Irreducible Actions Research Articles

Orbit equivalence rigidity of irreducible actions of right-angled Artin groups

Let $G_\Gamma \curvearrowright X$ and $G_\Lambda \curvearrowright Y$ be two free measure-preserving actions of one-ended right-angled Artin groups with trivial center on standard probability spaces. Assume they are irreducible, i.e. every element from a standard generating set acts ergodically. We prove that if the two actions are stably orbit equivalent (or merely stably $W^*$-equivalent), then they are automatically conjugate through a group isomorphism between $G_\Gamma$ and $G_\Lambda$. Through work of Monod and Shalom, we derive a superrigidity statement: if the action $G_\Gamma \curvearrowright X$ is stably orbit equivalent (or merely stably $W^*$-equivalent) to a free, measure-preserving, mildly mixing action of a countable group, then the two actions are virtually conjugate. We also use the works of Popa and Ioana, Popa and Vaes to establish the $W^*$-superrigidity of Bernoulli actions of all infinite conjugacy classes groups having a finite generating set made of infinite-order elements where two consecutive elements commute, and one has a nonamenable centralizer: these include one-ended nonabelian right-angled Artin groups, but also many other Artin groups and most mapping class groups of finite-type surfaces.

Invariant Jacobians for irreducible restricted action

The problem of invariant Jacobians stems from the study of isolated singularities in complex geometry. In this paper, we deal with analogous problem of invariant Jacobians proposed by Stephen Yau in the case of prime characteristic. We give a complete classification of (infinitesimal version) invariant Jacobians for an irreducible action of the restricted Lie algebra . In particular, unlike the characteristic 0 case, the invariant Jacobians for irreducible actions may not be irreducible. We hope that it will be useful for the study of singularities in positive characteristic.

Rigidity for Von Neumann Algebras Given by Locally Compact Groups and Their Crossed Products

We prove the first rigidity and classification theorems for crossed product von Neumann algebras given by actions of non-discrete, locally compact groups. We prove that for arbitrary free probability measure preserving actions of connected simple Lie groups of real rank one, the crossed product has a unique Cartan subalgebra up to unitary conjugacy. We then deduce a W* strong rigidity theorem for irreducible actions of products of such groups. More generally, our results hold for products of locally compact groups that are nonamenable, weakly amenable and that belong to Ozawa's class S.

Wilson作用 および 1PI 有効作用における波動関数くりこみの関係

Wilson作用 および 1PI 有効作用における波動関数くりこみの関係

RANK 3 BINGO

AbstractWe classify irreducible actions of connected groups of finite Morley rank on abelian groups of Morley rank 3.

Stabilizer rigidity in irreducible group actions

We consider irreducible actions of locally compact product groups, and of higher rank semi-simple Lie groups. Using the intermediate factor theorems of Bader-Shalom and Nevo-Zimmer, we show that the action stabilizers, and all irreducible invariant random subgroups, are co-amenable in their normal closure. As a consequence, we derive rigidity results on irreducible actions that generalize and strengthen the results of Bader-Shalom and Stuck-Zimmer.

On the wave function renormalization for Wilson actions and their one particle irreducible actions

On the wave function renormalization for Wilson actions and their one particle irreducible actions

Small representations of SL2in the finite Morley rank category

AbstractWe study definable irreducible actions of SL2on an abelian group of Morley rank ≤ 3rkand prove they are rational representations of the group.

IRREDUCIBLE ACTIONS AND COMPRESSIBLE MODULES

Any finite set of linear operators on an algebra A yields an operator algebra B and a module structure on A, whose endomorphism ring is isomorphic to a subring AB of certain invariant elements of A. We show that if A is a critically compressible left B-module, then the dimension of its self-injective hull  over the ring of fractions of AB is bounded by the uniform dimension of A and the number of linear operators generating B. This extends a known result on irreducible Hopf actions and applies in particular to weak Hopf action. Furthermore we prove necessary and sufficient conditions for an algebra A to be critically compressible in the case of group actions, group gradings and Lie actions.

