New Hopf algebras help classify 4D shapes.
problem Classifying 4D shapes up to deformations.
method Developed non-factorizable ribbon Hopf algebras.
result Some derived invariants are boundary-dependent.
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48 results for “ribbon Hopf algebras”
Refines a tangle invariant using XC-algebras.
problem Building a refined tangle invariant using XC-algebras.
method Constructing a canonical strict monoidal functor that refines the Kerler-Kauffman-Radford invariant.
result Preserves the braiding, twist, and open trace.
Modified Hennings invariant defined using quantum groups and integrals.
problem Defining a modified Hennings invariant using quantum groups.
method Topological ribbon Hopf algebra, discrete Fourier transforms, symmetrized graded integral, modified trace.
result Modified graded Hennings invariant defined and extended to empty manifolds.
We extend the construction of the Hennings TQFT for ribbon Hopf algebras to the case of ribbon quasi-Hopf algebras as defined by Drinfeld. Calculations proceed in a similar fashion to the ordinary Hopf algebra case, but also require the handling of the non-trivial coassociator in the triple tensor product of the algebr…
Knot invariants from XC-structures on Sweedler algebra are trivially determined.
problem Defining and characterizing knot invariants from XC-structures.
method Examining XC-structures on the Sweedler algebra and their relation to knot invariants.
result Knot invariants from XC-structures on Sweedler algebra are completely determined by the framing of the knot.
Paper extends quantum invariant to colored ideal triangulations.
problem Quantum invariants for colored ideal triangulations.
method Uses Heisenberg double and pentagon relation to extend invariant.
result Invariance of colored ideal triangulations under moves.
Compute central extension of mapping class group from stated skein algebra
problem Compute central extension of mapping class group from stated skein algebra
method Compute central extension of mapping class group from stated skein algebra
result Compute central extension of mapping class group from stated skein algebra
New non-semisimple TQFTs derived from quantum Hennings invariants.
problem Constructing non-semisimple 2+1-TQFTs with mapping class group representations.
method Generalized Hennings invariants, modified traces, and modified TQFT construction.
result Construction of non-semisimple TQFTs yielding mapping class group representations.
New categories from TQFTs interpret skein relations.
problem Interpreting skein relations in TQFTs.
method Constructing half-braided algebras and their bimodules.
result Stated skein relations correspond to a TQFT.
M. Hennings and G. Kuperberg defined quantum invariants Z_{Henn} and Z_{Kup} of closed oriented 3-manifolds based on certain Hopf algebras, respectively. We prove that |Z_{Kup}|=|Z_{Henn}|^2 for lens spaces when both invariants are based on factorizable finite dimensional ribbon Hopf algebras.
Constructs TQFTs for cobordisms with cohomology class decorations.
problem Creating TQFTs for cobordisms with cohomology class decorations.
method Starting from an abelian group and a factorizable ribbon Hopf -bialgebra , constructs a TQFT for connected framed cobordisms between connected surfaces with connected boundary decorated with cohomology classes with coefficients in .
result Our functor recovers a special case of Kerler-Lyubashenko TQFTs when restricted to trivial decorations.
New braided Frobenius algebras created from specific Hopf algebras.
problem Creating new algebraic structures from Hopf algebras.
method Heap operation and Yang-Baxter operator on tensor product.
result Heap operation induces a braiding compatible with Frobenius operations.
Quantum invariant derived from ternary cohomology of self-distributive structures.
problem Defining and proving a quantum invariant from ternary cohomology.
method Constructing a ribbon category from a TSD set, showing it coincides with the cocycle invariant.
result The ribbon cocycle invariant is a quantum invariant.
The general method of Reshetikhin and Turaev is followed to develop topological invariants of closed, connected, orientable 3-manifolds from a new class of algebras called pseudo-modular Hopf algebras. Pseudo-modular Hopf algebras are a class of Z_2-graded ribbon Hopf algebras that generalise the concept of a modular H…
A bottom tangle is a tangle in a cube consisting only of arc components, each of which has the two endpoints on the bottom line of the cube, placed next to each other. We introduce a subcategory B of the category of framed, oriented tangles, which acts on the set of bottom tangles. We give a finite set of generators of…
In GT/0006019 oriented quantum algebras were motivated and introduced in a natural categorical setting. Invariants of knots and links can be computed from oriented quantum algebras, and this includes the Reshetikhin-Turaev theory for Ribbon Hopf algebras. Here we continue the study of oriented quantum algebras from a m…
We define the notion of a Kirby element of a ribbon category C (not necessarily semisimple). Kirby elements lead to 3-manifolds invariants. We characterize (in terms of the structure maps of some categorical Hopf algebra) a set of Kirby elements of C which is sufficiently large to recover the known quantum invariants c…
The Reshetikhin-Turaev invariant, Turaev's TQFT, and many related constructions rely on the encoding of certain tangles (n-string links, or ribbon n-handles) as n-forms on the coend of a ribbon category. We introduce the monoidal category of Hopf diagrams, and describe a universal encoding of ribbon string links as Hop…
A new integral for tangles in handlebodies connects to Hopf algebras.
