Papers

We compute minimal $\mathfrak{sl}_N$ complexes for oriented rational tangles, denoted $[\![\tau_{u/v}^{O}]\!]_N$, in terms of the topological Fukaya category of the $3$-punctured disk. This demonstrates that the complexes stabilize, allowing us to define a HOMFLY-PT homology for rational tangles.

We study right comodule algebras and their connections with Hopf bi-Galois objects and right coideal subalgebras. For a Hopf $2$-cocycle $\sigma$ on $K$ and a unital right-colinear algebra map $\Gamma:A\to K$, we show that $\Gamma:A_\sigma\to K^\sigma$ is an algebra map precisely when $\sigma$ is trivial on $\operatorname{im}\Gamma\otimes\operatorname{im}\Gamma$. We describe comodule-algebra morphisms $A_\sigma\to K^\sigma$ by characters and establish injectivity criteria using the coradical and extensions of characters to convolution-invertible functionals. For coactions on Hopf quotients projecting to the coproduct, character-evaluation maps are injective, and equality of their images is determined by a unique algebra character. For the $\imath$-Hopf algebras $H^\imath_\tau$ of Chen–Lu–Pan–Ruan–Wang, with $H$ having a bijective antipode and a Hopf pairing, we obtain a Hopf bi-Galois realization $H^\imath_\tau\hookrightarrow F_\sigma(H\otimes H^{\mathrm{cop}})$. We identify algebra characters with nonzero $\tau$-twisted compatible maps $\chi$ and prove that the corresponding morphisms $\xi_{\tau,\chi}$ of Chen–Lu–Pan–Ruan–Wang are injective, yielding right coideal realizations in $(H\otimes H)^\imath$. We classify the $H^\imath_\tau$ up to comodule-algebra isomorphism and determine when these realizations have the same image.

We study the circle-relation subspace $O_n^\circ(V)$ in the defining relations of the level-$n$ Zhu algebra of a vertex operator algebra $V$. We assemble the residue relations generating $O_n^\circ(V)$ into generating functions and obtain an explicit recursion that replaces every sufficiently deep mode by retained modes with indices at least $-2n-1$, modulo $O_n^\circ(V)$. This recursion is encoded in a left truncated field $Y_L$. Since such mode replacements are valid only in the leftmost position of an operator product, we prove the replacement, derivative, commutator, and one-step iterate identities for $Y_L$ needed for finite-length calculations. For a finite homogeneous strong generating set $S$, we define the finite-length circle subspaces $O_n^\circ(N,S,V)$ and their plus subspaces. We formulate a finite-length conjecture under an explicit auxiliary-degree hypothesis and a separate termwise conjecture for the correction residues of one-sided iterates. We prove reductions to the $k=0$ strip and to last vacuum mode $-1$, recover the length-one relation spaces, and give explicit Virasoro computations at word length two for $n=1$ and $n=2$. The $Y_L$ identities are the input for the companion paper (Part II).

We construct quantum codes from codewords of a cyclic code and the minors of a polynomial matrix. The construction uses a two-row, three-column matrix whose minors yield a second matrix satisfying the CSS orthogonality condition. We derive the code dimension from the greatest common divisors of the entries and minors, with a simplified formula for odd lifting factors. A residue characterization identifies additional common factors of the minors and leads to a method for controlling the dimension through polynomial selection. We also give a condition for changing the minor generator without changing the code. Explicit polynomial constructions yield 13 quantum codes with verified minimum distances, including a $[[255,80,12]]$ code of rate $80/255$.

We study two explicit constructions of binary Calderbank–Shor–Steane quantum low-density parity-check (QLDPC) codes with check weight six. The first uses a fixed circulant shift pattern and yields parameters $\code{6N}{2N+2}{6}$ with Tanner graph girth twelve for $N=31$ and every $N\ge37$. A graph incidence interpretation gives the ranks and girths, and a uniform weight six logical operator establishes the exact distance. The second construction uses fixed matrix polynomials with circulant and affine permutation coefficients. Here, a polynomial commutation identity guarantees orthogonality for every cyclic lift. Four realizations give $[[192,16,8]]$, $[[288,16,12]]$, $[[384,16,14]]$, and $[[576,32,20]]$, with Tanner girth eight and computationally certified distances. We give reproducible construction data, distinguish classical cycle distance from quantum distance, and identify the limitations of both families.

