Papers

Let $G$ be a simply connected simple complex algebraic group. It is proved by Oya, Qin, and Yakimov that the quantized coordinate ring $\mathcal{O}_q(G)$ is generated by generalized quantum minors, and therefore carries a quantized cluster algebra structure, for all $G$ but type $F_4$. In this article, we settle the $F_4$ case by an argument uniform across $G_2$, $F_4$, and $E_8$. The main idea is to bootstrap the existing proof in type $E_8$, which relies on Lusztig's canonical basis of the quantum adjoint representation, and replace it with the crystal combinatorics of the quasi-minuscule representation. As a consequence, we prove that $\mathcal{O}_q(F_4)$ also has a quantized cluster algebra structure.

Given a spherical finite tensor category C, not necessarily semisimple, and a two-sided modified trace on its projective ideal, we define an open-closed three-dimensional topological field theory with values in vector spaces. The bordism category has as morphisms three-dimensional bordisms with corners, whose boundary is partitioned into the gluing boundary, parametrised by source and target surface, and the unparametrised free boundary. The free boundary is equipped with an embedded C-coloured graph satisfying an admissibility condition. Our construction starts from a new three-manifold invariant based on the multi-handlebody invariant of [arXiv:1809.07991] and the chromatic maps of [arXiv:2302.04509]. The open-closed topological field theory is then obtained via the universal construction.

We develop the cellular structures for the endomorphism algebras of cyclotomic $q$-webs, which form a new family of quantum algebras sitting in between cyclotomic Hecke algebras and cyclotomic $q$-Schur algebras. We show that the Grothendieck group of a module category of the cyclotomic $q$-webs for $q$ generic or a root of unity is isomorphic to an integrable highest weight module over quantum affine $\mathfrak{gl}_p$; moreover, the isomorphism maps the classes of projective indecomposable modules to the canonical basis. This substantially generalizes the classic works of Lascoux-Leclerc-Thibon, Ariki, and Varagnolo-Vasserot.

We establish a Duflo–Kontsevich-type theorem for differential graded (dg) manifolds. Specifically, we prove that the Hochschild–Kostant–Rosenberg maps twisted by the square root of the Todd class realize an isomorphism between the Tamarkin–Tsygan calculus and the Cartan calculus of the dg manifold. At the level of cohomology, this confirms the Kontsevich–Shoikhet conjecture formulated in arXiv:math/9812009.

Topological defects in quantum field theories are believed to assemble into higher categories with extra structure. This has been made precise for defects in $2$- and $3$-dimensional oriented topological quantum field theories by Davydov, Kong, and Runkel and by Carqueville, Meusburger, and Schaumann, respectively. In this paper we study the extra structure present on these categories when the topological field theory is additionally reflection positive. We define reflection defect TQFTs as symmetric monoidal functors out of a defect bordism category that intertwine orientation reversal with complex conjugation; they are reflection positive if they satisfy an additional positivity condition. Our main result is that in two dimensions the bicategory of defects $\mathcal{T}_\mathcal{Z}$ associated to a reflection defect TQFT carries the natural structure of an $O(2)$-dagger bicategory, a structure we define explicitly. If the theory is reflection positive, $\mathcal{T}_\mathcal{Z}$ can be equipped with additional structure closely related to the definition of a 3-Hilbert space (the two agree up to some finiteness and completeness conditions).

In this short article we verify the conjecture proposed by P. M. Sołtan and the author, by proving that a second countable compact quantum group whose scaling automorphisms are inner, must be of Kac type.