problem Constructing a functor for bottom tangles in handlebodies.
method Extension of Kontsevich integral to handlebodies, functor construction.
result Induces canonical isomorphism between bottom tangles and a specific PROP.
Functor connects 4D 2-handlebodies to ribbon categories, detecting non-deformation diffeomorphisms.
problem Detecting non-deformation diffeomorphisms in 4D 2-handlebodies.
method Constructs a braided monoidal functor from 4D 2-handlebodies to unimodular ribbon categories.
result Functor detects non-deformation diffeomorphisms when is not semisimple and is not factorizable.
New invariant for 3-manifolds from topological Hopf superalgebra.
problem Constructing an invariant for 3-manifolds.
method Using a ribbon superalgebra from a topological Hopf superalgebra, we construct an invariant of 3-manifolds of Hennings type.
result Constructs a new invariant for 3-manifolds from a topological Hopf superalgebra.
This paper defines ribbons and ribbon complexes in CW spaces and analyzes their topological properties.
problem Characterizing and analyzing topological structures in CW spaces.
method Introducing planar ribbons, ribbon complexes, and ribbon nerves in Alexandroff-Hopf-Whitehead CW spaces, and studying their topological properties.
result Characterization of ribbons and ribbon nerves by Betti numbers and homotopy types.
Introduces XC-tangles for quantum tangle invariants.
problem Quantum invariants of tangles and virtual tangles.
method Introduces XC-tangles and their equivalence to virtual upwards tangles.
result Extension of knot isotopy invariants to XC-tangles.
New 3D TQFTs derived from non-semisimple categories.
problem Constructing topological invariants from non-semisimple categories.
method Using modified traces and Lyubashenko's invariants, with additional assumptions for factorizability.
result Produces new 2+1-TQFTs and monoidal extensions of representations.
We present an invariant of connected and oriented closed 3-manifolds based on a coribbon Weak Hopf Algebra H with a suitable left-integral. Our invariant can be understood as the generalization to Weak Hopf Algebras of the Hennings-Kauffman-Radford evaluation of an unoriented framed link using a dual quantum-trace. Thi…
Abstract: Mapping class groups act on cohomology of surfaces via Hochschild cohomology.
problem Understanding the action of mapping class groups on cohomology of surfaces.
method Associate cochain complexes to surfaces, with mapping class groups acting projectively on cohomology.
result Projective action of mapping class groups on Hochschild cohomology of Hopf algebras.
We show that simple coverings of B^4 branched over ribbon surfaces up to certain local ribbon moves bijectively represent orientable 4-dimensional 2-handlebodies up to handle sliding and addition/deletion of cancelling handles. As a consequence, we obtain an equivalence theorem for simple coverings of S^3 branched over…
Involutive Hopf monoids yield surface invariants.
problem Involutive Hopf monoids in symmetric monoidal categories.
method Construction of invariants via (co)equalizers and images.
result Categorical generalization of quantum double models.
Alexander invariant for ribbon tangles connects to planar algebras.
problem Alexander invariant for ribbon tangles.
method Construction of Alexander invariant and its functorial properties.
result Alexander invariant commutes with compositions in a circuit algebra.
Researchers create explicit Hopf cyclic cohomology classes for a specific Hopf algebra.
problem Constructing Hopf cyclic cohomology classes for a bicrossed product Hopf algebra.
method Explicit construction of Hopf action and invariant trace on a convolution algebra.
result Explicit representative cocycles in cyclic cohomology of the convolution algebra.
Refined invariants for 4D 2-handlebodies, linking quantum groups and cohomology.
problem Constructing and studying new invariants for 4D 2-handlebodies.
method Defining invariants for pairs , using unimodular ribbon Hopf coalgebras.
result Decomposition formulas for original invariants in terms of refined ones.
