Nilpotent Actions Research Articles

Orthonormal bases arising from nilpotent actions

In this paper, we prove the existence of an orthonormal basis in at least one orbit of every generic irreducible representation of a simply connected and connected nilpotent Lie group. Our result has a wide-ranging impact, encompassing all irreducible representations of a nilpotent Lie group that are square-integrable modulo its center. This resolves a fundamental open problem in time-frequency analysis and frame theory, originally posed by Karlheinz Gröchenig. The implications of our findings are significant and far-reaching.

Fixed points of nilpotent actions on R²

Fixed points of nilpotent actions on R²

Fixed points of nilpotent actions on R²

Fixed points of nilpotent actions on R²

Local rigidity of higher rank non-abelian action on torus

In this paper, we show local smooth rigidity for higher rank ergodic nilpotent action by toral automorphisms. In former papers all examples for actions enjoying the local smooth rigidity phenomenon are higher rank and have no rank-one factors. In this paper we give examples of smooth rigidity of actions having rank-one factors. The method is a generalization of the KAM (Kolmogorov–Arnold–Moser) iterative scheme.

Finite orbits for nilpotent actions on the torus

A homeomorphism of the $2$-torus with Lefschetz number different from zero has a fixed point. We give a version of this result for nilpotent groups of diffeomorphisms. We prove that a nilpotent group of $2$-torus diffeomorphims has finite orbits when the group has some element with Lefschetz number different from zero.

Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras IV

Let $U'_q(\mathfrak{g})$ be a twisted affine quantum group of type $A_{N}^{(2)}$ or $D_{N}^{(2)}$ and let $\mathfrak{g}_{0}$ be the finite-dimensional simple Lie algebra of type $A_{N}$ or $D_{N}$. For a Dynkin quiver of type $\mathfrak{g}_{0}$, we define a full subcategory ${\mathcal C}_{Q}^{(2)}$ of the category of finite-dimensional integrable $U'_q(\mathfrak{g})$-modules, a twisted version of the category ${\mathcal C}_{Q}$ introduced by Hernandez and Leclerc. Applying the general scheme of affine Schur-Weyl duality, we construct an exact faithful KLR-type duality functor ${\mathcal F}_{Q}^{(2)}: Rep(R) \rightarrow {\mathcal C}_{Q}^{(2)}$, where $Rep(R)$ is the category of finite-dimensional modules over the quiver Hecke algebra $R$ of type $\mathfrak{g}_{0}$ with nilpotent actions of the generators $x_k$. We show that ${\mathcal F}_{Q}^{(2)}$ sends any simple object to a simple object and induces a ring isomorphism $K(Rep(R)) \simeq K({\mathcal C}_{Q}^{(2)})$.

Fixed points of nilpotent actions on

We prove that a nilpotent subgroup of orientation-preserving$C^{1}$diffeomorphisms of$\mathbb{S}^{2}$has a finite orbit of cardinality at most two. We also prove that a finitely generated nilpotent subgroup of orientation-preserving$C^{1}$diffeomorphisms of$\mathbb{R}^{2}$preserving a compact set has a global fixed point. These results generalize theorems of Frankset al for the abelian case. We show that a nilpotent subgroup of orientation-preserving$C^{1}$diffeomorphisms of$\mathbb{S}^{2}$that has a finite orbit of odd cardinality also has a global fixed point. Moreover, we study the properties of the 2-points orbits of nilpotent fixed-point-free subgroups of orientation-preserving$C^{1}$diffeomorphisms of$\mathbb{S}^{2}$.

P-Localizing Group Extensions with a Nilpotent Action on the Kernel

Assume P is a family of primes, and let () P represent the P-localization functor. If is a group extension giving rise to a nilpotent action of G on N, we prove that the sequence is exact. Moreover, in the case where Q satisfies a certain pair of homological conditions, we show that the map ι P is an injection. This generalizes the well-known result that () P is exact in the category of nilpotent groups. Applications are given to calculating P-localizations of virtually nilpotent groups.

Nilpotent actions on non-abelian tensor products of groups

Nilpotent actions on non-abelian tensor products of groups

Nilpotent action by an amenable group and Euler characteristic

We prove two types of vanishing results for the Euler characteristic.

Nilpotent action on the KDV variables and 2-dimensional Drinfeld-Sokolov reduction

We note that a version “with spectral parameter” of the Drinfeld-Sokolov reduction gives a natural mapping from the KdV phase space to the group of loops with values in $$\widehat{N}_ (+) /A, \widehat{N}_ (+) $$ : affine nilpotent andA principal commutative (or anisotropic Cartan) subgroup; this mapping is connected to the conserved densities of the hierarchy. We compute the Feigin-Frenkel action of $$\widehat{n}_ (+) $$ (defined in terms of screening operators) on the conserved densities in thesl 2 case.

