Involutory Hopf group-coalgebras and flat bundles over 3-manifolds

Involutory Hopf algebras and 3-manifold invariants

We establish a 3-manifold invariant for each finite-dimensional, involutory Hopf algebra. If the Hopf algebra is the group algebra of a group $G$, the invariant counts homomorphisms from the fundamental group of the manifold to $G$. The invariant can be viewed as a state model on a Heegaard diagram or a triangulation of the manifold. The computation of the invariant involves tensor products and contractions of the structure tensors of the algebra. We show that every formal expression involving these tensors corresponds to a unique 3-manifold modulo a well-understood equivalence. This raises th…

Generalized Kuperberg invariants of 3-manifolds

In the 90s, based on presentations of 3-manifolds by Heegaard diagrams, Kuperberg associated a scalar invariant of 3-manifolds to each finite dimensional involutory Hopf algebra over a field. We generalize this construction to the case of involutory Hopf algebras in arbitrary symmetric monoidal categories admitting certain pairs of morphisms called good pairs. We construct examples of such good pairs for involutory Hopf algebras whose distinguished grouplike elements are central. The generalized construction is illustrated by an example of an involutory super Hopf algebra.

对合Hopf群-余代数与4-流形上平坦丛的不变量

Involutory Hopf group-coalgebras and invariants of flat bundles over 4-manifolds

本文通过引入对合Hopf G-三元组结构，利用着色三剖分图构造了4-流形上平坦丛的不变量，推广了Chaidez-Cotler-Cui及Mochida的已有构造。

从交换霍普夫代数中获得3流形与嵌入带框架链接的不变量

Invariants from involutory Hopf algebras of 3-manifolds with embedded framed links

本文基于交换霍普夫代数和Drinfeld双代数定义了3流形与嵌入链接的不变量，推广了Kuperberg和HKR不变量，并通过Heegaard-Link图示方法构建了新的证明。

对合Hopf代数的Heisenberg双重与闭$3$-流形的不变量

The Heisenberg double of involutory Hopf algebras and invariants of closed $3$-manifolds

利用对合Hopf代数的Heisenberg双重典范元素，基于正规o-图构造闭3-流形的不变量，计算简单，且对群代数退化为同态计数。

Quantum invariants of closed framed $3$-manifolds based on ideal triangulations

We construct a new type of quantum invariant of closed framed $3$-manifolds with the vanishing first Betti number. The invariant is defined for any finite dimensional Hopf algebra, such as small quantum groups, and is based on ideal triangulations. We use the canonical element of the Heisenberg double, which satisfies a pentagon equation, and graphical representations of $3$-manifolds introduced by R. Benedetti and C. Petronio. The construction is simple and easy to be understood intuitively; the pentagon equation reflects the Pachner $(2,3)$ move of ideal triangulations and the non-involutive…