Product groups acting on manifolds

We analyse volume-preserving actions of product groups on Riemannian manifolds. To this end, we establish a new superrigidity theorem for ergodic cocycles of product groups ranging in linear groups. There are no a priori assumptions on the acting groups, except a spectral gap assumption on their action. Our main application to manifolds concerns irreducible actions of Kazhdan product groups. We prove the following dichotomy: Either the action is infinitesimally linear, which means that the derivative cocycle arises from unbounded linear representations of all factors. Otherwise, the action is measurably isometric, in which case there are at most two factors in the product group. As a first application, this provides lower bounds on the dimension of the manifold in terms of the number of factors in the acting group. Another application is a strong restriction for actions of non-linear groups.

On maximal actions and w-maximal actions of finite hypergroups

Sunder and Wildberger (J. Algebr. Comb. 18, 135–151, 2003) introduced the notion of actions of finite hypergroups, and studied maximal irreducible actions and *-actions. One of the main results of Sunder and Wildberger states that if a finite hypergroup K admits an irreducible action which is both a maximal action and a *-action, then K arises from an association scheme. In this paper we will first show that an irreducible maximal action must be a *-action, and hence improve Sunder and Wildberger’s result (Theorem 2.9). Another important type of actions is the so-called w-maximal actions. For a w-maximal action π:K→Aff (X), we will prove that π is faithful and |X|≥|K|, and |K| is the best possible lower bound of |X|. We will also discuss the strong connectivity of the digraphs induced by a w-maximal action.

Irreducible and end type group actions on Λ‐trees

Synopsis. Let Λ be an ordered abelian group. It is shown that groups in a certain class can have no non-trivial action of end type on a Λ-tree. A similar result is obtained for irreducible actions.

A Classification of Multiplicity Free Actions

The action of a complex algebraic groupGon an affine varietyVis said to be multiplicity free if each irreducible representation ofGoccurs at most once in the ring C[V] of regular functions onV. This paper concerns the classical setting whereVis a vector space andGacts linearly onV. The irreducible linear multiplicity free actions have been classified by V. Kac. The focus here is on nonirreducible linear multiplicity free actions. We exhibit many such actions which do not decompose as direct products of irreducible actions. Our main results provide a classification for all linear multiplicity free actions on a vector space.

Properly discontinuous actions on Λ-trees

The main result is a theorem giving several possibilities for the action of a 2-generator group acting on a Λ-tree, generalising the result that, if the action is free then the group is either free or free abelian. This involves investigation of several cases in which the action is shown to be properly discontinuous. This leads to a generalisation of results of Culler and Morgan, characterising abelian, dihedral and irreducible actions on ℝ-trees, to arbitrary Λ-trees.

Variational principle for effective action and prospects of nonlinear optimization

Building upon an analysis of effective versus one-particle irreducible actions by Dannenberg, we show that the Feynman-Jensen variational principle produces a manifestly Lorentz-covariant lower bound on the generating functional of 1PI Green's functions. The well-known gaussian effective action is reproduced with a gaussian trial action and linear transformations of the path-integral variables. The existence of exactly computable nongaussian trial actions, based on nonlinear shear-typ transformations, is pointed out. Application to the φ 2 4 effective potential shows a second-order phase transition at λ = 1.55.

Irreducible actions and faithful actions of hopf algebras

LetH be a Hopf algebra acting on an algebraA. We will examine the relationship betweenA, the ring of invariantsA H, and the smash productA # H. We begin by studying the situation whereA is an irreducibleA # H module and, as an application of our first main theorem, show that ifD is a division ring then [D : D H]≦dimH. We next show that prime rings with central rings of invariants satisfy a polynomial identity under the action of certain Hopf algebras. Finally, we show that the primeness ofA # H is strongly related to the faithfulness of the left and right actions ofA # H onA.

The Gromov topology on [formula omitted]-trees

We are interested in isometric actions of a fixed finitely generated group on R -trees. Using metric methods inspired by Gromov's work, we define a more geometric topology on sets of such objects. We prove it to be the same as the Morgan-Shalen topology, defined by the translation lengths of the group elements, in the case of minimal irreducible actions.