We calculate the chromatic spherical invariant $\mathscr{K}_{\mathrm{Rep}(H)}(L(p,q))$ of lens spaces $L(p, q)$ for arbitrary finite-dimensional unibalanced, unimodular Hopf algebras $H$. In the case of the index-2 Nichols Hopf algebra $\mathcal{K}_2$ and $q=1$, we find that the invariant of $L(p, 1)$ depends only on the parity of $p$ for $p\geq 1$. We also describe the duals $\mathcal{S_I}(M)^*$ of the admissible skein modules of handlebodies $M=H_g$ and of $M=L(p, q)$ for arbitrary tensor ideals $\mathcal{I}$ of a ribbon tensor category. In particular, $\mathcal{S_I}(L(p, q))^*$ corresponds with cyclic traces on $\mathcal{I}$ satisfying a property reminiscent of the modified trace condition.

Chiral topological order is incompatible with strictly two-dimensional local commuting-projector Hamiltonians on finite-dimensional tensor-product Hilbert spaces. Invertible subalgebras (ISAs) provide an alternative framework. We construct an ISA supporting chiral semions, together with its anti-semion commutant, on a square lattice with one qubit per edge. We give explicit non-Pauli generators for both algebras and prove that together they generate all local operators on the Hilbert space. Starting from the $\mathbb{Z}_4$ toric code, we condense the boson $e^2m^2$ and identify the semion and anti-semion algebras. We then decompose each $\mathbb{Z}_4$ qudit into two qubits and use a finite-depth circuit to disentangle one qubit per edge fixed by the condensation constraints. Pumping the semion ISA along the $z$ direction defines an explicit three-dimensional non-Clifford quantum cellular automaton (QCA). We show that pumping eight times is equivalent to a translation followed by a finite-depth quantum circuit with ancillary qubits. We also give an explicit finite-depth circuit relating this construction to the semion QCA previously obtained from topological quantum field theory. A similar condensation construction yields a $U(1)_4$ ISA on a square lattice. Together with the semion and earlier odd-prime ISAs, finite stacking realizes every pointed Witt class of bosonic Abelian anyon theories. Finally, we extend both constructions to all spatial dimensions $d=4k{-}2$, with one qubit or $\mathbb{Z}_4$ qudit on each $(2k{-}1)$-cell for the semion or $U(1)_4$ ISA, respectively.

We introduce a diagram algebra $\mathcal{K}_L$ based on Kuperberg's $A_2$-webs and prove a Schur–Weyl duality $\mathcal{K}_L \cong \operatorname{End}_{U_q(\mathfrak{sl}_3)}((V^+ \oplus V^-)^{\otimes L})$, where $V^\pm$ are the fundamental representations of $U_q(\mathfrak{sl}_3)$. Thus, $\mathcal{K}_L$ is a natural $A_2$ analogue of the Temperley–Lieb algebra. We construct an explicit cellular structure for $\mathcal{K}_L$, with the web basis as its cellular basis and cell modules indexed by dominant $U_q(\mathfrak{sl}_3)$-weights. The bases of these modules are parametrized by dominant lattice paths via a recursive growth algorithm. Using this structure, we determine local generators of $\mathcal{K}_L$ and establish web formulations of the first and second fundamental theorems of invariant theory. In particular, we show that the braid-group image, augmented by local idempotents, does not generate $\mathcal{K}_L$, and identify an additional local generator. Finally, we derive a bimodule decomposition of $(V^+ \oplus V^-)^{\otimes L}$, yielding a complete set of irreducible $\mathcal{K}_L$-modules for formal or generic $q$.

We prove the Labastida–Mariño–Ooguri–Vafa conjecture for oriented links with arbitrary integer framings. Consequently, the framed reformulated invariants have finite integral LMOV expansions for every integer framing vector.

