Action Of Finite Abelian Group Research Articles

Stable Separation of Orbits for Finite Abelian Group Actions

Stable Separation of Orbits for Finite Abelian Group Actions

Equivariant birational types and Burnside volume

We introduce equivariant Burnside groups, new invariants in equivariant birational geometry, generalizing birational symbols groups for actions of finite abelian groups, due to Kontsevich, Pestun, and the second author, and study their properties. We establish a specialization map for the equivariant birational type of a smooth algebraic variety with an action of a finite group.

The dual actions, equivariant autoequivalences and stable tilting objects

Pour une catégorie linéaire munie d’une action d’un groupe abélien fini, on étudie l’action duale du groupe des caractères de ce groupe abélien sur la catégorie des objets équivariants. On montre que les groupes des auto-équivalences équivariantes de ces deux catégories sont isomorphes. Dans la situation où la catégorie linéaire est triangulée, les objets basculants stables dans ces deux catégories sont en bijection naturelle compatible avec l’isomorphisme ci-dessus. On applique ces résultats aux complexes basculants stables sur les droites projectives à poids de type tubulaire.

ALMOST FIXED POINTS OF FINITE GROUP ACTIONS ON MANIFOLDS WITHOUT ODD COHOMOLOGY

If X is a smooth manifold and $$ \mathcal{G} $$ is a subgroup of Diff(X) we say that (X, $$ \mathcal{G} $$ ) has the almost fixed point property if there exists a number C such that for any finite subgroup G ≤ $$ \mathcal{G} $$ there is some x ∈ X whose stabilizer Gx ≤ $$ \mathcal{G} $$ satisfies [G : Gx] ≤ C. We say that X has no odd cohomology if its integral cohomology is torsion free and supported in even degrees. We prove that if X is compact and possibly with boundary and has no odd cohomology then (X, Diff(X)) has the almost fixed point property. Combining this with a result of Petrie and Randall we conclude that if Z is a non-necessarily compact smooth real affine variety, and Z has no odd cohomology, then (Z, Aut(Z)) has the almost fixed point property, where Aut(Z) is the group of algebraic automorphisms of Z lifting the identity on Spec ℝ. The main ingredients in the proof are: (1) the Jordan property for diffeomorphism groups of compact manifolds with nonzero Euler characteristic, and (2) the study of λ-stability, a condition on actions of finite abelian groups on manifolds that we introduce in this paper.

Abelian varieties with finite abelian group action

An automorphism of an abelian variety induces a decomposition of the variety up to isogeny. There are two such results, namely the isotypical decomposition and Roan’s decomposition theorem. We show that they are essentially the same. Moreover, we generalize in a sense this result to abelian varieties with action of an arbitrary finite abelian group. An early version of this article was inadvertently published before all the revisions had been completed and then retracted [ https://doi.org/10.1007/s00013-018-1244-3 ]. This article is the final peer reviewed version.

RETRACTED ARTICLE: Abelian varieties with finite abelian group action

An automorphism of an abelian variety induces a decomposition of the variety up to isogeny. There are two such results, namely the isotypical decomposition and Roan’s decomposition theorem. We show that they are essentially the same. Moreover, we generalize in a sense this result to abelian varieties with action of an arbitrary finite abelian group. An early version of this article was inadvertently published before all the revisions had been completed and then retracted [ https://doi.org/10.1007/s00013-018-1244-3 ]. This article is the final peer reviewed version.

On dualities of actions and inclusions

Following the results known in the case of a finite abelian group action on C⁎-algebras we prove the following two theorems;•an inclusion P⊂A of (Watatani) index-finite type has the Rokhlin property (is approximately representable) if and only if the dual inclusion is approximately representable (has the Rokhlin property).•an inclusion P⊂A of (Watatani) index-finite type has the tracial Rokhlin property (is tracially approximately representable) if and only if the dual inclusion is tracially approximately representable (has the tracial Rokhlin property). Moreover, we provide an alternate proof of Phillips' theorem about the relations between tracial Rokhlin action and tracially approximate representable dual action using a new conceptual framework suggested by authors.