A $q$-deformed real number, or “$q$-real”, was defined by Morier-Genoud and the second author. When $x\in\mathbb{R}$ such that $x\geq0$, the $q$-analogue $[x]_q$ is a power series with integer coefficients in one formal variable~$q$. In general a $q$-real is a formal Laurent series. The main goal of this paper is to study the coefficients of $q$-reals as functions on~$\mathbb{R}$ and give a combinatorial interpretation of these coefficients. This allows us to prove a conjecture studied by several authors stating that the $q$-deformed golden ratio has the smallest radius of convergence among the radii of the $q$-reals associated with positive real numbers. This is a $q$-analogue of the classical Hurwitz theorem. Our approach is combinatorial. We prove that for every real number $x$ in the interval $(1,2)$ the absolute value of each coefficient of the power series representing the $q$-real $[x]_q$ is dominated by the absolute value of the corresponding coefficient of the $q$-deformed golden ratio. The main notion is a certain collection of ordered rooted trees associated with a $q$-real. We prove that the golden ratio corresponds to a universal class of trees.

We prove that two finite-dimensional linear representations of the braid group are isomorphic. One representation comes from the monodromy of the boundary Casimir connection, and the other comes from the $\iota$quantum Weyl group. Both representations are defined for any split symmetric pair $\mathfrak{k}\subset \mathfrak{g}$ and for any integrable representation of $\mathfrak{k}$. Our proof uses $(O_m,\mathfrak{so}_{2n})$ spin Howe duality, so it is only for the pair $\mathfrak{so}_m\subset \mathfrak{sl}_m$.

We construct relative modular categories from the weighted representation theory of the unrolled quantum group of the orthosymplectic Lie superalgebra $\mathfrak{osp}(2 \vert 2n)$.

Much work in knot theory has consisted of categorifying, and thereby strengthening, knot invariants. We take the opposite approach: decategorification, more commonly called combinatorics. We introduce a technique that relates the dual braid group generators, the Hecke images of pure braids, and factorization problems in reflection groups to knot invariants. We prove that the $(a,z{=}0)$-HOMFLYPT polynomial can be computed as a solution to such a problem. This technique was motivated by Coxeter–Catalan combinatorics. For example, we give a new proof of EL-shellability of the noncrossing partition lattice using the image of the full twist in the Hecke algebra; in a surprising sort of combinatorial reciprocity, its inverse computes the homotopy type. Similarly, noncrossing partitions naturally arise from our construction applied to positive powers of the full twist, while cluster complexes come from the same construction applied to negative powers. In crystallographic type, we exploit the conjugacy of all Coxeter elements to give the first reflection subword models for rational noncrossing Catalan objects. Our reciprocity gives two models: one generalizing noncrossing partitions, and one generalizing clusters. Applying the same method produces two (rational) noncrossing parking models.

A fusion category $\mathcal{C}$ is said to be $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$-quadratic if the group $G(\mathcal{C})$ of invertible objects is isomorphic to $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, and the remaining simple objects form an orbit under the action of $G(\mathcal{C})$. In this paper, we give a partial classification of $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$-quadratic fusion categories of rank six. More precisely, we show that its Grothendieck ring $\mathcal{K}_0(\mathcal{C})$ must be one of nine fusion rings if the fusion rule multiplicities are less than $20$, and the categorifications of five of them are previously known. We prove that one of the last four fusion rings can be realized as de-equivariantization of a near-group fusion category of type $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}+8$.

We study the linearisation of formal vector fields and homotopy algebras using graded coalgebras. We prove that a formal vector field is linearisable if and only if it satisfies a splitting property, and we give an explicit recursive construction of a linearising coordinate transformation. This provides a direct coalgebraic proof of Bandiera's characterisation of linearisable, equivalently homotopy abelian, $L_{\infty}[1]$ algebras. We establish an associative analogue, characterising linearisability of $A_{\infty}[1]$ algebras by a splitting property for one-sided Hochschild cochains. We also examine the corresponding splitting property for standard Hochschild cochains and show that it is equivalent to $A_{\infty}[1]$ commutativity in the sense of Briggs and Gélinas. Finally, we establish a corresponding criterion for formal endomorphisms and give sufficient conditions for splitting that recover Basto-Gonçalves' linearisation theorem for admissible formal vector fields.