The modular data of the doubles of the Haagerup–Izumi categories is a grafting of two finite metric groups (conjectured by Evans–Gannon, proved for untwisted categories by Gannon–Schopieray–Yadav). Gang and Kim found non-unitary modular data of this shape in the topological twist of rank-$0$ $S$-fold theories, from the Bethe vacua of an abelianised gauge theory. The aim is to understand rigorously the significance of the handle-gluing and fibering operators, and to construct the Bethe vacua. For every hyperbolic once-punctured-torus monodromy $\varphi$, a word in $L$ and $R$, the two groups are $A_\mp=\coker(\varphi\mp1)$, with the torsion linking forms of the mapping tori of $\varphi$ and $-\varphi$. Over all words of trace $m^2+2$ these forms reproduce the Evans–Gannon table of Q-systems, one printed entry being corrected, and the distinct modular data in the list of Gannon–Schopieray–Yadav. The Bethe data are given in closed form by the word: the level matrix, the positive vacuum, $c_{\eff}=\#L$, and the handle-gluing values $2^{\#L}\lambda^{\pm1}|A_-||A_+|$ at the identity classes, $\lambda$ the stretch factor. Each kernel vector of $\varphi\mp1$ modulo $|A_\mp|$ at which an explicit trigonometric formula does not vanish gives a solution of the Bethe equations in closed form. Under the half-index dictionary, with the unit in the class of $\sigma V^*$, $c\equiv\#L-6\#R\pmod{24}$.

We give two types of examples where two non isomorphic finite quandles have isomorphic enveloping groups. This implies that two non-isomorphic finite quandles can share isomorphic representation categories (as additive and tensor categories simultaneously).

We study certain simple affine vertex algebras $L_{-1-h/6}(\mathfrak{g})$ from rank-one Argyres-Douglas theories, where $\mathfrak{g}$ is a non-simply-laced simple Lie algebra and $h$ is its Coxeter number. For $L_{-3}(F_4)$, we determine the maximal ideal of $V^{-3}(F_4)$, Zhu's algebra, and the associated variety. We also classify the irreducible highest-weight $L_{-3}(F_4)$-modules and determine the characters of its irreducible ordinary modules. The associated variety is $\overline{\mathbb{O}}_{\widetilde A_1}$, and the level $-3$ is totally collapsing for the orbit $\mathbb{O}_{\widetilde A_1}$. The Poisson coinvariants of the $C_2$-algebra of $L_{-3}(F_4)$ are isomorphic to $\mathbb{C}[\ell]/(\ell^5)$, where $\ell$ is induced by the conformal vector. Consequently, the characters of ordinary $L_{-3}(F_4)$-modules satisfy a fifth-order modular linear differential equation. We also give modular linear differential equations of orders five and eleven satisfied by the characters of ordinary modules for $L_{-2}(B_3)$ and $L_{-2}(G_2)$ respectively.

By equipping the category $\mathcal{M}$ of modules over a finite dimensional algebra with the action of a finite tensor category $\mathcal{C}$, we introduce a subcategory $\mathcal{PM}$ satisfying numerous remarkable properties: it is generating and cogenerating, functorially finite, closed under Auslander–Reiten translations (i.e. precluster tilting), and it computes the finitistic dimension. Through a functor category perspective, $\mathcal{PM}$ is identified with the $\mathcal{C}$-module functors that are projective, and we also prove a $\mathcal{C}$-equivariant version of the Auslander(–Solberg) correspondence. As examples, we show that $\mathcal{PM}$ contains only finitely many indecomposable modules for some tensor product algebras. For some family of 1-Iwanaga–Gorenstein algebras introduced by Geiss–Leclerc–Schröer, we show that $\mathcal{PM}$ is exactly the subcategory of $\tau$-locally free modules.

A tensor product is known for commutative Rota–Baxter algebras through a universal construction, but its underlying algebra need not be the ordinary tensor product $A\otimes B$ and can be substantially larger. We show that, after passing to augmented Rota–Baxter algebras, the ordinary tensor product itself carries a natural Rota–Baxter operator which makes the category monoidal with the usual associator and unit constraints. The resulting structure is intrinsically nonsymmetric: over a field of characteristic zero, this operator and its flip-conjugated counterpart are the only possibilities among linear combinations of five natural terms, while over any field the full category admits no braiding. Its comonoid objects are precisely Rota–Baxter cocycle bialgebras. Moreover, the free Rota–Baxter adjunction yields a bimonad and free cocycle bialgebras on ordinary bialgebras, with compatible Hopf structures for connected filtered inputs, in both the associative and commutative settings.