Uniqueness of orientation-preserving free actions of finite abelian groups on surfaces

Let G be a finite abelian group. We list all the cases where the topological equivalence class of orientation-preserving free G-actions on a closed surface is unique. Moreover, we obtain the classification of topological equivalence classes of orientation-preserving free actions of finite abelian groups of rank 2.

Fourier transforms and bent functions on faithful actions of finite abelian groups

Let G be a finite abelian group acting faithfully on a finite set X. The G-bentness and G-perfect nonlinearity of functions on X are studied by Poinsot and co-authors (Discret Appl Math 157:1848–1857, 2009; GESTS Int Trans Comput Sci Eng 12:1–14, 2005) via Fourier transforms of functions on G. In this paper we introduce the so-called \(G\)-dual set \(\widehat{X}\) of X, which plays the role similar to the dual group \(\widehat{G}\) of G, and develop a Fourier analysis on X, a generalization of the Fourier analysis on the group G. Then we characterize the bentness and perfect nonlinearity of functions on X by their own Fourier transforms on \(\widehat{X}\). Furthermore, we prove that the bentness of a function on X can be determined by its distance from the set of G-linear functions. As direct consequences, many known results in Logachev et al. (Discret Math Appl 7:547–564, 1997), Carlet and Ding (J Complex 20:205–244, 2004), Poinsot (2009), Poinsot et al. (2005) and some new results about bent functions on G are obtained. In order to explain the theory developed in this paper clearly, examples are also presented.

Computation of invariants of finite abelian groups

We investigate the computation and applications of rational invariants of the linear action of a finite abelian group in the nonmodular case. By diagonalization, such a group action can be described by integer matrices of orders and exponents. We make use of integer linear algebra to compute a minimal generating set of invariants along with the substitution needed to rewrite any invariant in terms of this generating set. In addition, we show how to construct a minimal generating set that consists only of polynomial invariants. As an application, we provide a symmetry reduction scheme for polynomial systems whose solution set is invariant by a finite abelian group action. Finally, we also provide an algorithm to find such symmetries given a polynomial system.

Cyclic covers of affine [formula omitted]-varieties

We consider normal affine T-varieties X endowed with an action of finite abelian group G commuting with the action of T. For such varieties we establish the existence of G-equivariant geometrico-combinatorial presentations in the sense of Altmann and Hausen. As an application, we determine explicit presentations of the Koras–Russell threefolds as bi-cyclic covers of A3 equipped with a hyperbolic Gm-action.

Central exact sequences of tensor categories, equivariantization and applications

We define equivariantization of tensor categories under tensor group scheme actions and give necessary and sufficient conditions for an exact sequence of tensor categories to be an equivariantization under a finite group or finite group scheme action. We introduce the notion of central exact sequence of tensor categories and use it in order to present an alternative formulation of some known characterizations of equivariantizations for fusion categories, and to extend these characterizations to equivariantizations of finite tensor categories under finite group scheme actions. In particular, we obtain a simple characterization of equivariantizations under actions of finite abelian groups. As an application, we show that if ${\mathcal C}$ is a fusion category and $F: {\mathcal C} \to {\mathcal D}$ is a dominant tensor functor of Frobenius-Perron index $p$, then $F$ is an equivariantization if $p=2$, or if ${\mathcal C}$ is weakly integral and $p$ is the smallest prime factor of FPdim ${\mathcal C}$.

On abelian group actions and Galois quantizations

Quantizations of actions of finite abelian groups G are explicitly described by elements in the tensor square of the group algebra of G. Over algebraically closed fields of characteristic 0 these are in one to one correspondence with the second cohomology group of the dual of G. With certain adjustments this result is applied to group actions over any field of characteristic 0. In particular we consider the quantizations of Galois extensions, which are quantized by “deforming” the multiplication. For the splitting fields of products of quadratic polynomials this produces quantized Galois extensions that all are Clifford type algebras.

