## Involutory Hopf algebras and 3-manifold invariants (1991)

Venue: | Intern. J. Math |

Citations: | 39 - 4 self |

### BibTeX

@ARTICLE{Kuperberg91involutoryhopf,

author = {Greg Kuperberg},

title = {Involutory Hopf algebras and 3-manifold invariants},

journal = {Intern. J. Math},

year = {1991},

pages = {41--66}

}

### Years of Citing Articles

### OpenURL

### Abstract

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 the possibility of an algorithm which can determine whether two given 3-manifolds are homeomorphic. 1

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Citation Context ...tensor: M is a non-degenerate bilinear form. We say that H has invertible dimension if dimH ̸= 0 in the ground field of H. In an involutory Hopf algebra, the trace T is also a left and right integral =-=[14]-=-. To prove this, we use the fact that there exists some non-zero integral, and we show that T is a multiple of it: Lemma 3.3 (Radford and Larson). The tensor: is always a right integral (which may be ... |

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private communication
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