Furusho, Hirose and Komiyama reformulated the pentagon equation using an involution called the flip, which led to a new characterisation of the graded Grothendieck-Teichmüller group $\mathsf{GRT}_1$. We show that the flip is a reflection in the cyclic structure of the operad of framed Drinfeld-Kohno Lie algebras, with the framings removed. Applying the reflection twice to the pentagon gives the two-cycle relation for every Drinfeld associator. This relation is precisely the condition for cyclic compatibility on the associativity morphism, giving a new proof that $\mathsf{GRT}_1$ acts by cyclic automorphisms of parenthesized ribbon chord diagrams.

In this work, we study restricted Laistrygonian algebras over a field of positive characteristic $p$. We compute their Drinfeld double $\mathtt{D_p(q,a)}$ and give explicit presentations by generators and relations. We prove that $\mathtt{D_p(q,a)}$ is a pointed Hopf algebra. For $\mathtt{D_p(1,a)}$, we describe an abelian exact sequence of Hopf algebras and identify one of its quotients as a restricted enveloping Lie algebra. This exact sequence enables us to classify the finite-dimensional simple $\mathtt{D_p(1,a)}$-modules. We also introduce an infinite-dimensional Hopf algebra covering the restricted Drinfeld double and study its relation with the corresponding finite-dimensional case. Using this covering together with a Frobenius map, we construct a $3\times 3$ commutative diagram whose rows and columns are exact sequences of Hopf algebras. In particular, our approach provides a unified treatment of the Drinfeld doubles associated with restricted Laistrygonian algebras in arbitrary positive characteristic.

In this paper, we study derivation based noncommutative geometry of path algebras arising from quivers by identifying a Lie algebra of derivations from which a differential calculus is constructed. We show that properties of the differential calculus are intimately connected with the structure of the quiver. In particular, by introducing the notion of a separating quiver, we find a condition on the quiver which is equivalent to the module of differential forms being a projective bimodule. For such quivers, one can construct bimodule connections, as well as bimodule forms which serve as noncommutative analogues of the Riemannian metric. As an illustration of the above concepts, we present a class of quivers for which one can construct Levi-Civita connections, i.e. torsion free connections compatible with a bimodule form.

We study the rank-twelve potential modular datum denoted $12^{39,406}_{4,1276.}$ in the low-rank classification tables. Its global dimension is $(1287+351\sqrt{13})/2$, and its fusion ring agrees with that of the known rank-twelve Haagerup modular category. The datum satisfies the modular relations, has nonnegative integral Verlinde coefficients, and passes several familiar genus-one necessary tests. We prove, nevertheless, that it cannot be realized by a unitary modular tensor category. The obstruction already occurs in the representation of the three-strand braid group on $\operatorname{Hom}(C_{[10]},B^{\otimes3})$. Generalized Frobenius–Schur indicators determine the eigenvalues of the two-strand generator on this twelve-dimensional space. The central relation in $B_3$ then reduces the problem to a product of three prescribed unitary conjugacy classes. A sixfold symmetry reduces the $1302$ determinant-compatible cases to $217$ representatives. Rank-one inequalities leave only $17$ of these, and ten explicitly displayed degree-zero rank-two Schubert inequalities exclude the remaining cases. A violated inequality with gap $2/117$ is worked out explicitly. All calculations used in the obstruction are exact. The argument uses unitarity essentially and leaves non-unitary categorification open.

Let $p = 4q + 1$ be a prime integer, where $q$ is an odd prime. Let $L = \mathbb{Q}(\zeta_p)^+$ denote the maximal totally real subfield of the $p$-th cyclotomic field $\mathbb{Q}(\zeta_p)$. In this paper, we prove that if $U_L^+=U_L^2$ (i.e., every totally positive unit is a square in $L$), then for any non-trivial subfield $K \subseteq L$ and any non-trivial totally positive unit $\varepsilon \in U_K^+$, the absolute trace satisfies the sharp lower bound $\Tr_{K/\mathbb{Q}}(\varepsilon) \ge p - 4$. As an application, we show that a spherical fusion category $\C$ with global dimension $\dim(\mathcal{C})=p=4q+1$ is pointed, where $2$ is a primitive root modulo $q$, or $p=4q+1<2^{22}$ and $p\neq 29$.

