Stable Equivalence Research Articles

Relatively stable equivalences of Morita type for blocks

Based on the fact that the relatively stable category of a p-block B is equivalent to the relatively stable category of its Brauer correspondent b as triangulated category, we introduce the notion of relatively stable equivalence of Morita type and show that there is a relatively stable equivalence of Morita type between B and b. Some invariants under stable equivalence of Morita type can be generalized to this relative case. In particular, we put forward the generalized Alperin–Auslander conjecture and prove it in special cases.

On the classification problem of free ergodic actions of nonamenable groups

We show that, for any countable discrete nonamenable group Γ, the relations of conjugacy, orbit equivalence, stable orbit equivalence, von Neumann equivalence, and stable von Neumann equivalence of free ergodic pmp actions of Γ on the standard atomless probability space are not Borel. This answers a question of Kechris.

The Grothendieck Groups and Stable Equivalences of Mesh Algebras

We deal with the finite-dimensional mesh algebras given by stable translation quivers. These algebras are self-injective, and thus the stable categories have a structure of triangulated categories. Our main result determines the Grothendieck groups of these stable categories. As an application, we give an complete classification of the mesh algebras up to stable equivalences.

Deformations of complexes for finite dimensional algebras

Let k be a field and let Λ be a finite dimensional k-algebra. We prove that every bounded complex V• of finitely generated Λ-modules has a well-defined versal deformation ring R(Λ,V•) which is a complete local commutative Noetherian k-algebra with residue field k. We also prove that nice two-sided tilting complexes between Λ and another finite dimensional k-algebra Γ preserve these versal deformation rings. Additionally, we investigate stable equivalences of Morita type between self-injective algebras in this context. We apply these results to the derived equivalence classes of the members of a particular family of algebras of dihedral type that were introduced by Erdmann and shown by Holm to be not derived equivalent to any block of a group algebra.

The Auslander–Reiten Components Seen as Quasi-Hereditary Categories

Quasi-hereditary algebras were introduced by E. Cline, B. Parshall and L. Scott in order to deal with highest weight categories as they arise in the representation theory of semi-simple complex Lie algebras and algebraic groups. These categories have been a very important tool in the study of finite-dimensional algebras. On the other hand, functor categories were introduced in representation theory by M. Auslander, and used in his proof of the first Brauer–Thrall conjecture and later used systematically in his joint work with I. Reiten on stable equivalence, as well as many other applications. Recently, functor categories were used by Martinez-Villa and Solberg to study the Auslander–Reiten components of finite-dimensional algebras. The aim of the paper is to introduce the concept of quasi-hereditary category. We can think of the Auslander–Reiten components as quasi-hereditary categories. In this way, we have applications to the functor category $$\mathrm {Mod}(\mathcal {C} )$$ , with $$\mathcal C$$ a component of the Auslander–Reiten quiver.

Topology and convexity in the space of actions modulo weak equivalence

We analyze the structure of the quotient $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ of the space of measure-preserving actions of a countable discrete group by the relation of weak equivalence. This space carries a natural operation of convex combination. We introduce a variant of an abstract construction of Fritz which encapsulates the convex combination operation on $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$. This formalism allows us to define the geometric notion of an extreme point. We also discuss a topology on $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ due to Abért and Elek in which it is Polish and compact, and show that this topology is equivalent to others defined in the literature. We show that the convex structure of $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ is compatible with the topology, and as a consequence deduce that $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ is path connected. Using ideas of Tucker-Drob, we are able to give a complete description of the topological and convex structure of $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ for amenable $\unicode[STIX]{x1D6E4}$ by identifying it with the simplex of invariant random subgroups. In particular, we conclude that $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ can be represented as a compact convex subset of a Banach space if and only if $\unicode[STIX]{x1D6E4}$ is amenable. In the case of general $\unicode[STIX]{x1D6E4}$ we prove a Krein–Milman-type theorem asserting that finite convex combinations of the extreme points of $\text{A}_{{\sim}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ are dense in this space. We also consider the space $\text{A}_{{\sim}_{s}}(\unicode[STIX]{x1D6E4},X,\unicode[STIX]{x1D707})$ of stable weak equivalence classes and show that it can always be represented as a compact convex subset of a Banach space. In the case of a free group $\mathbb{F}_{N}$, we show that if one restricts to the compact convex set $\text{FR}_{{\sim}_{s}}(\mathbb{F}_{N},X,\unicode[STIX]{x1D707})\subseteq \text{A}_{{\sim}_{s}}(\mathbb{F}_{N},X,\unicode[STIX]{x1D707})$ consisting of the stable weak equivalence classes of free actions, then the extreme points are dense in $\text{FR}_{{\sim}_{s}}(\mathbb{F}_{N},X,\unicode[STIX]{x1D707})$.