Hadamard Matrices and Their Designs: A Coding-Theoretic Approach

To every finite dimensional algebraic coefficient system (defined below) $(\Theta ,V)$ over the De Rham algebra $\Omega (M)$ of a manifold $M$, Sullivan builds a local system ${\rho _\Theta }:{\pi _1}(M) \to V$, in the topological sense, such that the two cohomologies $H_{{\rho _\Theta }}^{\ast }(M;V)$ and $H_\Theta ^{\ast }(\Omega (M);V)$ are isomorphic. In this paper, if ${\mathbf {K}}$ is a simplicial set and $(\Theta ,V)$ an algebraic system over the ${C^\infty }$ forms ${A_\infty }({\mathbf {K}})$, we prove a similar result. We use it to extend the Hirsch lemma to the case of fibration whose fiber is an Eilenberg-Mac Lane space with certain non nilpotent action of the fundamental group of the basis. We apply this to a model of the hyperbolic torus; different from the nilpotent one, this new model is a better mirror of the topology.

Theorie De Sullivan Pour La Cohomologie a Coefficients Locaux

To every finite dimensional algebraic coefficient system (defined below) (θ, V) over the De Rham algebra Ω(M) of a manifold M, Sullivan builds a local system ρθ : π 1 (M→V, in the topological sense, such that the two cohomologies H* ρθ (M; V) and H* θ (Ω(M); V) are isomorphic. In this paper, if K is a simplical set and (θ, V) an algebraic system over the C ∞ forms A ∞(K), we prove a similar result. We use it to extend the Hirsch lemma to the case of fibration whose fiber is an Eilenberg-Mac Lane spece with certain non nilpotent action of the fundamental group of the basis. We apply this to a model of the hyperbolic torus; different from the nilpotent one, this new model is a better mirror of the topology

Hadamard matrices and their designs: a coding-theoretic approach

To every finite dimensional algebraic coefficient system (defined below) ( Θ , V ) (\Theta ,V) over the De Rham algebra Ω ( M ) \Omega (M) of a manifold M M , Sullivan builds a local system ρ Θ : π 1 ( M ) → V {\rho _\Theta }:{\pi _1}(M) \to V , in the topological sense, such that the two cohomologies H ρ Θ ∗ ( M ; V ) H_{{\rho _\Theta }}^{\ast }(M;V) and H Θ ∗ ( Ω ( M ) ; V ) H_\Theta ^{\ast }(\Omega (M);V) are isomorphic. In this paper, if K {\mathbf {K}} is a simplicial set and ( Θ , V ) (\Theta ,V) an algebraic system over the C ∞ {C^\infty } forms A ∞ ( K ) {A_\infty }({\mathbf {K}}) , we prove a similar result. We use it to extend the Hirsch lemma to the case of fibration whose fiber is an Eilenberg-Mac Lane space with certain non nilpotent action of the fundamental group of the basis. We apply this to a model of the hyperbolic torus; different from the nilpotent one, this new model is a better mirror of the topology.

Combined symmetries in curved space-times

The exact combination of internal and of geometrical symmetries in curved space-times is discussed. It is seen that it is possible to define locally a Lie group S containing two noninvariant subgroups E and G, such that in the flat limit of the space-time, S decomposes in the product of the Poincaré group resulting from a contraction of E and an internal group with point dependent parameters resulting fom G. Furthermore, E does not have necessarily a nilpotent action on S except in the flat limit. An explicit squared mass-splitting expression for S is derived and its behavior in a local gravitational field and in the flat limit of the space-time is analyzed.

On G-spaces, Serre classes and G-nilpotency

0. Introduction. The concept of a nilpotent action on a nilpotent group was studied in (2), and was crucial to the definition of a nilpotent map (or nilpotent fibration) in (4). These ideas had, of course, already been treated systematically in (1)†. The concept found a new, and related, application to homotopy theory in (3), where the following situation was studied. Let X be a nilpotent space and let G be a group which ‘acts’on X. Here it is not strictly speaking necessary to specify the nature of the action, provided that there is an induced action of G on the homotopy and homology groups of X. Thus G might be a group of base-point preserving homeomorphisms of X, or a group of (based) homotopy classes of self-homotopy-equivalences of X. We then discussed in (3) questions related to the nilpotency of the induced actions of G on the homotopy and homology groups of X and also relativized the theory to a discussion of fibrations on which the group G acts; to obtain interesting results related to nilpotency, it turned out to be at least necessary to suppose the fibration F → E → B quasinilpotent (7), meaning that all spaces are connected and π1B acts nilpotently on the homology groups of F.

A functorial version of a construction of Hochschild and Mostow for representations of Lie algebras

Let be a Lie algebra, a complemented ideal of , and W an -module. Hochschild and Mostow have described the construction of a -module “induced” from W, which is finite-dimensional provided W is finite-dimensional and satisfies a nilpotent action condition. This note describes a modification of their construction which is functorial and a weak adjoint to the restriction functor from –modules to -modules.