The tube algebra, which carries the structure of a $C^*$ weak Hopf algebra, is a fundamental tool for characterizing topological excitations in topological phases. In this work, we generalize the tube algebra framework to codimension-2 defects in $(2+1)$D gapped phases described by Turaev–Viro–Barrett–Westbury TQFTs, with particular emphasis on Levin–Wen string-net models. We show that such codimension-2 defects are described by a defect tube algebra, which naturally carries the structure of a comodule algebra over the weak Hopf tube algebra associated with the topological excitations. Boundary and domain wall defects are then characterized by the representations of the corresponding boundary and domain wall defect tube algebras. In particular, a domain wall defect tube algebra can be regarded as a generalized Drinfeld double of the tube algebras associated with the two adjacent boundary defects. More generally, for a domain wall joining $N$ bulk phases, the associated domain wall defect tube algebra can be viewed as an $N$-tuple algebra. This perspective extends naturally to general $k$-defects, namely codimension-2 defects joining $k$ codimension-1 defects, for which the resulting generalized tube algebra carries the structure of a multicomodule algebra.

We show that every symmetric fusion category over $\mathbb{R}$ is equivalent to the category of finite-dimensional semi-linear representations of a $\mathbb{Z}_2$-graded finite super group. The proof uses Galois descent for tensor categories over $\mathbb{C}/\mathbb{R}$, reducing the classification to semi-linear $\mathbb{Z}_2$-actions on symmetric fusion categories over $\mathbb{C}$. As a further structural result, we establish a Tannaka-Krein type correspondence between symmetric fusion categories over $\mathbb{R}$ and finite groupoids with a $\mathbb{Z}_2 \times \mathrm{B} \mathbb{Z}_2$-action. This gives a complete real analogue of Deligne's classification result.

We prove a number of Jacobson/solvable/nilpotent-radical (co)invariance results for finite-dimensional associative or Lie (co)module-algebras over Hopf algebras $H$, provided the characteristic, if positive, is large relative to the dimension of the algebra and the (co)actions satisfy trace-preservation conditions automatic when $H$ is involutory. This generalizes a number of such radical-invariance criteria in the literature, due to A. Gordienko, V. Linchenko, Pagon-Repovš-Zaicev and others, providing a uniform enriched-categorical framework for those results.

Composition algebras in symmetric tensor categories are defined. The classical results on standard unital composition algebras, as well as the classification of the unital composition superalgebras, are reviewed in light of this definition, and the unital composition algebras in the category $Ver_4^+$ (the counterpart to the category of vector superspaces in characteristic 2) are classified.

In this paper, we introduce the notion of four-angle Hopf modules over a BiHom-Hopf algebra $H$. We show that the category ${}_{H}^{H}\mathfrak{M}_{H}^{H}$ of BiHom-four-angle Hopf modules admits a strict monoidal category structure with respect to either the BiHom-tensor product $\otimes_{H}$ or the BiHom-cotensor product $\square_{H}$ as its monoidal product. We prove that the category $\mathcal{YD}_{H}^{H}(m,n,p,q)$ of BiHom-$(m,n,p,q)$-Yetter-Drinfel'd modules with parameters $m,n,p,q\in\mathbb{Z}$ forms a strict braided monoidal category equipped with a new monoidal product. Furthermore, we establish monoidal equivalences between the monoidal categories $\mathcal{YD}_{H}^{H}(m,n,p,q)$ and ${}_{H}^{H}\mathfrak{M}_{H}^{H}$, where ${}_{H}^{H}\mathfrak{M}_{H}^{H}$ carries either $\otimes_{H}$ or $\square_{H}$ as its monoidal product. Finally, we construct braiding structures for the monoidal categories $({}_{H}^{H}\mathfrak{M}_{H}^{H},\otimes_{H})$ and $({}_{H}^{H}\mathfrak{M}_{H}^{H},\square_{H})$.