We show that for any n > 3 there exists an equivalence functor from the category of n-fold connected simple coverings of B^3 x [0, 1] branched over ribbon surface tangles up to certain local ribbon moves, to the category Chb^{3+1} of orientable relative 4-dimensional 2-handlebody cobordisms up to 2-deformations. As a c…
A closer look at an example introduced by Livingston & Melvin and later studied by Miyazaki shows that a plumbing of two fibered ribbon knots (along their fiber surfaces) may be algebraically slice yet not ribbon.
New algebraic structure derived from Hopf algebra and Drinfel'd twist.
problem Developing a new algebraic structure from existing mathematical concepts.
method Extending -algebra to a Hopf algebra, twisting with Drinfel'd twist, and identifying Hopf morphisms and braided morphisms.
result Braided -algebra is derived from the process.
Analyses cohomology relations for moving frames and coframes.
problem Relating Hopf cyclic cohomology of moving frames and coframes.
method Uses van Est analogy for DG Hopf algebras.
result Establishes cohomology isomorphism for DG Hopf algebras.
We associate to each infinite primitive Lie pseudogroup a Hopf algebra of `transverse symmetries', by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed as a `quantum group' counterpart of the infinite-dimensional primitive Lie algeb…
Let be a nonnegative integer, we use ribbon graph diagrams and the Yamada polynomial skein relations to construct an algebra which is shown to be closely related to the Temerley-Lieb Algebra. We prove that the algebra is isomorphic to some quotient of a three variables polynomi…
A bottom tangle is a tangle in a cube consisting of arc components whose boundary points are on a line in the bottom square of the cube. A ribbon bottom tangle is a bottom tangle whose closure is a ribbon link. For every n-component ribbon bottom tangle T, we prove that the universal invariant J_T of T associated to th…
Invariants for 4-manifolds from Hopf group-algebras.
problem Constructing invariants for flat connections on 4-manifolds.
method Using finite type involutory quasitriangular Hopf -algebras and coloring Kirby diagrams.
result Invariants defined for 4-manifolds and connections.
Analogous exponential map defined for Hopf algebras.
problem Defining an exponential map for Hopf algebras.
method Analogy with Lie groups, interpretation as states, Hilbert C* bimodules, dual Hopf algebra elements.
result Multiple interpretations of exponential map values in Hopf algebras.
This paper develops 4-manifold invariants using Hopf algebras.
problem Creating 4-manifold invariants from Hopf algebras.
method Using Hopf triplets and trisection diagrams, the authors construct 4-manifold invariants.
result Every Hopf triplet yields a diffeomorphism invariant of closed 4-manifolds.
This paper studies finite type invariants for welded string links and ribbon tubes, showing characterizations and algebraic structures.
problem Finite type invariants for ribbon knotted surfaces and their relation to welded string links.
method Developed a theory of finite type invariants for welded string links up to -equivalence, studied algebraic structures, and showed characterizations.
result Characterizes the information contained by finite type invariants in low degrees for welded string links.
Paper categorifies a polynomial related to ribbon graphs.
problem Enumerating partial duals of ribbon graphs.
method Using an extended Frobenius algebra in unoriented topological quantum field theory.
result A categorification of the partial-dual genus polynomial.
This paper is the second part of our work on 4-dimensional 2-handlebodies. In the first part (arXiv:math.GT/0407032) it is shown that up to certain set of local moves, connected simple coverings of B^4 branched over ribbon surfaces, bijectively represent connected orientable 4-dimensional 2-handlebodies up to 2-deforma…
We present a differential calculus on the extension of the quantum plane obtained considering that the (bosonic) generator is invertible and furthermore working polynomials in instead of polynomials in . We call quantum Lie algebra to this extension and we obtain its Hopf algebra structure and its dual H…
Generalizes Kuperberg's invariant to broader algebraic structures.
problem Extending Kuperberg's invariant to a wider class of algebraic objects.
method Constructing a scalar invariant for 3-manifolds using involutory Hopf algebras in arbitrary symmetric monoidal categories with good pairs of morphisms.
result Illustrated examples of the generalized construction for specific algebraic structures.
Currents on Lie groups form a Hopf algebra structure.
problem Understanding algebraic structure of currents on Lie groups.
method Defined Hopf algebra structure on currents using convolution and wedge product.
result Explicit formulas for Hopf algebra operations on currents are derived.
New restrictions found on triple linking numbers of knot derivatives.
problem Understanding triple linking numbers of knot derivatives.
method Defined a new obstruction for Z[Z]-homology ribbon knots.
result New non-doubly slice knots discovered.