On Morita equivalences with endopermutation source and isotypies

Hausdorff Morita equivalence is an equivalence relation on singular foliations, which induces a bijection between their leaves. Our main statement is that linearizability along a leaf is invariant under Hausdorff Morita equivalence. The proof relies on a characterization of tubular neighborhood embeddings using Euler-like vector fields.

The Johnson–Morita theory for the handlebody group

ABSTRACT In several articles, McCall and Lowe have claimed that endurantism and perdurantism are “equivalent.” From this, they conclude that there is no fact of the matter as to whether we live in an endurantist world or in a perdurantist world. In this paper, I use the notion of Morita equivalence to show in which precise sense, McCall and Lowe's equivalence claim turns out to be true.

Preprojective algebras, skew group algebras and Morita equivalences

A bstract We investigate domain walls between 2 d gapped phases of Turaev-Viro type topological quantum field theories (TQFTs) by constructing domain wall tube algebras. We begin by analyzing the domain wall tube algebra associated with bimodule categories, and then extend the construction to multimodule categories over N base fusion categories. We prove that the resulting tube algebra is naturally equipped with a C ∗ weak Hopf algebra structure. We show that topological excitations localized on domain walls are classified by representations of the corresponding domain wall tube algebra, in the sense that the functor category of bimodules admits a fusion-preserving embedding into the representation category of the domain wall tube algebra. We further establish the folding trick and Morita theory in this context. Then, most crucially, we provide a rigorous construction of the Drinfeld quantum double from weak Hopf boundary tube algebras using a skew-pairing, and establish an isomorphism between domain wall tube algebra and Drinfeld quantum double of boundary tube algebras. Motivated by the correspondence between the domain wall tube algebra and the quantum double of the boundary tube algebras, we introduce the notion of an N -tuple algebra and demonstrate how it arises in the multimodule domain wall setting. Finally, we consider defects between two domain walls, showing that such defects can be characterized by representations of a domain wall defect tube algebra. We briefly outline how these defects can be systematically treated within this representation-theoretic framework.

Semilinear clannish algebras have been recently introduced by the first author and Crawley-Boevey as a generalization of Crawley-Boevey’s clannish algebras. In the present paper, we associate semilinear clannish algebras to the (colored) triangulations of a surface with marked points and orbifold points, and exhibit a Morita equivalence between these algebras and the Jacobian algebras constructed a few years ago by Geuenich and the second author.

This paper develops a theory of pretriangulated 2 -representations of dg 2 -categories. We characterize cyclic pretriangulated 2 -representations, under certain compactness assumptions, in terms of dg modules over dg algebra 1 -morphisms internal to associated dg 2 -categories of compact objects. Further, we investigate the Morita theory and quasi-equivalences for such dg 2 -representations. We relate this theory to various classes of examples of dg categorifications from the literature.

Here all rings have identities. Let R be a ring and let R-mod denote the additive category of left finitely generated R-modules. Note that if R is a noetherian ring, then R-mod is an abelian category and every R-module is a finite direct sum of indecomposable R-modules. Finite Group Modular Representation Theory concerns the study of left finitely generated OG-modules where G is a finite group and O is a complete discrete valuation ring with O/J(O) a field of prime characteristic p. Thus OG is a noetherian O-algebra. The Green Theory in this area yields for each isomorphism type of finitely generated indecomposable (and hence for each isomorphism type of finitely generated simple OG-module) a theory of vertices and sources invariants. The vertices are derived from the set of p-subgroups of G. As suggested by the above, in Basic Definition and Main Results for Rings Section, let Σ be a fixed subset of subrings of the ring R and we develop a theory of Σ-vertices and sources for finitely generated R-modules. We conclude Basic Definition and Main Results for Rings Section with examples and show that our results are compatible with a ring isomorphic to R. For Idempotent Morita Equivalence and Virtual Vertex-Source Pairs of Modules of a Ring Section, let e be an idempotent of R such that R=ReR. Set B=eRe so that B is a subring of R with identity e. Then, the functions eR⊗R∗:R−mod→B−mod and Re⊗B∗:B−mod→R−mod form a Morita Categorical Equivalence. We show, in this Section, that such a categorical equivalence is compatible with our vertex-source theory. In Two Applications with Idemptent Morita Equivalence Section, we show such compatibility for source algebras in Finite Group Block Theory and for naturally Morita Equivalent Algebras.

