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
AI总结：
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。
摘要：
我们利用有限维、对合、幺模且余幺模的Hopf代数$H$构造了闭定向$3$-流形的一个不变量。我们采用R. Benedetti和C. Petronio引入的正规o-图框架，在该框架中，可以通过定向虚纽结图来表示分支理想三角剖分。我们将$H$的Heisenberg双重$\mathcal{H}(H)$的一个典范元素分配给每个实交叉点，该交叉点代表一个分支理想四面体。该不变量取值于循环商$\mathcal{H}(H)/{[\mathcal{H}(H),\mathcal{H}(H)]}$，该商同构于基域。在构造中，我们仅使用$H$的典范元素和结构常数，而不使用$H$的任何表示。这一点，连同正规o-图移动的有限性和局部性条件，使得我们不变量的计算相当简单且易于理解。当$H$是有限群的群代数时，该不变量计数从$3$-流形的基本群到该群的群同态的数量。
英文摘要：
We construct an invariant of closed oriented $3$-manifolds using a finite dimensional, involutory, unimodular and counimodular Hopf algebra $H$. We use the framework of normal o-graphs introduced by R. Benedetti and C. Petronio, in which one can represent a branched ideal triangulation via an oriented virtual knot diagram. We assign a copy of a canonical element of the Heisenberg double $\mathcal{H}(H)$ of $H$ to each real crossing, which represents a branched ideal tetrahedron. The invariant takes values in the cyclic quotient $\mathcal{H}(H)/{[\mathcal{H}(H),\mathcal{H}(H)]}$, which is isomorphic to the base field. In the construction we use only the canonical element and structure constants of $H$ and we do not use any representations of $H$. This, together with the finiteness and locality conditions of the moves for normal o-graphs, makes the calculation of our invariant rather simple and easy to understand. When $H$ is the group algebra of a finite group, the invariant counts the number of group homomorphisms from the fundamental group of the $3$-manifold to the group.
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