Pro-Species of Algebras I: Basic Properties

In this paper, we generalise part of the theory of hereditary algebras to the context of pro-species of algebras. Here, a pro-species is a generalisation of Gabriel’s concept of species gluing algebras via projective bimodules along a quiver to obtain a new algebra. This provides a categorical perspective on a recent paper by Geis et al. (2016). In particular, we construct a corresponding preprojective algebra, and establish a theory of a separated pro-species yielding a stable equivalence between certain functorially finite subcategories.

Inner amenability, property Gamma, McDuff factors and stable equivalence relations

We say that a countable group $G$ is McDuff if it admits a free ergodic probability measure preserving action such that the crossed product is a McDuff $\text{II}_{1}$ factor. Similarly, $G$ is said to be stable if it admits such an action with the orbit equivalence relation being stable. The McDuff property, stability, inner amenability and property Gamma are subtly related and several implications and non-implications were obtained in Effros [Property $\unicode[STIX]{x1D6E4}$ and inner amenability. Proc. Amer. Math. Soc.47 (1975), 483–486], Jones and Schmidt [Asymptotically invariant sequences and approximate finiteness. Amer. J. Math.109 (1987), 91–114], Vaes [An inner amenable group whose von Neumann algebra does not have property Gamma. Acta Math.208 (2012), 389–394], Kida [Inner amenable groups having no stable action. Geom. Dedicata173 (2014), 185–192] and Kida [Stability in orbit equivalence for Baumslag–Solitar groups and Vaes groups. Groups Geom. Dyn.9 (2015), 203–235]. We complete the picture with the remaining implications and counterexamples.

A survey on Heegaard Floer homology and concordance

In this survey paper, we discuss several different knot concordance invariants coming from the Heegaard Floer homology package of Ozsváth and Szabó. Along the way, we prove that if two knots are concordant, then their knot Floer complexes satisfy a certain type of stable equivalence.

Stable equivalences of Morita type do not preserve tensor products and trivial extensions of algebras

Stable equivalences of Morita type do not preserve tensor products and trivial extensions of algebras

Behaviour of Entropy Under Bounded and Integrable Orbit Equivalence

Let G and H be infinite finitely generated amenable groups. This paper studies two notions of equivalence between actions of such groups on standard Borel probability spaces. They are defined as stable orbit equivalences in which the associated cocycles satisfy certain tail bounds. In ‘integrable stable orbit equivalence’, the length in H of the cocycle-image of an element of G must have finite integral over its domain (a subset of the G-system), and similarly for the reverse cocycle. In ‘bounded stable orbit equivalence’, these functions must be essentially bounded in terms of the length in G. ‘Integrable’ stable orbit equivalence arises naturally in the study of integrable measure equivalence of groups themselves, as introduced recently by Bader, Furman and Sauer. The main result is a formula relating the Kolmogorov–Sinai entropies of two actions which are equivalent in one of these ways. Under either of these tail assumptions, the entropies stand in a proportion given by the compression constant of the stable orbit equivalence. In particular, in the case of full orbit equivalence subject to such a tail bound, entropy is an invariant. This contrasts with the case of unrestricted orbit equivalence, under which all free ergodic actions of countable amenable groups are equivalent. The proof uses an entropy-bound based on graphings for orbit equivalence relations, and in particular on a new notion of cost which is weighted by the word lengths of group elements.

