Equivariant Morita equivalences between Podles' spheres

The Spectral Geometry of the Equatorial Podles Sphere

We propose a slight modification of the properties of a spectral geometry a la Connes, which allows for some of the algebraic relations to be satisfied only modulo compact operators. On the equatorial Podles sphere we construct suq2-equivariant Dirac operator and real structure which satisfy these modified properties.

Morita invariance of equivariant and Hopf-cyclic cohomology of module algebras over Hopf algebroids

We investigate the equivariant and Hopf-cyclic cohomology of module algebras over Hopf algebroids and derive their Morita invariance. For this, we use the tools developed by McCarthy for $k$-linear categories and subsequently by Kaygun and Khalkhali for a certain class of categories, called as $H$-categories, for a Hopf algebra $H$.

The resolvent cocycle in twisted cyclic cohomology and a local index formula for the Podles sphere

We continue the investigation of twisted homology theories in the context of dimension drop phenomena. This work unifies previous equivariant index calculations in twisted cyclic cohomology. We do this by proving the existence of the resolvent cocycle, a finitely summable analogue of the JLO cocycle, under weaker smoothness hypotheses and in the more general setting of `modular' spectral triples. As an application we show that using our twisted resolvent cocycle, we can obtain a local index formula for the Podles sphere. The resulting twisted cyclic cocycle has non-vanishing Hochschild class w…

Morita "equivalences" of equivariant torus spectral triples

In general, Morita equivalence of spectral triples need not be a symmetric relation. In this paper, we show that Morita equivalence of spectral triples is an equivalence relation for equivariant torus spectral triples.

Classification of Invariant Star Products up to Equivariant Morita Equivalence on Symplectic Manifolds

In this paper we investigate equivariant Morita theory for algebras with momentum maps and compute the equivariant Picard groupoid in terms of the Picard groupoid explicitly. We consider three types of Morita theory: ring-theoretic equivalence, *-equivalence and strong equivalence. Then we apply these general considerations to star product algebras over symplectic manifolds with a Lie algebra symmetry. We obtain the full classification up to equivariant Morita equivalence.

融合等变稳定性条件与森田对偶

Fusion-equivariant stability conditions and Morita duality

本研究探讨带融合范畴作用的三角范畴上的融合等变Bridgeland稳定性条件模空间，证明其为普通稳定性条件模空间的闭复子流形，推广了有限群作用下的相关结果，并给出在McKay箭图与几何稳定性条件中的应用。

Tensor functors between Morita duals of fusion categories

Given a fusion category $\mathcal{C}$ and an indecomposable $\mathcal{C}$-module category $\mathcal{M}$, the fusion category $\mathcal{C}^*_\mathcal{M}$ of $\mathcal{C}$-module endofunctors of $\mathcal{M}$ is called the (Morita) dual fusion category of $\mathcal{C}$ with respect to $\mathcal{M}$. We describe tensor functors between two arbitrary duals $\mathcal{C}^*_\mathcal{M}$ and $\mathcal{D}^*_\mathcal{N}$ in terms of data associated to $\mathcal{C}$ and $\mathcal{D}$. We apply the results to $G$-equivariantizations of fusion categories and group-theoretical fusion categories. We describe…

Strictification of weakly equivariant Hopf algebras

A weakly equivariant Hopf algebra is a Hopf algebra A with an action of a finite group G up to inner automorphisms. We show that each weakly equivariant Hopf algebra can be replaced by a Morita equivalent algebra B with a strict action of G and with a coalgebra structure that leads to a tensor equivalent representation category. However, the coproduct of this strictification cannot, in general, be chosen to be unital, so that a strictification of the G-action can only be found on a weak Hopf algebra B.

Twist star products and Morita equivalence

We present a simple no-go theorem for the existence of a deformation quantization of a homogeneous space M induced by a Drinfel'd twist: we argue that equivariant line bundles on M with non-trivial Chern class and symplectic twist star products cannot both exist on the same manifold M. This implies, for example, that there is no symplectic star product on the complex projective spaces induced by a twist based on U(gl(n,C))[[h]] or any sub-bialgebra, for every n greater or equal than 2.

动力冯·诺依曼代数的Morita理论

Morita theory for dynamical von Neumann algebras

本文研究动力冯·诺依曼代数的等变Morita等价，证明其保持顺从性等动力性质，并用于改进已有结果、回答关于余顺从性与相对顺从性等价的问题，同时探讨紧情形下与C*-代数Morita等价的关系。

Tannaka-Krein duality for compact quantum homogeneous spaces. I. General theory

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 structure over the tensor category Rep(G) of finite dimensional representations of G. We show that there is a one-to-one correspondence between the quantum G-homogeneous spaces up to equivariant Morita equivalence, and indecomposable module C*-categories over Rep(G) up to natural equivalence. This gives a global approach to the duality theory for ergodic actions…

On transitive action on quiver varieties

Associated with each finite subgroup $Γ$ of $\rm{SL}_2(\mathbb{C})$ there is a family of noncommutative algebras $O_τ(Γ)$ quantizing $\mathbb{C}^2/\!\!/Γ$. Let $G_Γ$ be the group of $Γ$-equivariant automorphisms of $O_τ$. One of the authors earlier defined and studied a natural action of $G_Γ$ on certain quiver varieties associated with $Γ$. He established a $G_Γ$-equivariant bijective correspondence between quiver varieties and the space of isomorphism classes of $O_τ$-ideals. The main theorem in this paper states that when $Γ$ is a cyclic group, the action of $G_Γ$ on each quiver variety is …

动力 von Neumann 代数的 Morita 等价的范畴论解释

A categorical interpretation of Morita equivalence for dynamical von Neumann algebras

本文通过构造表示范畴与对应范畴之间的典范函子，证明了动力 von Neumann 代数的等变 Morita 等价等价于其表示范畴作为模 $W^*$-范畴的等价，并在紧量子群情形下得到范畴等价，给出 Eilenberg-Watts 定理的等变版本。