Trivial Action Research Articles

Permanent Weak Module Amenability of Semigroup Algebras

Abstract We employ the fact that L1(G) is n-weakly amenable for each n ≥ 1 to show that for an inverse semigroup S with the set of idempotents E, ℓ1(S) is n- weakly module amenable as an ℓ1(E)-module with trivial left action. We study module amenability and weak module amenability of the module projective tensor products of Banach algebras.

REGULAR INJECTIVITY AND EXPONENTIABILITY IN THE SLICE CATEGORIES OF ACTIONS OF POMONOIDS ON POSETS

For a pomonoid S, let us denote Pos-S the category of S-posets and S-poset maps. In this paper, we consider the slice category Pos-S/B for an S-poset B, and study some categorical ingredients. We first show that there is no non-trivial injective object in Pos-S/B. Then we investigate injective objects with respect to the class of regular monomorphisms in this category and show that Pos-S/B has enough regular injective objects. We also prove that regular injective objects are retracts of exponentiable objects in this category. One of the main aims of the paper is to draw attention to characterizing injectivity in the category Pos-S/B under a particular case where B has trivial action. Among other things, we also prove that the necessary condition for a map (an object) here to be regular injective is being convex and present an example to show that the converse is not true, in general.

Givental action and trivialisation of circle action

In this paper, we show that the Givental group action on genus zero cohomological field theories, also known as formal Frobenius manifolds or hypercommutative algebras, naturally arises in the deformation theory of Batalin–Vilkovisky algebras. We prove that the Givental action is equal to an action of the trivialisations of the trivial circle action. This result relies on the equality of two Lie algebra actions coming from two apparently remote domains: geometry and homotopical algebra.

On Torsion in the Cohomology of Locally Symmetric Varieties

The main result of this paper is the existence of Galois representations associated with the mod p (or mod pm) cohomology of the locally symmetric spaces for GLn over a totally real or CM field, proving conjectures of Ash and others. Following an old suggestion of Clozel, recently realized by Harris-Lan-Taylor-Thorne for characteristic 0 cohomology classes, one realizes the cohomology of the locally symmetric spaces for GLn as a boundary contribution of the cohomology of symplectic or unitary Shimura varieties, so that the key problem is to understand torsion in the cohomology of Shimura varieties. Thus, we prove new results on the p-adic geometry of Shimura varieties (of Hodge type). Namely, the Shimura varieties become perfectoid when passing to the inverse limit over all levels at p, and a new period map towards the flag variety exists on them, called the Hodge-Tate period map. It is roughly analogous to the embedding of the hermitian symmetric domain (which is roughly the inverse limit over all levels of the complex points of the Shimura variety) into its compact dual. The Hodge-Tate period map has several favorable properties, the most important being that it commutes with the Hecke operators away from p (for the trivial action of these Hecke operators on the flag variety), and that automorphic vector bundles come via pullback from the flag variety.

Erratum to: Inverse-Closed Algebras of Integral Operators on Locally Compact Groups

It was pointed out by Detlev Poguntke that the values of the map θ in the proof of Proposition 2.7 of original article may not be equivalence classes of Bochner integrable functions. Consequently the statements of Proposition 2.7, Corollary 2.8, Theorem 3.10, Theorem 5.2, and Corollary 5.3 of original article are not correct, and an extra hypothesis is needed in these statements in order to obtain the corresponding conclusions. Details in this connection can be found below. One of the situations when the proof of Proposition 2.7 of original article applies is that of discrete groups, so that hypothesis will be added in the statement as follows. Proposition 0.1. Let G be a discrete group and (A, G, α) be a C∗-dynamical system. Then there exists a C∗-dynamical system (Ā, G, ᾱ) with trivial action ᾱ and an isometric ∗-homomorphism θ : 1(G,A;α) → (G, Ā; ᾱ) = (G) ⊗Ā.