We provide a simplified and more human-friendly version of the proof of the Deligne–Drinfeld conjecture recently given by OpenAI. No originality is claimed.

We prove that a Hopf algebra over an arbitrary field has a bijective antipode if either its left or right regular module has finite uniform dimension. In particular, this solves Skryabin's conjecture, posed in 2006, that every noetherian Hopf algebra has a bijective antipode. Starting from Skryabin's injectivity theorem for weakly finite Hopf algebras, we associate to each finite-dimensional subcoalgebra an essential left ideal of common denominators that is also a right coideal in the antipode image. An annihilator identity makes this image essential in the original algebra as a left module over the image, yielding finite left uniform dimension from finite right uniform dimension. A second denominator construction inside the antipode image, together with a restricted inverse-antipode identity, proves surjectivity.

Emil Artin asked whether the left and right dimensions of an extension of skew fields must agree. Cohn and Schofield gave counterexamples through elaborate constructions. It has been a long-standing open question whether there is any division ring, infinite-dimensional over its center, for which the left and right indices agree whenever at least one is finite. We give the first such examples. Call a division ring index-symmetric if every division subring with at least one finite index has equal finite indices. Using a duality formula for bimodules over division algebras, we prove that every twisted Calabi–Yau division algebra is index-symmetric. The examples include the Weyl skew fields and, more generally, the quotient division rings of rings of differential operators on smooth affine varieties in characteristic zero. Further examples are the quotient division rings of enveloping algebras of finite-dimensional Lie algebras, Ore extensions of twisted Calabi–Yau noetherian domains, quantum affine spaces, and group algebras of torsion-free polycyclic-by-finite groups.

Let $F$ be an algebraically closed field of characteristic $7$, and let $V=L(\frac12,0)_F$. For the degree-four singular vector $q=(L(-2)^2-2L(-4))\mathbf1$ in the universal vacuum vertex operator algebra $\bar V$, set $V_q=\bar V/I(q)$. We prove that the natural quotient map induces an isomorphism $A_1(V_q)\cong A_1(V)\cong F[T]/(T(T-4)(T-5))\cong F^3$. Explicit $O_1(V_q)$-relations and four families of triangular PBW straightening relations give the upper bound, while three one-dimensional homogeneous subspaces give the matching lower bound.

We investigate set-theoretic solutions of the Yang–Baxter equation from a categorical perspective. The paper has two main objectives. On the one hand, several existing constructions and classification results in the literature are recast in categorical terms, thereby showing that certain hypotheses previously imposed on such solutions are in fact superfluous. On the other hand, the categorical approach leads naturally to new classes of Yang-Baxter solutions. Our approach consists in identifying suitable subcategories of the category $\mathcal{YB}$ of Yang-Baxter solutions and providing categorical characterisations of the solutions they encode. As an application, we prove that the category of Płonka bi-magmas is isomorphic to the subcategory of $\mathcal{YB}$ consisting of solutions of the Baaj-Long-Skandalis equation. Along the way, we introduce and study a new class of Lyubashenko type solutions and examine the extent to which non-degenerate bijective solutions of the braid equation arise from skew left braces.

We introduce and study trigonometric weight functions of type $B$ and type $C$ arising from integrable lattice models. These functions give $q$-deformations of the factorial $K$-theoretic Schur $P$- and $Q$-functions of Ikeda and Naruse, with the $P$-functions specialized to the type $B$ case. We realize them as specializations of partition functions of a six-vertex model with triangular boundary conditions and derive their algebraic properties using the Yang-Baxter equation and properties of the $R$- and $K$-matrices. We introduce normalized weight functions and establish diagonal evaluation formulas, divisibility properties, and bounds on Newton polytopes analogous to those in type $A$, together with recursive relations and stability. We further derive branching formulas relating these functions to type $A$ weight functions, and prove equal-cardinality orthogonality and all-cardinality biorthogonality relations.

