On 3-manifold invariants arising from finite-dimensional Hopf algebras.

Tian, Jianjun Paul

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By allowing the coproduct to be non-unital and weakening the counit and antipode axioms of a C*-Hopf algebra too, we obtain a selfdual set of axioms describing a coassociative quantum group, that we call a weak C*-Hopf algebra, which is sufficiently general to describe the symmetries of essentially arbitrary fusion rules. It is the same structure that can be obtained by replacing the multiplicative unitary of Baaj and Skandalis with a partial isometry. The algebraic properties, the existence...

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Let V be a ℂ-space, $σ \in E n d (V^{\otimes 2})$ be a pre-braid operator and let $F \in l i n (V^{\otimes 2}, ℂ) .$ This paper offers a sufficient condition on (σ,F) that there exists a Clifford algebra Cl(V,σ,F) as the Chevalley F-dependent deformation of an exterior algebra $C l (V, σ, 0) \equiv V^{\land} (σ)$ . If $σ \neq σ^{- 1}$ and F is non-degenerate then F is not a σ-morphism in σ-braided monoidal category. A spinor representation as a left Cl(V,σ,F)-module is identified with an exterior algebra over F-isotropic ℂ-subspace of V. We give a sufficient condition on braid σ that the spinor...

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The purpose of this paper is two fold: we study the behaviour of the forgetful functor from $𝕊$ -modules to graded vector spaces in the context of algebras over an operad and derive the construction of combinatorial Hopf algebras. As a byproduct we obtain freeness and cofreeness results for those Hopf algebras. Let $𝒪$ denote the forgetful functor from $𝕊$ -modules to graded vector spaces. Left modules over an operad $𝒫$ are treated as $𝒫$ -algebras in the category of $𝕊$ -modules. We generalize...

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A gauge-field approach to 3- and 4-manifold invariants

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An approach to construction of topological invariants of the Reshetikhin-Turaev-Witten type of 3- and 4-dimensional manifolds in the framework of SU(2) Chern-Simons gauge theory and its hidden (quantum) gauge symmetry is presented.