We recall the notion of a singular foliation (SF) on a manifold M , viewed as an appropriate submodule of \mathfrak{X}(M) , and adapt it to the presence of a Riemannian metric g , yielding a module version of a singular Riemannian foliation (SRF). Following Garmendia–Zambon on Hausdorff Morita equivalence of SFs, we define the Morita equivalence of SRFs (both in the module sense as well as in the more traditional geometric one of Molino) and show that the leaf spaces of Morita equivalent SRFs are isomorphic as pseudo-metric spaces.In a second part, we introduce the category of \mathcal{I} -Poisson manifolds. Its objects and morphisms generalize Poisson manifolds and morphisms in the presence of appropriate ideals \mathcal{I} of the smooth functions on the manifold such that two conditions are satisfied: (i) The category of Poisson manifolds becomes a full subcategory when choosing \mathcal{I}=0 and (ii) there is a reduction functor from this new category to the category of Poisson algebras, which generalizes coistropic reduction to the singular setting.Every SF on M gives rise to an \mathcal{I} -Poisson manifold on T^{*}M and g enhances this to an SRF if and only if the induced Hamiltonian lies in the normalizer of \mathcal{I} . This perspective provides, on the one hand, a simple proof of the fact that every module SRF is a geometric SRF and, on the other hand, a construction of an algebraic invariant of singular foliations: Hausdorff Morita equivalent SFs have isomorphic reduced Poisson algebras.

We characterise the Morita equivalence classes of blocks with extraspecial defect groups p+1+2 for p≥5, and so show that Donovan’s conjecture and the Alperin-McKay conjecture hold for such p-groups. For p=3 we reduce Donovan’s conjecture for blocks with defect group 3+1+2 to bounding the Cartan invariants for such blocks of quasisimple groups. We apply the characterisation to the case p=5 as an example, to list the Morita equivalence classes of such blocks.

Abstract Given a locally compact quantum group $\mathbb{G}$ and two $\mathbb{G}$-$W^{*}$-algebras $\alpha : A\curvearrowleft \mathbb{G}$ and $\beta : B\curvearrowleft \mathbb{G}$, we study the notion of equivariant $W^{*}$-Morita equivalence $(A, \alpha )\sim _{\mathbb{G}} (B, \beta )$, which is an equivariant version of Rieffel’s notion of $W^{*}$-Morita equivalence. We prove that important dynamical properties of $\mathbb{G}$-$W^{*}$-algebras, such as (inner) amenability, are preserved under equivariant Morita equivalence. For a coideal von Neumann algebra $L^\infty (\mathbb{K}\backslash \mathbb{G})\subseteq L^\infty (\mathbb{G})$ with dual coideal von Neumann algebra $L^\infty (\check{\mathbb{K}})\subseteq L^\infty (\check{\mathbb{G}})$, we use a natural $\check{\mathbb{G}}$-$W^{*}$-Morita equivalence $L^\infty (\mathbb{K}\backslash \mathbb{G})\rtimes _\Delta \mathbb{G} \sim _{\check{\mathbb{G}}} L^\infty (\check{\mathbb{K}})$ to relate dynamical properties of $L^\infty (\mathbb{K}\backslash \mathbb{G})$ with dynamical properties of $L^\infty (\check{\mathbb{K}})$. We use this to refine some recent results established by Anderson-Sackaney and Khosravi. This refinement allows us to answer a question of Kalantar, Kasprzak, Skalski, and Vergnioux, namely that for $\mathbb{H}$ a closed quantum subgroup of the compact quantum group $\mathbb{G}$, coamenability of $\mathbb{H}\backslash \mathbb{G}$ and relative amenability of $\ell ^\infty (\check{\mathbb{H}})$ in $\ell ^\infty (\check{\mathbb{G}})$ are equivalent. Moreover, if $\mathbb{G}$ is compact, we study the relation between $\mathbb{G}$-$W^{*}$-Morita equivalence of $(A, \alpha )$ and $(B, \beta )$ and $\mathbb{G}$-$C^{*}$-Morita equivalence of the associated $\mathbb{G}$-$C^{*}$-algebras $(\mathcal{R}(A), \alpha )$ and $(\mathcal{R}(B), \beta )$ of regular elements.

We call a von Neumann algebra with finite-dimensional center a multifactor. We introduce an invariant of bimodules over \mathrm{II}_{1} multifactors that we call modular distortion, and use it to formulate two classification results.We first classify finite depth finite index connected hyperfinite \mathrm{II}_{1} multifactor inclusions A\subset B in terms of the standard invariant (a unitary planar algebra), together with the restriction to A of the unique Markov trace on B . The latter determines the modular distortion of the associated bimodule. Three crucial ingredients are Popa’s uniqueness theorem for such inclusions which are also homogeneous, for which the standard invariant is a complete invariant, a generalized version of the Ocneanu Compactness Theorem, and the notion of Morita equivalence for inclusions.Second, we classify fully faithful representations of unitary multifusion categories into bimodules over hyperfinite \mathrm{II}_{1} multifactors in terms of the modular distortion. Every possible distortion arises from a representation, and we characterize the proper subset of distortions that arise from connected \mathrm{II}_{1} multifactor inclusions.