In this paper, we study modular fusion categories whose mapping class group representations are trivial on the Torelli groups, with particular emphasis on the congruence properties of the resulting representations of $\mathrm{Sp}(2g, \mathbb{Z})$. We prove that, for any $g \ge 3$, the Torelli group is contained in the kernel of the genus-$g$ mapping class group representation $\rho_g$ associated with a modular fusion category $\mathcal{C}$ if and only if $\mathcal{C}$ is pointed. This refines the corresponding result in \cite{MW25}. We also characterize the modular fusion categories for which the genus-$2$ Torelli group is contained in $\ker \rho_2$; in particular, categories with isotropic adjoint subcategories belong to this class. Finally, we give a complete classification of modular fusion categories with isotropic adjoint subcategories, and determine the groups $G$ satisfying $\mathcal{Z}(\mathrm{Vect}_G)$ is 2-Torelli trivial but $\mathcal{Z}(\mathrm{Vect}_G)_{\mathrm{ad}}$ is not isotropic are precisely 2-Engel groups of nilpotent class 3.

We study the simplicial boundary $\partial\mathcal{FS}$ of Culler-Vogtmann Outer space via thickenings defined by graph-theoretic connectivity. Let $C'$ be the subcomplex of the free splitting complex obtained from $\partial\mathcal{FS}$ by adding all stable graphs that are not $3$-edge connected, together with their faces. We prove that the inclusion $\partial\mathcal{FS}\hookrightarrow C'$ is $(2n-3)$-connected. The proof shows, more precisely, that adding graphs with cut vertices is a homotopy equivalence, while the only non-contractible fibres in the $2$-bond thickening occur over $\theta$-graphs. The result gives further evidence that $\partial\mathcal{FS}$ may be $(2n-3)$-spherical, an $\operatorname{Out}(F_n)$-analogue of Rognes's connectivity conjecture for the common basis complex. It also gives a topological, universal-cover perspective that unifies several existing results about the commutative graph complex.

Nichols algebras are among the most important and mysterious objects in quantum algebra. We continue the systematic program of studying them through their point modules. We study truncated point modules of Nichols algebras associated with racks and 2-cocycles over non-abelian groups. We determine the truncated point spaces for all currently known finite-dimensional Nichols algebras arising from simple Yetter-Drinfeld modules over groups. In particular, for all such algebras the space of 3-truncated point modules is empty. We further find all truncated point modules for other important families, including affine racks equipped with the constant 2-cocycle -1, and for the non-trivial indecomposable rack of size three with the constant 2-cocycle a cubic root of one.

Self-dual corings and self-dual algebra extensions are classified and these (self-)dualities are compared with Ringel (self-)duality, revealing fundamental differences. To each coring $\mathcal C$, two algebras are associated, known as the left and the right dual algebra of $\mathcal C$. It is shown that these two algebras coincide in a natural way if and only if $\mathcal C$ is a Frobenius coring. Various other equivalent characterisations of $\mathcal C$ being self-dual are given, in terms of certain algebra extensions being Frobenius extensions and in terms of certain forgetful or restriction functors being Frobenius functors. Ringel self-duality however is shown to be rather different, for which a homological explanation is given.

Let $M_\gamma=T^2\times_\gamma S^1$, where $\gamma\in\mathrm{SL}_2(\mathbb{Z})$ is hyperbolic. For every $N\geq2$, we compute the empty-skein, or quantum-torus, direct summand in Kinnear's decomposition of the $\mathrm{SL}_N$-skein module of $M_\gamma$, thereby answering his centralizer question for this summand in the hyperbolic case. Its dimension is expressed in terms of the periodic Nielsen numbers $N_k=|\det(I-\gamma^k)|$ and the visible Nielsen numbers $V_{e\mathbb{Z}^2}(f_\gamma)$ carried by torsion in the Weyl coinvariant lattices. Only moduli $e\mid N$ occur, so the rank-$N$ summand is determined by $N_1,\ldots,N_N$ together with the visible Nielsen numbers at the divisors of $N$. On the $\mathrm{GL}_N$ permutation lattice these coinvariants are torsion-free, so no such correction occurs; the observer corrections arise precisely upon passage to the $\mathrm{SL}_N$ character lattice. For $N=3$, we obtain an explicit formula with the single correction $V_{3\mathbb{Z}^2}(f_\gamma)$, and construct infinitely many pairs of non-homeomorphic hyperbolic torus bundles whose $\mathrm{GL}_N$-skein-module dimensions agree for every $N$, while their $\mathrm{SL}_3$ quantum-torus summands differ in dimension by six. These pairs also have identical periodic Nielsen data and finite-cover visibility profiles at every iterate. We do not compute the additional endomorphism-algebra summands of the full $\mathrm{SL}_N$-skein module.