Lorentzian compact manifolds: Isometries and geodesics

In this work we investigate families of compact Lorentzian manifolds in dimension four. We show that every lightlike geodesic on such spaces is periodic, while there are closed and non-closed spacelike and timelike geodesics. Also their isometry groups are computed. We also show that there is a non trivial action by isometries of H3(R) on the nilmanifold S1×(Γk∖H3(R)) for Γk, a lattice of H3(R).

Equivariant algebraic kk-theory and adjointness theorems

We introduce an equivariant algebraic kk-theory for G-algebras and G-graded algebras. We study some adjointness theorems related with crossed product, trivial action, induction and restriction. In particular we obtain an algebraic version of the Green–Julg Theorem which gives us a computational tool.

Fusion systems and group actions with abelian isotropy subgroups

AbstractWe prove that if a finite group G acts smoothly on a manifold M such that all the isotropy subgroups are abelian groups with rank ≤ k, then G acts freely and smoothly on M × $\mathbb{S}^{n_1}\$ × … × $\mathbb{S}^{n_k}$ for some positive integers n1, …, nk. We construct these actions using a recursive method, introduced in an earlier paper, that involves abstract fusion systems on finite groups. As another application of this method, we prove that every finite solvable group acts freely and smoothly on some product of spheres, with trivial action on homology.

Depth One Extensions of Semisimple Algebras and Hopf Subalgebras

Depth one extensions of finite dimensional semisimple algebras are completely characterized in terms of their algebra centers. For extensions of semisimple Hopf algebras this characterization translates into a trivial monoidal action of the dual fusion category Rep(A*) on Rep(B).

Deformation of extremal metrics, complex manifolds and the relative Futaki invariant

Let (X,\Omega) be a closed polarized complex manifold, g be an extremal metric on X that represents the K\"ahler class \Omega, and G be a compact connected subgroup of the isometry group Isom(X,g). Assume that the Futaki invariant relative to G is nondegenerate at g. Consider a smooth family $(M \to B)$ of polarized complex deformations of (X,\Omega)\simeq (M_0,\Theta_0) provided with a holomorphic action of G with trivial action on B. Then for every t\in B sufficiently small, there exists an h^{1,1}(X)-dimensional family of extremal Kaehler metrics on M_t whose K\"ahler classes are arbitrarily close to \Theta_t. We apply this deformation theory to show that certain complex deformations of the Mukai-Umemura 3-fold admit Kaehler-Einstein metrics.

Bernoulli actions are weakly contained in any free action

AbstractLet Γ be a countable group and let f be a free probability measure-preserving action of Γ. We show that all Bernoulli actions of Γ are weakly contained in f. It follows that for a finitely generated group Γ, the cost is maximal on Bernoulli actions for Γ and that all free factors of i.i.d. (independent and identically distributed) actions of Γ have the same cost. We also show that if f is ergodic, but not strongly ergodic, then f is weakly equivalent to f×I, where Idenotes the trivial action of Γ on the unit interval. This leads to a relative version of the Glasner–Weiss dichotomy.

Quasilinear actions on products of spheres

For some small values of f, we prove that if G is a group having a complex (real) representation with fixity f, then it acts freely and smoothly on a product of f+1 spheres with trivial action on homology.

Orientable homotopy modules

We prove a conjecture of Morel identifying Voevodsky’s homotopy invariant sheaves with transfers with spectra in the stable homotopy category which are concentrated in degree zero for the homotopy $t$-structure and have a trivial action of the Hopf map. This is done by relating these two kind of objects to Rost’s cycle modules. Applications to algebraic cobordism and construction of cycle classes are given.

Poverty of Cause in Mythological Narrative

Many myths recount how a single trivial action (for example, a simple misunderstanding) had cosmic consequences for the human race (for example, the entry of death into the world). Such “poverty-of-cause” narratives are reflective neither of a supposed primitive mentality, as earlier theorists suggested, nor of a widespread folk-idea of the so-called Butterfly Effect. Rather, the disproportion arises from the confluence of two general characteristics of traditional aetiological narrative, namely, the treatment of causation as local and immediate, and the portrayal of protagonists as representative of the species.