Rationality of motivic zeta functions for curves with finite abelian group actions

Let VarkG denote the category of pairs (X,σ), where X is a variety over k and σ is a group action on X. We define the Grothendieck ring for varieties with group actions as the free abelian group of isomorphism classes in the category VarkG modulo a cutting and pasting relation. The multiplication in this ring is defined by the fiber product of varieties. This allows for motivic zeta-functions for varieties with group actions to be defined. This is a formal power series ∑n=0∞[Symn(X,σ)]tn with coefficients in the Grothendieck ring. The main result of this paper asserts that the motivic zeta-function for an algebraic curve with a finite abelian group action is rational. This is a partial generalization of Weil’s First Conjecture.

Codes as Fractals and Noncommutative Spaces

We consider the CSS algorithm relating self-orthogonal classical linear codes to q-ary quantum stabilizer codes and we show that to such a pair of a classical and a quantum code one can associate geometric spaces constructed using methods from noncommutative geometry, arising from rational noncommutative tori and finite abelian group actions on Cuntz algebras and fractals associated to the classical codes.

The tracial Rokhlin property for actions of finite groups on C*-algebras

We define ``tracial'' analogs of the Rokhlin property for actions of finite groups, approximate representability of actions of finite abelian groups, and approximate innerness. We prove the following four analogs of related ``nontracial'' results. $\bullet$ The crossed product of an infinite dimensional simple separable unital C*-algebra with tracial rank zero by an action of a finite group with the tracial Rokhlin property again has tracial rank zero. $\bullet$ An outer action of a finite abelian group on an infinite dimensional simple separable unital C*-algebra has the tracial Rokhlin property if and only if its dual is tracially approximately representable, and is tracially approximately representable if and only if its dual has the tracial Rokhlin property. $\bullet$ If a strongly tracially approximately inner action of a finite cyclic group on an infinite dimensional simple separable unital C*-algebra has the tracial Rokhlin property, then it is tracially approximately representable. $\bullet$ An automorphism of an infinite dimensional simple separable unital C*-algebra $A$ with tracial rank zero is tracially approximately inner if and only if it is the identity on $K_0 (A)$ mod infinitesimals.

A global mirror symmetry framework for the Landau–Ginzburg/Calabi–Yau correspondence

We show how the Landau–Ginzburg/Calabi–Yau correspondence for the quintic three-fold can be cast into a global mirror symmetry framework. Then we draw inspiration from Berglund–Hubsch mirror duality construction to provide an analogue picture featuring all Calabi–Yau hypersurfaces and certain quotients by finite abelian group actions.

Smooth finite abelian uniformizations of projective spaces and Calabi–Yau orbifolds

We give a classification of smooth complex manifolds with a finite abelian group action, such that the quotient is isomorphic to a projective space. The case where the manifold is a Calabi–Yau is studied in detail.

Transitive actions of finite abelian groups of sup-norm isometries

There is a long-standing conjecture of Nussbaum which asserts that every finite set in R n on which a cyclic group of sup-norm isometries acts transitively contains at most 2 n points. The existing evidence supporting Nussbaum’s conjecture only uses abelian properties of the group. It has therefore been suggested that Nussbaum’s conjecture might hold more generally for abelian groups of sup-norm isometries. This paper provides evidence supporting this stronger conjecture. Among other results, we show that if Γ is an abelian group of sup-norm isometries that acts transitively on a finite set X in R n and Γ contains no anticlockwise additive chains, then X has at most 2 n points.

FREE ACTIONS OF FINITE ABELIAN GROUPS ON 3-DIMENSIONAL NILMANIFOLDS

We study free actions of finite abelian groups on 3­dimensional nilmanifolds. By the works of Bieberbach and Waldhausen, this classification problem is reduced to classifying all normal nilpotent subgroups of almost Bieberbach groups of finite index, up to affine conjugacy. All such actions are completely classified.