We show that the computational complexity of Reshetikhin-Turaev invariants of simply colored links is preserved when their underlying ribbon fusion categories are related by the zesting construction. Zesting modifies an $A$-graded ribbon fusion category $\mathcal{C}$ with additional algebraic data $\zeta$ to produce a new category $\mathcal{C}^{\zeta}$ whose link invariants are known to differ from those of $\mathcal{C}$ by an invariant of $A$-colored links $\mathcal{J}_{\zeta}$ depending only on $\zeta$. Building on this understanding and on earlier work on quantum braid group representations under zesting, our result suggests how zesting contributes to the organization of (2+1)D topological quantum field theories and topological phases into complexity-theoretic hierarchies. To prove our main result we develop a local formalism analogous to the Reshetikhin-Turaev construction to compute tangle invariants $\mathcal{J}_{\zeta}(T)$, which leads to a polynomial time algorithm to compute invariants of links $\mathcal{J}_{\zeta}(L)$. A byproduct of our construction is an identification (up to a sign) of the link invariants $\mathcal{J}_{\zeta}(L)$ as rack cocycle invariants, which may be of independent interest. Our formalism also extends to define invariants of closed $3$-manifolds with $A$-structure and we obtain similar complexity results for homotopy quantum field theories built from $A$-modular fusion categories.

Analytic calculations of zero- and single-scale quantities in perturbative quantum field theory result into special numbers and functions, the first of which have been revealed during the last decades. These are generalizations of the polylogarithm in form of Kummer-Poincaré iterative integrals over special alphabets and extensions thereof.With growing order in the coupling constant, the polylogarithms, Nielsen integrals, the iterated integrals over linear denominator terms, cyclotomic letters, letters induced by quadratic forms, square-root valued letters, and more general functions contribute. For the nested sums we consider nested harmonic sums, generalized harmonic sums, nested sums implied by quadratic forms, cyclotomic harmonic sums, and nested sums containing central binomials. We construct the $q$-extensions of these special functions and of the nested sums, which are associated to them by the series expansion at $x=0$, and their Mellin transform in the $q$-free case. These functions are expected to play a role in perturbative calculations in the case of $q$-deformed commutation relations. For the simpler function spaces closed form solutions are presented. For more involved alphabets we present the algorithmic steps leading to the $q$-extension for the individual cases. We also derive the determining differential and difference equations of these higher transcendental functions. The $q$-extended special functions are quite different form the corresponding $\mu$-extended functions.

We show that the category of twisted/untwisted representations of the Heisenberg conformal net $\text{Heis}$ is a continuous Tambara-Yamagami category for the group $\mathbb{R}$. We compute the associators and the $\mathbb{Z}/2$-crossed braiding. By taking a $\mathbb{Z}/2$-equivariantization, we obtain an explicit computation of the braided continuous tensor category of representations of the fixed-points conformal net $\text{Heis}^{\mathbb{Z}/2}$. This provides the first explicit computation of a category of representations of a conformal net containing irreducible representations whose tensor product is a direct integral of irreducible representations.