On finite simple and nonsolvable groups acting on closed 4–manifolds

We show that the only finite nonabelian simple groups to admit a locally linear, homologically trivial action on a closed simply connected 4-manifold M (or on a 4-manifold with trivial first homology) are the alternating groups A 5 , A 6 and the linear fractional group PSL(2, 7). (We note that for homologically nontrivial actions all finite groups occur.) The situation depends strongly on the second Betti number b 2 (M) of M and was known before if b 2 (M) is different from two, so the main new result concerns the case b 2 (M) = 2. We prove that the only simple group that occurs in this case is A 5 , and then deduce a short list of finite nonsolvable groups which contains all candidates for actions of such groups.

The automorphism group of certain factorial threefolds and a cancellation problem

A new class of counterexamples to a generalized cancellation problem for affine varieties is presented. Each member of the class is an affine factorial complex threefold admitting a locally trivial action of the additive group, hence the total space for a principal G a bundle over a quasiaffine base. The automorphism groups for these varieties are also determined.

Automorphisms of groups

The main purpose of the present paper is to give an explicit description of the Wells map of a given group extension to the case of automorphisms acting trivially on the quotient group. From this we obtain some of new necessary and sufficient conditions for an automorphism of a normal subgroup of a group to extend to the group itself, with trivial action on the quotient group.

Additive group actions with finitely generated invariants

Every locally trivial action of the additive group of complex numbers on a factorial affine variety has finitely generated ring of invariants. A criterion is given for such an action on complex four space to be conjugate to a translation. Restrictions on the nature of the singularities of the variety defined by the ring of invariants of triangular actions are noted.

Generalized Burnside rings and group cohomology

We define the cohomological Burnside ring B n ( G , M ) of a finite group G with coefficients in a Z G -module M as the Grothendieck ring of the isomorphism classes of pairs [ X , u ] where X is a G-set and u is a cohomology class in a cohomology group H X n ( G , M ) . The cohomology groups H X ∗ ( G , M ) are defined in such a way that H X ∗ ( G , M ) ≅ ⊕ i H ∗ ( H i , M ) when X is the disjoint union of transitive G-sets G / H i . If A is an abelian group with trivial action, then B 1 ( G , A ) is the same as the monomial Burnside ring over A, and when M is taken as a G-monoid, then B 0 ( G , M ) is equal to the crossed Burnside ring B c ( G , M ) . We discuss the generalizations of the ghost ring and the mark homomorphism and prove the fundamental theorem for cohomological Burnside rings. We also give an interpretation of B 2 ( G , M ) in terms of twisted group rings when M = k × is the unit group of a commutative ring.

The Drinfel’d Double for Group-cograded Multiplier Hopf Algebras

Let G be any group and let K(G) denote the multiplier Hopf algebra of complex functions with finite support in G. The product in K(G) is pointwise. The comultiplication on K(G) is defined with values in the multiplier algebra M(K(G) ⊗K(G )) by the formula \((\Delta(f\,)) (p,q)\!\! =\!\! f(pq)\) for all \(f\! \in\! K(G)\) and \(p, q \!\in\! G\). In this paper we consider multiplier Hopf algebras B (over \(\Bbb C\)) such that there is an embedding I: K(G) →M(B). This embedding is a non-degenerate algebra homomorphism which respects the comultiplication and maps K(G) into the center of M(B). These multiplier Hopf algebras are called G-cograded multiplier Hopf algebras. They are a generalization of the Hopf group-coalgebras as studied by Turaev and Virelizier. In this paper, we also consider an admissible action π of the group G on a G-cograded multiplier Hopf algebra B. When B is paired with a multiplier Hopf algebra A, we construct the Drinfel’d double Dπ where the coproduct and the product depend on the action π. We also treat the *-algebra case. If π is the trivial action, we recover the usual Drinfel’d double associated with the pair \(\langle A, B \rangle\). On the other hand, also the Drinfel’d double, as constructed by Zunino for a finite-type Hopf group-coalgebra, is an example of the construction above. In this case, the action is non-trivial but related with the adjoint action of the group on itself. Now, the double is again a G-cograded multiplier Hopf algebra.