We study properties of non-Lorentzian geometries arising from BPS decoupling limits of string theory that are central to matrix theory and the AdS/CFT correspondence. We focus on duality transformations between ten-dimensional non-Lorentzian geometries coupled to matrix theory on D-branes. We demonstrate that T- and S-duality transformations exhibit novel asymmetric properties: depending not only on the choice of transformation but also on the value of the background fields, the codimension of the foliation structure of the dual non-Lorentzian background may be different or the same. This duality asymmetry underlies features observed in the study of non-commutativity and Morita equivalence in matrix and gauge theory. Finally, we show how the holographic correspondence involving non-commutative Yang-Mills fits into our framework, from which we further obtain novel holographic examples with non-Lorentzian bulk geometries.

We define and study RoCK blocks for double covers of symmetric groups. We prove that RoCK blocks of double covers are Morita equivalent to standard ‘local’ blocks. The analogous result for blocks of symmetric groups, a theorem of Chuang and Kessar, was an important step in Chuang and Rouquier ultimately proving Broué’s abelian defect group conjecture for symmetric groups. Indeed we prove Broué’s conjecture for the RoCK blocks defined in this article. Our methods involve translation into the quiver Hecke superalgebras setting using the Kang-Kashiwara-Tsuchioka isomorphism and categorification methods of Kang-Kashiwara-Oh. As a consequence we construct Morita equivalences between more general objects than blocks of finite groups. In particular, our results extend to certain blocks of level one cyclotomic Hecke-Clifford superalgebras. We also study imaginary cuspidal categories of quiver Hecke superalgebras and classify irreducible representations of quiver Hecke superalgebras in terms of cuspidal systems.

Labelled graphs as Morita equivalence invariants for a class of inverse semigroups

We present a state sum construction that assigns a scalar to a skeleton in a closed oriented three-dimensional manifold. The input datum is the pivotal bicategory \mathbf{Mod}^{\mathrm{sph}}(\mathcal{A}) of spherical module categories over a spherical fusion category \mathcal{A} . The interplay of algebraic structures in this pivotal bicategory with moves of skeleta ensures that our state sum is independent of the skeleton on the manifold. We show that the bicategorical invariant recovers the value of the standard Turaev–Viro invariant associated to \mathcal{A} , thereby proving the independence of the Turaev–Viro invariant under pivotal Morita equivalence without recurring to the Reshetikhin–Turaev construction. A key ingredient for the construction is the evaluation of graphs on the sphere with labels in \mathbf{Mod}^{\mathrm{sph}}(\mathcal{A}) that we develop in this article. A central tool is Nakayama-twisted traces on pivotal bimodule categories, which we study beyond semisimplicity.

Abstract We prove that the Drinfeld center of a fusion 2-category is invariant under Morita equivalence. We go on to show that the concept of Morita equivalence between connected fusion 2-categories corresponds to a notion of Witt equivalence between braided fusion 1-categories. A strongly fusion 2-category is a fusion 2-category whose braided fusion 1-category of endomorphisms of the monoidal unit is $\mathbf{Vect}$ or $\mathbf{SVect}$ . We prove that every fusion 2-category is Morita equivalent to the 2-Deligne tensor product of a strongly fusion 2-category and an invertible fusion 2-category. We proceed to show that every fusion 2-category is Morita equivalent to a connected fusion 2-category. As a consequence, we find that every rigid algebra in a fusion 2-category is separable. This implies in particular that every fusion 2-category is separable. Conjecturally, separability ensures that a fusion 2-category is 4-dualizable. We define the dimension of a fusion 2-category, and prove that it is always non-zero. Finally, we show that the Drinfeld center of any fusion 2-category is a finite semisimple 2-category.

Abstract In this paper, we investigate locally finitely presented pure semisimple (hereditary) Grothendieck categories. We show that every locally finitely presented pure semisimple (resp., hereditary) Grothendieck category $\mathscr {A}$ is equivalent to the category of left modules over a left pure semisimple (resp., left hereditary) ring when $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ is a QF-3 category, and every representable functor in $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ has finitely generated essential socle. In fact, we show that there exists a bijection between Morita equivalence classes of left pure semisimple (resp., left hereditary) rings $\Lambda $ and equivalence classes of locally finitely presented pure semisimple (resp., hereditary) Grothendieck categories $\mathscr {A}$ that $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ is a QF-3 category, and every representable functor in $\mathrm {Mod}(\mathrm {fp}(\mathscr {A}))$ has finitely generated essential socle. To prove this result, we study left pure semisimple rings by using Auslander’s ideas. We show that there exists, up to equivalence, a bijection between the class of left pure semisimple rings and the class of rings with nice homological properties. These results extend the Auslander and Ringel–Tachikawa correspondence to the class of left pure semisimple rings. As a consequence, we give several equivalent statements to the pure semisimplicity conjecture.

We classify up to Morita equivalence all blocks whose defect groups are Suzuki 2-groups. The classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field, and in fact holds up to basic Morita equivalence. As a consequence Donovan's conjecture holds for Suzuki 2-groups. A corollary of the proof is that Suzuki Sylow 2-subgroups of finite groups with no nontrivial odd order normal subgroup are trivial intersection.