Let $N^k(\mathfrak{sl}_2)$ be the universal parafermion vertex algebra and $N_k(\mathfrak{sl}_2)$ its simple quotient. We classify all irreducible highest weight $N^k(\mathfrak{sl}_2)$-modules for every $k\neq 0$. We give a presentation of Zhu's algebra $A(N^k(\mathfrak{sl}_2))$ as a quotient of a polynomial algebra in four variables by an ideal generated by three explicit polynomials. This presentation also shows that $A(N^k(\mathfrak{sl}_2))$ is a free module of rank three over the polynomial subalgebra generated by the classes of the fields of weights two and three. The irreducible highest weight $N^k(\mathfrak{sl}_2)$-modules are parametrized by a two-parameter family $L_k[x,y]$, $(x,y)\in\mathbb C^2$, constructed using a free-field realization. We also prove that each $N^k(\mathfrak{sl}_2)$-module $L_k[x,y]$ can be realized as an $N^k(\mathfrak{sl}_2)$-submodule of an irreducible weight module $M$ for the universal affine vertex algebra $V^k(\mathfrak{sl}_2)$. At non-integral admissible levels, we prove that $L_k[x,y]$ is an $N_k(\mathfrak{sl}_2)$-module if and only if the associated $V^k(\mathfrak{sl}_2)$-module $M$ is an $L_k(\mathfrak{sl}_2)$-module. This gives the classification of irreducible highest weight $N_k(\mathfrak{sl}_2)$-modules at all non-integral admissible levels. We also classify the irreducible highest weight modules for the simple parafermion algebra $N_{-2}(\mathfrak{sl}_2)$ at the critical level. At positive integral levels, the irreducible $N_k(\mathfrak{sl}_2)$-modules were previously classified by Arakawa, Lam, and Yamada.

Let $k$ be an algebraically closed field of characteristic $p\geq 5$, and let $\mathrm{Ver}_p^+$ be the even part of the Verlinde fusion category $\mathrm{Ver}_p$, the semisimplification of $\mathrm{Rep}_k(\mathbb Z/p)$. Let $\mathfrak g$ be a linearly reductive Lie algebra in $\mathrm{Ver}_p^+$, i.e., one whose finite-dimensional representations are semisimple. A basic class of examples is obtained by semisimplifying a simple Lie algebra over $k$ equipped with the action of $\mathbb Z/p$ by a principal unipotent element, when $p$ exceeds its Coxeter number. We prove that $\mathfrak g$ is invariantless, i.e., that the unit object is not a summand of $\mathfrak g$. For odd $m$ with $3\leq m\leq p-2$, set $\mathfrak{g}_m:=\operatorname{Hom}_{\mathrm{Ver}_p^+}(L_m,\mathfrak g)$ and $E_{\mathfrak g}:=\bigoplus_{3\leq m\leq p-2,\ m\ {\rm odd}}\mathfrak g_m^{(1)}[m]$, where $(1)$ denotes Frobenius twist. Our main result is an isomorphism of graded algebras $H^\bullet_{\mathrm{CE}}(\mathfrak g)\cong\bigwedge^\bullet E_{\mathfrak g}^*$. We also identify this algebra with the de Rham cohomology $H^\bullet_{\mathrm{dR}}(G)$ of the group scheme $G=\exp(\mathfrak g)$ and show that the induced graded Hopf algebra structure agrees with the standard one on the exterior algebra. Moreover, if $V$ is a simple $\mathfrak g$-module on which $\mathfrak g$ acts nontrivially, then $H^\bullet_{\mathrm{CE}}(\mathfrak g,V)=0$. Hence for every finite-dimensional $\mathfrak g$-module $V$ one has $H^\bullet_{\mathrm{CE}}(\mathfrak g,V)\cong\bigwedge^\bullet E_{\mathfrak g}^*\otimes V^{\mathfrak g}$. This recovers the theorem of Borel and Chevalley on the cohomology of complex semisimple Lie algebras and its analogue in sufficiently large positive characteristic. We also prove similar results for relative cohomology.