Nonlinear Automorphisms Research Articles

Nonlinear automorphism of the conformal algebra in 2D and continuous sqrt{Toverline{T}} deformations

The conformal algebra in 2D (Diff(S1)⨁Diff(S1)) is shown to be preserved under a nonlinear map that mixes both chiral (holomorphic) generators T and overline{T} . It depends on a single real parameter and it can be regarded as a “nonlinear SO(1, 1) automorphism.” The map preserves the form of the momentum density and naturally induces a flow on the energy density by a marginal sqrt{Toverline{T}} deformation. In turn, the general solution of the corresponding flow equation of the deformed action can be analytically solved in closed form, recovering the nonlinear automorphism. The deformed CFT2 can also be described through the original theory on a field-dependent curved metric whose lapse and shift functions are given by the variation of the deformed Hamiltonian with respect to the energy and momentum densities, respectively. The conformal symmetries of the deformed theories can then also be seen to arise from diffeomorphisms that fulfill suitably deformed conformal Killing equations. Besides, Cardy formula is shown to map to itself under the nonlinear automorphism. As a simple example, the deformation of N free bosons is briefly addressed, making contact with recent related results and the dimensional reduction of the ModMax theory. Furthermore, the nonlinear map between the conformal algebra in 2D and its ultra/non-relativistic versions (BMS3≈CCA2≈GCA2), including the corresponding finite sqrt{Toverline{T}} deformation, are recovered from a limiting case of the nonlinear automorphism. The extension to a three-parameter nonlinear ISO(1, 1) automorphism of the conformal algebra, and a discrete nonlinear automorphism of BMS3 are also briefly discussed.

On Beloshapka’s rigidity conjecture for real submanifolds in complex space

A well known Conjecture due to Beloshapka asserts that all totally nondegenerate polynomial models with the length $l \geq 3$ of their Levi–Tanaka algebra are rigid, that is, any point preserving automorphism of them is completely determined by the restriction of its differential at the fixed point onto the complex tangent space. For the length $l = 3$, Beloshapka’s Conjecture was proved by Gammel and Kossovskiy in 2006. In this paper, we prove the Conjecture for arbitrary length $l \geq 3$. As another application of our method, we construct polynomial models of length $l \geq 3$, which are not totally nondegenerate and admit large groups of point preserving nonlinear automorphisms.

Permutation Groups Induced by Derksen Groups in Characteristic Two

We consider the so-called Derksen group which is a subgroup of the polynomial automorphism group of the polynomial ring in n variables over a field. The Derksen group is generated by affine automorphisms and one particular non-linear automorphism. Derksen (1994) proved that if the characteristic of the underlying field is zero and n ≥ 3, then the Derksen group is equal to the entire tame subgroup. The result is called Derksen’s Theorem. It is quite natural to ask whether the same property holds for positive characteristic. In this paper, we point out that the question can be easily answered negatively when the underlying field is of characteristic two. We shall also prove that the permutation group induced by the Derksen group over a finite field of characteristic two is a proper subgroup of the alternating group on the n-dimensional linear space over the finite field. This is a stronger result that Derksen’s Theorem does not hold when the underlying field is a finite field of characteristic two.

Nonlinear CR automorphisms of Levi degenerate hypersurfaces and a new gap phenomenon

We give a complete classification of polynomial models for smooth real hypersurfaces of finite Catlin multitype in C-3, which admit nonlinear infinitesimal CR automorphisms. As a consequence, we obtain a sharp 1-jet determination result for any smooth hypersurface with such a model. The results also prove a conjecture of the first author about the origin of such nonlinear automorphisms (AIM list of problems, 2010). As another consequence, we describe all possible dimensions of the Lie algebra of infinitesimal CR automorphisms, which leads to a new secondary gap phenomenon.

Generalization of a counterexample to Derksen\u2019s theorem in characteristic two

The Derksen group is the subgroup of the tame subgroup which is generated by affine automorphisms and one particular non-linear automorphism. It is known that the Derksen group is equal to the entire tame subgroup in characteristic zero and in dimension greater than or equal to three (Derksen’s Theorem). Recently, Maubach and Willems (Serdica Math J 37:305–322, 2011) proved that if the defining field of polynomial automorphisms is prime field of characteristic two, then Derksen’s Theorem does not hold in dimension three. In this paper, we generalize the above counterexample to the case of higher dimension.

Higher order symmetries of real hypersurfaces in ℂ³

We study nonlinear automorphisms of Levi degenerate hypersurfaces of finite multitype. By results of Kolar, Meylan, and Zaitsev in 2014, the Lie algebra of infinitesimal CR automorphisms may contain a graded component consisting of nonlinear vector fields of arbitrarily high degree, which has no analog in the classical Levi nondegenerate case, or in the case of finite type hypersurfaces in C 2 \mathbb C^2 . We analyze this phenomenon for hypersurfaces of finite Catlin multitype with holomorphically nondegenerate models in complex dimension three. The results provide a complete classification of such manifolds. As a consequence, we show on which hypersurfaces 2-jets are not sufficient to determine an automorphism. The results also confirm a conjecture about the origin of nonlinear automorphisms of Levi degenerate hypersurfaces, formulated by the first author for an AIM workshop in 2010.

