Hopf algebra
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Topological semimetals carrying arbitrary Hopf numbers: Fermi surface topologies of a Hopf link, Solomon's knot, trefoil knot, and other linked nodal varieties Open
We propose a new type of Hopf semimetals indexed by a pair of numbers\n$(p,q)$, where the Hopf number is given by $pq$. The Fermi surface is given by\nthe preimage of the Hopf map, which is nontrivially linked for a nonzero Hopf\nnumber. T…
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On lattice models of gapped phases with fusion category symmetries Open
A bstract We construct topological quantum field theories (TQFTs) and commuting projector Hamiltonians for any 1+1d gapped phases with non-anomalous fusion category symmetries, i.e. finite symmetries that admit SPT phases. The construction…
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Lectures on Yangian symmetry Open
In these introductory lectures we discuss the topic of Yangian symmetry from\nvarious perspectives. Forming the classical counterpart of the Yangian and an\nextension of ordinary Noether symmetries, first the concept of nonlocal charges\ni…
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Logarithmic conformal field theory, log-modular tensor categories and modular forms Open
The two pillars of rational conformal field theory and rational vertex\noperator algebras are modularity of characters on the one hand and its\ninterpretation of modules as objects in a modular tensor category on the other\none. Overarchin…
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Non-degeneracy conditions for braided finite tensor categories Open
For a braided finite tensor category C with unit object 1∈C, Lyubashenko considered a certain Hopf algebra F∈C endowed with a Hopf pairing ω:F⊗F→1 to define the notion of a 'non-semisimple' modular tensor category. We say that C is non-deg…
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Hopf braces and Yang-Baxter operators Open
This paper introduces Hopf braces, a new algebraic structure related to the Yang-Baxter equation which include Rump's braces and their non-commutative generalizations as particular cases. Several results of classical braces are still valid…
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DEFORMATIONS OF SHUFFLES AND QUASI-SHUFFLES Open
We investigate deformations of the shuffle Hopf algebra structure Sh(A) which can be defined on the tensor algebra over a commutative algebra A. Such deformations, leading for example to the quasi-shuffle algebra QSh(A), can be in-terprete…
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A second proof of the Shareshian--Wachs conjecture, by way of a new Hopf algebra Open
This is a set of working notes which give a second proof of the Shareshian--Wachs conjecture, the first (and recent) proof being by Brosnan and Chow in November 2015. The conjecture relates some symmetric functions constructed combinatoria…
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Genealogy of Non-perturbative Quantum-Invariants of 3-Manifolds: The Surgical Family Open
We study the relations between the invariants $τ_{RT}$, $τ_{HKR}$, and $τ_L$ of Reshetikhin-Turaev, Hennings-Kauffman-Radford, and Lyubashenko, respectively. In particular, we discuss explicitly how $τ_L$ specializes to $τ_{RT}$ for semisi…
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Non-perturbative completion of Hopf-algebraic Dyson-Schwinger equations Open
For certain quantum field theories, the Kreimer-Connes Hopf-algebraic approach to renormalization reduces the Dyson-Schwinger equations to a system of non-linear ordinary differential equations for the expansion coefficients of the renorma…
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Quivers with loops and generalized crystals Open
In the context of varieties of representations of arbitrary quivers, possibly carrying loops, we define a generalization of Lusztig Lagrangian subvarieties. From the combinatorial study of their irreducible components arises a structure ri…
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The characteristic polynomial of the Adams operators on graded connected Hopf algebras Open
The Adams operators [math] on a Hopf algebra [math] are the convolution powers of the identity of [math] . They are also called Hopf powers or Sweedler powers. We study the Adams operators when [math] is graded connected. The main result i…
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Feynman Categories Open
In this paper we give a new foundational, categorical formulation for operations and relations and objects parameterizing them. This generalizes and unifies the theory of operads and all their cousins including but not limited to PROPs, mo…
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Mathematical Aspects of Scattering Amplitudes Open
In these lectures we discuss some of the mathematical structures that appear when computing multi-loop Feynman integrals. We focus on a specific class of special functions, the so-called multiple polylogarithms, and discuss introduce their…
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The Hopf Algebra of Fliess Operators and Its Dual Pre-lie Algebra Open
We study the Hopf algebra H of Fliess operators coming from Control Theory in\nthe one-dimensional case. We prove that it admits a graded, finte-dimensional,\nconnected gradation. Dually, the vector space IR is both a pre-Lie algebra for\n…
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-Poincaré invariant quantum field theories with Kubo-Martin-Schwinger weight Open
A natural star product for 4-d $\\kappa$-Minkowski space is used to\ninvestigate various classes of $\\kappa$-Poincar\\'e invariant scalar field\ntheories with quartic interactions whose commutative limit coincides with the\nusual $\\phi^4…
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Connected Hopf algebras of Gelfand-Kirillov dimension four Open
We classify connected Hopf algebras of Gelfand-Kirillov dimension 4 over an algebraically closed field of characteristic zero.
