that each root of \(X\) with respect to \(T\) can be obtained by restriction of some root of \(X\) with respect to \(\mathbb {T}\). This allows to give an elementary proof of the description of roots of the affine Cremona group. Several results on restriction of roots in the case of a subtorus action on an affine toric variety are obtained.

Roots of the affine Cremona group

Let k [n] = k[x 1,…, x n ] be the polynomial algebra in n variables and let \( {\mathbb{A}^n} = {\text{Spec}}\;{{\bold{k}}^{\left[ n \right]}} \). In this note we show that the root vectors of \( {\text{Au}}{{\text{t}}^*}\left( {{\mathbb{A}^n}} \right) \), the subgroup of volume preserving automorphisms in the affine Cremona group \( {\text{Aut}}\left( {{\mathbb{A}^n}} \right) \), with respect to the diagonal torus are exactly the locally nilpotent derivations x α(∂/∂x i ), where x α is any monomial not depending on x i . This answers a question posed by Popov.

Locally nilpotent derivations and Nagata-type utomorphisms of a polynomial algebra

We study locally nilpotent derivations belonging to a Lie algebra sa n of a special affine Cremona group in connection with the root decompositions of sa n relative to the maximum standard torus. It is proved that all root locally nilpotent derivations are elementary. As a continuation of this research, we describe two- and three-root derivations. By using the results obtained by Shestakov and Umirbaev, it is shown that the exponents of almost all obtained three-root derivations are wild automorphisms of a polynomial algebra in three variables.

Every regular automorphism of the affine Cremona group is inner

It is shown that all regular automorphisms of the affine Cremona group over an algebraically closed field of characteristic zero are inner.

On automorphisms of block-triangular polynomial translation groups

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.

A note on automorphisms of the affine Cremona group

Let $\mathcal{G}$ be an ind-group and let $\mathcal{U} \subseteq \mathcal{G}$ be a unipotent ind-subgroup. We prove that an abstract group automorphism $\theta \colon \mathcal{G} \to \mathcal{G}$ maps $\mathcal{U}$ isomorphically onto a unipotent ind-subgroup of $\mathcal{G}$, provided that $\theta$ fixes a closed torus $T \subseteq \mathcal{G}$, which normalizes $\mathcal{U}$ and the action of $T$ on $\mathcal{U}$ by conjugation fixes only the neutral element. As an application we generalize a result by Hanspeter Kraft and the author as follows: If an abstract group automorphism of the affine Cremona group $\mathcal{G}_3$ in dimension 3 fixes the subgroup of tame automorphisms $T{\mathcal G}_3$, then it also fixes a whole family of non-tame automorphisms (including the Nagata automorphism).