On Generators of the Tame Automorphism Group of Free Metabelian Lie Algebras

We prove that the tame automorphism group TAut(M n ) of a free metabelian Lie algebra M n in n variables over a field k is generated by a single nonlinear automorphism modulo all linear automorphisms if n ≥ 4 except the case when n = 4 and char(k) ≠ 3. If char(k) = 3, then TAut(M 4) is generated by two automorphisms modulo all linear automorphisms. We also prove that the tame automorphism group TAut(M 3) cannot be generated by any finite number of automorphisms modulo all linear automorphisms.

Supersymmetry of the Chiral de Rham Complex 2: Commuting Sectors

We construct two commuting N = 2 structures on the space of sections of the chiral de Rham complex (CDR) of a Calabi-Yau manifold. We use this extra supersymmetry to construct a nonlinear automorphism of CDR preserving these N = 2 structures. Finally, we show how to extend these results to construct a commuting pair of N = 4 super vertex algebras in the HyperKahler case.

The envelope of holomorphy of a model third-degree surface and the rigidity phenomenon

The structures of the graded Lie algebra aut Q infinitesimal automorphisms of a cubic (a model surface in ℂN) and the corresponding group Aut Q of its holomorphic automorphisms are studied. It is proved that for any nondegenerate cubic, the positively graded components of the algebra aut Q are trivial and, as a consequence, Aut Q has no subgroups consisting of nonlinear automorphisms of the cubic that preserve the origin (the so-called rigidity phenomenon). In the course of the proof, the envelope of holomorphy for a nondegenerate cubic is constructed and shown to be a cylinder with respect to the cubic variable whose base is a Siegel domain of the second kind.

Nonlinear automorphisms of plane domains

We prove that the number of holomorphic nonlinear polynomials mapping a plane domain one-to-one onto itself is at most countable.

New classes of test polynomials of polynomial algebras

A polynomialp in a polynomial algebra over a field is called a test polynomial if any endomorphism of the polynomial algebra that fixesp is an automorphism. some classes of new test polynomials recognizing nonlinear automorphisms of polynomial algebras are given. In the odd prime characteristic case, test polynomials recognizing non-semisimple automorphisms are also constructed.

Quadrics of codimension 4 in ℂ7 and their automorphisms

Automorphisms of nondegenerate (4, 3)-quadrics are studied. We classify quadrics of this kind with nonlinear automorphisms and find the dimensions of their groups of automorphisms.

Nonlinear equivariant automorphisms

We show that the natural representation of SL3 × SL5 × SL13 allows nonlinear equivariant automorphisms; more exactly, the group of polynomial automorphisms on ℂ3 ⊗ ℂ5 ⊗ ℂ13 commuting with the simple SL3 × SL5 × SL13-action is isomorphic to ℂ ⋉ ℂ. This is the first example of a simple module with nonlinear equivariant automorphisms.

About C-rigid quadrics

In the article strongly nondegenerate (k, n)-quadrics all of whose linear automorphisms are of the formz→μz, ω → ¦μ¦2ω, μ ∈ ℂ {0} are considered. Quadrics all of whose linear automorphisms are of this form were calledc-rigid by V. Beloshapka. The main result of the article is the following: anyc-rigid strongly nondegenerate (k, n)-quadric has no nonlinear automorphisms. A table indicating the relationship between linear and nonlinear automorphisms for (k, n)-quadrics is presented.

Structure of groups of nonlinear automorphisms for (3,3) quadrics

We study the group of nonlinear automorphisms for (k, n) quadrics. We prove that this group is nilpotent and describe the multiplication law for this group in terms of the corresponding Lie algebra. For nonlinear automorphisms of (3,3) quadrics, explicit formulas are obtained.

Real quadrics of codimension 3 in and their non-linear automorphisms

In this paper, non-degenerate (3,3)-quadrics are considered. A list of quadrics with non-linear automorphisms is obtained up to equivalence. All nullquadrics of codimension 3 in are determined. We give an example of a quadric with a non-linear automorphism not representable as a Poincare automorphism.

Q rotations and other Q transformations as unitary nonlinear automorphisms of quantum algebras

For the quantum algebras with three generators [SUq(2), SUq(1,1), q oscillator and others] the unitary operator U(s;q) is found in terms of q exponents from algebra’s generators. Being the q analog of the linear automorphism for the Lie algebras, the operator U(s;q) is shown to generate nonlinear automorphisms of the quantum algebras (q rotations, q shifts, Bogoliubov q transformations, etc.). The matrix elements Tnn′ of the operator U(s;q) are explicitly expressed in terms of Askey–Wilson polynomials. The operator U(s;q) is shown to give rise the new systems of q-coherent states of the quantum algebras.