The BRST complex of homological Poisson reduction

BRST complexes are differential graded Poisson algebras. They are associated to a coisotropic ideal $J$ of a Poisson algebra $P$ and provide a description of the Poisson algebra $(P/J)^J$ as their cohomology in degree zero. Using the notion of stable equivalence introduced by Felder and Kazhdan, we prove that any two BRST complexes associated to the same coisotropic ideal are quasi-isomorphic in the case $P = \mathbb{R}[V]$ where $V$ is a finite-dimensional symplectic vector space and the bracket on $P$ is induced by the symplectic structure on $V$. As a corollary, the cohomology of the BRST complexes is canonically associated to the coisotropic ideal $J$ in the symplectic case. We do not require any regularity assumptions on the constraints generating the ideal $J$. We finally quantize the BRST complex rigorously in the presence of infinitely many ghost variables and discuss uniqueness of the quantization procedure.

On the equivalent classification of three-dimensional competitive Leslie/Gower models via the boundary dynamics on the carrying simplex.

We study the asymptotic behavior of the competitive Leslie/Gower model (map) [Formula: see text]It is shown that T unconditionally admits a globally attracting 1-codimensional invariant hypersurface [Formula: see text], called carrying simplex, such that every nontrivial orbit is asymptotic to one in [Formula: see text]. More general and easily checked conditions to guarantee the existence of carrying simplex for competitive maps are provided. An equivalence relation is defined relative to local stability of fixed points on [Formula: see text] (the boundary of [Formula: see text]) on the space of all three-dimensional Leslie/Gower models. Using a formula on the sum of the indices of all fixed points on the carrying simplex for three-dimensional maps, we list the 33 stable equivalence classes in terms of simple inequalities on the parameters [Formula: see text] and [Formula: see text] and draw their orbits on [Formula: see text]. In classes 1-18, every nontrivial orbit tends to a fixed point on [Formula: see text]. In classes 19-25, each map possesses a unique positive fixed point which is a saddle on [Formula: see text], and hence Neimark-Sacker bifurcations do not occur. Neimark-Sacker bifurcation does occur within each of classes 26-31, while it does not occur in class 32. Each map from class 27 admits a heteroclinic cycle, which forms the boundary of [Formula: see text]. The criteria on the stability of heteroclinic cycles are also given. This classification makes it possible to further investigate various dynamical properties in respective class.

Invariance of the restricted p-power map on integrable derivations under stable equivalences

We show that the p-power maps in the first Hochschild cohomology space of finite-dimensional selfinjective algebras over a field of prime characteristic p commute with stable equivalences of Morita type on the subgroup of classes represented by integrable derivations. We show, by giving an example, that the p-power maps do not necessarily commute with arbitrary transfer maps in the Hochschild cohomology of symmetric algebras.

Double coverings of twisted links

Twisted links are a generalization of virtual links. As virtual links correspond to abstract links on orientable surfaces, twisted links correspond to abstract links on (possibly non-orientable) surfaces. In this paper, we introduce the notion of the double covering of a twisted link. It is defined by considering the orientation double covering of an abstract link or alternatively by constructing a diagram called a double covering diagram. We also discuss links in thickened surfaces, their diagrams and their stable equivalence classes. Bourgoin’s twisted knot group is understood as the virtual knot group of the double covering.

On the Adams isomorphism for equivariant orthogonal spectra

We give a natural construction and a direct proof of the Adams isomorphism for equivariant orthogonal spectra. More precisely, for any finite group G, any normal subgroup N of G, and any orthogonal G-spectrum X, we construct a natural map A of orthogonal G/N-spectra from the homotopy N-orbits of X to the derived N-fixed points of X, and we show that A is a stable weak equivalence if X is cofibrant and N-free. This recovers a theorem of Lewis, May, and Steinberger in the equivariant stable homotopy category, which in the case of suspension spectra was originally proved by Adams. We emphasize that our Adams map A is natural even before passing to the homotopy category. One of the tools we develop is a replacement-by-Omega-spectra construction with good functorial properties, which we believe is of independent interest.

