Related papers: Functoriality of groupoid quantales. II

We provide an explicit and elementary construction of the Morita $(\infty,2)$-category of a monoidal category which satisfies minimal conditions. We construct it as a $3$-coskeletal $2$-complicial set, in which the vertices encode the…

We consider the problem of existence of representations of topological groupoids on a principal bundle and the classification of such representations up to gauge transformation. Such representations naturally occur in various contexts such…

We define an equivalence relation between bimodules over maximal abelian selfadjoint algebras (masa bimodules) which we call spatial Morita equivalence. We prove that two reflexive masa bimodules are spatially Morita equivalent iff their…

We show that every involutive Hopf monoid in a complete and finitely cocomplete symmetric monoidal category gives rise to invariants of oriented surfaces defined in terms of ribbon graphs. For every ribbon graph this yields an object in the…

We construct equivariant vector bundles over quantum projective spaces making use of parabolic Verma modules over the quantum general linear group. Using an alternative realization of the quantized coordinate ring of projective space as a…

Let $\mathcal{G}$ be an algebraic quantum group and $\mathcal{U}$ a compact quantum subgroup. Given a left $\hat{\mathcal{U}}$-module algebra A with unit, we can endow $A\otimes\mathcal{G}$ with a structure of a right…

Given a Lie groupoid, we can form its orbit space, which carries a natural diffeology. More generally, we have a quotient functor from the Hilsum-Skandalis category of Lie groupoids to the category of diffeological spaces. We introduce the…

We apply the One-Sided Action Theorem from the first paper in this series to prove that Rieffel's Morita equivalence between the reduced crossed product by a proper saturated action and the generalized fixed-point algebra is a quotient of a…

We show how to use divisors on the projectivized Hodge bundle to construct special vector-valued modular forms and then apply invariant theory to construct all vector-valued Siegel modular forms of level two and degree two. Thus we…

In this paper we provide descriptions of the Whitehead groups with coefficients in a ring of the Hilbert modular group and its reduced version, as well as for the topological K-theory of $C^*$-algebras, after tensoring with $\mathbb{Q}$, by…

An ergodic action of a compact quantum group G on an operator algebra A can be interpreted as a quantum homogeneous space for G. Such an action gives rise to the category of finite equivariant Hilbert modules over A, which has a module…

We develop Morita theory for finitary additive 2-representations of finitary 2-categories. As an application we describe Morita equivalence classes for 2-categories of projective functors associated to finite dimensional algebras and for…

We show that, if one allows for curved deformations, the canonical map introduced in [KL09] between Morita deformations and second Hochschild cohomology of a dg algebra becomes a bijection. We also show that a bimodule induces an…

It is well known that rings are the objects of a bicategory, whose arrows are bimodules, composed through the bimodule tensor product. We give an analogous bicategorical description of C*-algebras, von Neumann algebras, Lie groupoids,…

In this paper we generalize the theory of multiplicative $G$-Higgs bundles over a curve to pairs $(G,\theta)$, where $G$ is a reductive algebraic group and $\theta$ is an involution of $G$. This generalization involves the notion of a…

We introduce an infinite family of quiver representation-valued invariants of classical, virtual and surface-knots and links associated to a choice of finite biquandle, commutative unital ring, biquandle module and set of biquandle…

Let $G$ be the complex general linear group and $g$ its Lie algebra equipped with a factorizable Lie bialgebra structure; let $U_h$ be the corresponding quantum group. We construct explicit $U_h$-equivariant quantization of Poisson orbit…

Let $G$ be a group and $\Bbbk$ a commutative ring. All categories and functors are assumed to be $\Bbbk$-linear. We define a $G$-invariant bimodule ${}_SM_R$ over $G$-categories $R, S$ and a $G$-graded bimodule ${}_BN_A$ over $G$-graded…

We use classical invariant theory to solve the biholomorphic equivalence problem for two families of plane curve singularities previously considered in the literature. Our calculations motivate an intriguing conjecture that proposes a way…

We develop a $\mathtt{q}$-analogue of the theory of conjugation equivariant $\mathcal D$-modules on a complex reductive group $G$. In particular, we define quantum Hotta-Kashiwara modules and compute their endomorphism algebras. We use the…