Consider an $n$-dimensional sum of the form $S_k=\sum{q^{f({\bf z})}}$ over lattice points ${\bf z}$ which lie in $K{\mathcal P}$, the $K$ scaling of an $n$-dimensional rational polytope $\mathcal P$. We formulate a conjecture on the structure of $S_K$ considered for $K$ prime, as a power series in $h=q-1$ modulo the $K^{th}$ cyclotomic polynomial, and in particular the existence of a decomposition in powers of the Gauss sum $\mathbb G$, up to the $n^{th}$, whose coefficients are almost independent of $K$, where the leading coefficient is related to the volume of $\mathcal P$ and the coefficient of ${\mathbb G}^r$ has contributions from the codimension $r$ skeleton of $\mathcal P$. The conjecture is proved for $n=1,2$.

Gauging fermion parity in a $(2+1)$-dimensional fermionic topological order requires lifting its symmetries to the resulting bosonic theory. We give a direct tensor-categorical proof that, for a slightly degenerate braided fusion category $\mathcal{C}$ and a fixed minimal nondegenerate extension ${\cal E}$, the restriction homomorphism $r\colon\operatorname{Aut}_{\mathrm{LR}}(\mathcal{E})\to \operatorname{Aut}_{\mathrm{LR}}(\mathcal{C})$ on classes of complex-linear locality-respecting (LR) braided autoequivalences is surjective with central kernel $\mathbb{Z}/2$, generated by fermion attachment. Thus each symmetry class has exactly two LR lift classes, related by attaching the fermion to every vortex. We apply graded Morita theory to reduce the kernel to two module types on the vortex sector, corresponding to tensoring with the vacuum or the fermion. We then prove relative rigidity to show that each type determines a unique LR class, which are distinguished by fermionic self-braiding even when their actions on simple-object labels coincide. We prove surjectivity by extending graph algebras in $\mathcal{C}^{\mathrm{rev}}\boxtimes\mathcal{C}$ to Lagrangian algebras in $\mathcal{E}^{\mathrm{rev}}\boxtimes\mathcal{E}$. Our results require neither unitarity nor a ribbon structure. We illustrate the lifting ambiguity in the Laughlin state, the $SO(3)_3$ theory and the Ising topological order.

We construct a chiral algebra $U$ of central charge eight inside the lattice vertex operator algebra $V_{E_6\oplus A_2}$. It is generated by two commuting affine $\mathfrak{sl}_2$ current algebras at levels $1$ and $39$, the stress tensor of the lattice theory, and a single weight-two field built from three lattice exponentials. We announce that $U$ is strongly rational and completely unitary, and that $Rep(U)$ is equivalent, as a unitary ribbon category, to the Drinfeld center $Z(\mathcal{H}_3)$ of the Haagerup fusion category $\mathcal{H}_3$. Its characters are the $\gamma=0$ Evans–Gannon vector. From $U$ we obtain a diagonal full CFT, and also a holomorphic $E_8$ extension on which $\mathcal{H}_3$ acts by topological defects.

We announce an explicit construction of a unitary, strongly rational vertex operator algebra $U$ of central charge $8$ inside $V_{E_6\oplus A_2}$, with a ribbon equivalence $\mathrm{Rep}(U)\simeq Z(\mathcal H_3)$ realizing the Haagerup double. The vertex algebra is generated by six affine currents and three specified weight-two fields; its weight-one Lie algebra is $\mathfrak{sl}_2\oplus\mathfrak{sl}_2$, at levels $1$ and $39$. We outline the argument, which combines exact lattice calculations, an all-weight strong-generation induction and $C_2$-cofiniteness, semisimple descent for a full corner of Zhu's algebra, modular Galois symmetry, and holomorphic $E_8$ extensions.

Parameterized Quantum Circuits have emerged as a central framework for variational quantum algorithms in the NISQ regime, with major applications in quantum simulation, combinatorial optimization, and, more recently, quantum machine learning (QML). By developing a new approach for computing dynamical Lie algebras associated with quantum circuits combining parametrized gates and non-parametrized entangling gates, we aim to extend the applicability of recent algebraic results to architectures that are actually used in quantum machine learning.