On equivalences for cohomological Mackey functors

By results of Rognerud, a source algebra equivalence between two p p -blocks of finite groups induces an equivalence between the categories of cohomological Mackey functors associated with these blocks, and a splendid derived equivalence between two blocks induces a derived equivalence between the corresponding categories of cohomological Mackey functors. We prove this by giving an intrinsic description of cohomological Mackey functors of a block in terms of the source algebras of the block, and then using this description to construct explicit two-sided tilting complexes realising the above mentioned derived equivalence. We show further that a splendid stable equivalence of Morita type between two blocks induces an equivalence between the categories of cohomological Mackey functors which vanish at the trivial group. We observe that the module categories of a block, the category of cohomological Mackey functors, and the category of cohomological Mackey functors which vanish at the trivial group arise in an idempotent recollement. Finally, we extend a result of Tambara on the finitistic dimension of cohomological Mackey functors to blocks.

Stable Green equivalences in bounded derived categories

In this paper, we will establish a Green correspondence theorem for bounded derived categories of group algebras. This theorem is a generalization of the Green correspondence theorem in the case of module categories.

Bar-Natan’s geometric complex and a categorification of the Dye–Kauffman–Miyazawa polynomial

In this paper, we construct a categorification of the two-variable Dye–Kauffman–Miyazawa polynomial by utilizing Bar-Natan’s construction of the Khovanov homology and homotopy quantum field theories (HQFTs) given by Turaev. In particular, for any stable equivalence class, we construct a [Formula: see text]-graded link homology over [Formula: see text] whose graded Euler characteristic is the two-variable Dye–Kauffman–Miyazawa polynomial. Moreover, we show that it is isomorphic to a special case of Dye–Kauffman–Manturov’s categorification. In this sense, we explain the special case of Dye–Kauffman–Manturov’s homology in terms of Bar-Natan’s construction.

Four small limit cycles around a Hopf singular point in 3-dimensional competitive Lotka–Volterra systems

The 3-dimensional competitive Lotka–Volterra (LV) systems have been studied for more than two decades, and particular attention has been focused on bifurcation of limit cycles. For such a system, Zeeman (1993) identified 33 stable equivalence classes on a carrying simplex, among which only classes 26–31 may have limit cycles. It has been shown that all these 6 classes may possess two limit cycles, and the existence of three limit cycles was claimed in some of these classes. Recently, Gyllenberg and Yan (2009) studied the existence of four limit cycles, three of them are small-amplitude limit cycles due to Hopf bifurcation and one additional limit cycle, enclosing all the three small-amplitude limit cycles, is due to the existence of a heteroclinic cycle, and proposed a new conjecture including: (i) There exists a 3-d competitive LV system with at least 5 limit cycles. (ii) In the case of a heteroclinic cycle on the boundary of the carrying simplex of a 3-d competitive LV system, the vanishing of the first four focus values (the vanishing of the zero-order focus value means that there is a pair of purely imaginary eigenvalues at the positive equilibrium) does not imply that the heteroclinic cycle is neutrally stable, and hence it does not imply that the positive equilibrium is a center. (iii) In the case of a heteroclinic cycle on the boundary of the carrying simplex of a 3-d competitive LV system, the vanishing of the first three focus values and that the heteroclinic cycle is neutrally stable do not imply the vanishing of the third-order focus value, and hence they do not imply that the positive equilibrium is a center. In this paper, we will present two examples belonging to class 27 and another two examples belonging to class 26, which exhibit at least four small-amplitude limit cycles in the vicinity of the positive equilibrium due to Hopf bifurcations, and prove that the items (ii) and (iii) in the conjecture are true. Moreover, showing the existence of four small-amplitude limit cycles is a necessary step towards proving item (i) of the conjecture.