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Weak multiplier bialgebras Open
A non-unital generalization of weak bialgebra is proposed with a multiplier-valued comultiplication. Certain canonical subalgebras of the multiplier algebra (named the ‘base algebras’) are shown to carry coseparable co-Frobenius coalgebra …
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Modularization of small quantum groups Open
We construct a large family of ribbon quasi-Hopf algebras related to small quantum groups, with a factorizable R-matrix. Our main purpose is to obtain non-semisimple modular tensor categories for quantum groups at even roots of unity, wher…
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On finite GK-dimensional Nichols algebras over abelian groups Open
We contribute to the classification of Hopf algebras with finite Gelfand-Kirillov dimension,for short, through the study of Nichols algebras over abelian groups. We deal first with braided vector spaces overwith the generator acting as a s…
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Free Actions of Compact Quantum Groups on Unital C*-Algebras Open
Let F be a field, Γ a finite group, and Map(Γ, F ) the Hopf algebra of all set-theoretic maps Γ → F . If E is a finite field extension of F and Γ is its Galois group, the extension is Galois if and only if the canonical map E ⊗F E → E ⊗F M…
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Actions of some pointed Hopf algebras on path algebras of quivers Open
We classify Hopf actions of Taft algebras [math] on path algebras of quivers, in the setting where the quiver is loopless, finite, and Schurian. As a corollary, we see that every quiver admitting a faithful [math] -action (by directed grap…
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Interacting Hopf Algebras: the theory of linear systems Open
As first main contribution, this thesis characterises the PROP SVk of linear subspaces over a field k - an important domain of interpretation for circuit diagrams appearing in diverse research areas. We present by generators and equations …
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Algebraic renormalisation of regularity structures Open
We give a systematic description of a canonical renormalisation procedure of stochastic PDEs containing nonlinearities involving generalised functions. This theory is based on the construction of a new class of regularity structures which …
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Kappa-deformations: historical developments and recent results Open
I shall recall in historical perspective some results from nineties and show further how $\kappa$-deformed symmetries and $\kappa$-Minkowski space inspired DSR (Doubly of Deformed Special Relativity) approach proposed after 2000. As very r…
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Associative realizations of the extended Snyder model Open
The star product usually associated to the Snyder model of noncommutative\ngeometry is nonassociative, and this property prevents the construction of a\nproper Hopf algebra. It is however possible to introduce a well-defined Hopf\nalgebra …
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Hopf algebra gauge theory on a ribbon graph Open
We generalize gauge theory on a graph so that the gauge group becomes a finite-dimensional ribbon Hopf algebra, the graph becomes a ribbon graph, and gauge-theoretic concepts such as connections, gauge transformations and observables are r…
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Finite-dimensional pointed or copointed Hopf algebras over affine racks Open
We study the pointed or copointed liftings of Nichols algebras associated to affine racks and constant cocycles for any finite group admitting a principal YD-realization of these racks. In the copointed case we complete the classification …
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Infinity Operads and Monoidal Categories with Group Equivariance Open
This monograph provides a coherent development of operads, infinity operads,\nand monoidal categories, equipped with equivariant structures encoded by an\naction operad. A group operad is a planar operad with an action operad\nequivariant …
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Monoidal categorification of cluster algebras II Open
We prove that the quantum unipotent coordinate algebra $A_q(\mathfrak{n}(w))\ $ associated with a symmetric Kac-Moody algebra and its Weyl group element $w$ has a monoidal categorification as a quantum cluster algebra. As an application of…