Papers

In this paper, we study the asymmetric phases of the quartic type (0, 1) and (1, 0) Dirac en- sembles via three approaches: the Riemann-Hilbert approach, bootstrapping with positivity, and Hamiltonian Monte Carlo (HMC) simulations. The focus of this work is on asymmetric solutions to the Schwinger-Dyson and saddle point equations of these models, whose solution spaces prove deeply intricate. Via the Riemann-Hilbert approach, we are able to give explicit formulae for the eigenvalue distributions and free energy of various solutions. Using Hamiltonian Monte Carlo simulations, we are able to reconstruct the phase structure. Lastly, using bootstrapping with positivity, we are able to reconstruct the eigenvalue distribution of these models from their bootstrapped moments. All three methods show excellent agreement for a large matrix size.

We introduce the notion of a unitary disk-like $n$-category, which is a disk-like $n$-category equipped with a reflection structure and a sphere trace inducing positive-definite pairings. Since a finite unitary disk-like $n$-category is defined to be the local field data of a fully extended $(n+1)$D unitary TQFT, we propose a complete finite unitary disk-like $n$-category as the definition of a finite $(n+1)$-Hilbert space for all $n$. For $n=1$ and $n=2$ we verify this proposal, proving that complete finite unitary disk-like 1- and 2-categories are isometrically equivalent to finite 2- and 3-Hilbert spaces respectively, and that these equivalences are functorial. For $n=1$ we recover a unitary refinement of Schommer-Pries' classification of oriented $(1+1)$D TQFTs in terms of $\mathrm{H}^*$-Morita equivalence classes of $\mathrm{H}^*$-algebras, and for $n=2$ we categorify this to classify oriented $(2+1)$D unitary TQFTs by $\mathrm{H}^*$-Morita equivalence classes of $\mathrm{H}^*$-multifusion categories. Along the way, we investigate fully incomplete 3-Hilbert spaces, prove a strictification result for pivotal dagger 2-categories, and define functors and higher transformations between disk-like $n$-categories.

Let $\mathpzc K$ be an arbitrary commutative ring containing an invertible element $\varepsilon$. Let ${\mathcal H}_{{\!\vartriangle\!}}(r)_{\mathpzc K}$ be the extended affine Hecke algebra of type $A$ with Hecke parameter $\varepsilon$, let $\Omega_{\mathpzc K}^{\otimes r}$ be the affine tensor space, and let ${\mathcal S}_{{\!\vartriangle\!}}(n,r)_{\mathpzc K}$ be the corresponding affine quantum Schur algebra. We first prove that the natural right action of ${\mathcal H}_{{\!\vartriangle\!}}(r)_{\mathpzc K}$ on $\Omega_{\mathpzc K}^{\otimes r}$ is always faithful. Assume further that $\mathpzc K$ is a field of characteristic $0$ and that $\varepsilon$ is not a root of unity. We prove that, for any $n\geq 2$, the natural algebra homomorphism $\xi_r:{\mathcal H}_{{\!\vartriangle\!}}(r)_{\mathpzc K}\rightarrow\operatorname{End}_{{\mathcal S}_{{\!\vartriangle\!}}(n,r)_{\mathpzc K}}(\Omega_{\mathpzc K}^{\otimes r})^{\mathrm{op}}$ is an isomorphism. This proves Conjecture~3.8.8 of \cite{DDF}. As an application, we prove the conjecture formulated in \cite[5.2.4]{DDF} concerning the center of the affine quantum Schur algebra. We also prove that ${\mathcal S}_{{\!\vartriangle\!}}(n,r)_{\mathpzc K}$ is left and right Noetherian whenever $\mathpzc K$ is a Noetherian commutative ring, which verify a conjecture in \cite[Rem. 1.7]{DY}.

We construct braided Hopf algebroids in a braided monoidal category over a field $k$ and study its properties using graphical representation. Then we study Ehresmann-Schauenburg Hopf algebroids assocaited to braided Hopf Galois extensions. We introduce braided Lie-Rinehart algebras which generalize the definition in \cite{ALP24, ALP23, ALP25} and study its universal enveloping algebra under symmetric condition. In particular, we introduce Lie-Rinehart algebras (braided Lie-algebroids) associated with braided Hopf algebroids in the category of the module of a triangular Hopf algebra. We then focus on the case for braided Ehresmann-Schauenburg Hopf algebroids and study their right invariant vector fields, which turn out to be isomorphic to the $H$-equivariant vector fields on the quantum principal bundle (Hopf Galois extension) as Lie-Rinehart algebras. Finally, we introduce braided jet Hopf algebroids associate to braided Hopf algebroids in the case that the source and target subalgebra belong to the braided center. Moreover, we show there is a dual pairing between the $k$-order universal enveloping algebra of the braided right invariant vector fields and the $k$-th jet space of a braided Hopf algebroid and further more a skew pairing between the universal enveloping algebra of its braided right invariant vector fields and its jet Hopf algebroid under a finiteness condition.

We use skein theoretic `recognition' theorems to classify deformations of certain pivotal or ribbon monoidal categories. We first show that a slight generalization of Kuperberg's characterization of quantum $G_2$ shows that any infinitesimal deformation of quantum $G_2$ as a pivotal category arises by varying $q$. This result applies both at generic $q$, and for the category of tilting modules at $q$ a root of unity (provided we exclude a few small roots of unity). Second, we prove a new Kuperberg-like characterization of Deligne's $S_t$ and use it to show that any infinitesimal deformation of $S_t$ (excluding $t=0$) as a ribbon category comes from varying $t$.

We construct skein theory for 3-manifolds with embedded line defects, starting from the data of a ribbon tensor category and its balanced braided module category. We focus on line defects arising from quantum symmetric pairs, and prove finiteness properties for their defect skein modules. As an application, we consider $\mathbf{Z}_2$-equivariant skein theory and establish an equivalence with defect skein theory in certain settings, leading to a skein theoretical construction of a family of $\mathrm{C}^\vee\mathrm{C}$ DAHA-modules in the Type AIII case.

The flip symmetry on link diagrams induces an involution on Khovanov-Rozansky ${\frak{gl}}_{N}$ homology. We prove that this involution is diagonalizable with eigenvalues $\pm1$. On the one hand, it is the identity over $\Bbb{F}_2$, generalizing a previous result of Chen and the author. On the other hand, it is expected to be nontrivial over $\Bbb{Z}$ in general. The key ingredients of the proof are (1) a homotopy perturbation argument via a detailed study of the fork twist, which allows us to reduce the computation to planar ${\frak{gl}}_{N}$ webs, and (2) the computation of the flip map for planar ${\frak{gl}}_{N}$ webs via diagrammatics of Soergel bimodules. The latter computation can also be interpreted as a naturality result for the half twist action on type $A$ Soergel bimodules, which might be of independent interest.

We explain how to derive the Robinson-Schensted-Knuth (RSK) correspondence, a fundamental bijection in algebraic combinatorics, from the Yang-Baxter equation. The Yang-Baxter equation arose in the study of quantum many-body systems and later became a cornerstone of the theory of solvable lattice models, particularly vertex models. In a vertex model, arrows occupy the edges of a grid, and each vertex carries a Boltzmann weight determined by the arrows on the four edges meeting at it. The weight of a configuration is the product of these local weights, and a partition function is the sum of the weights of all configurations with prescribed boundary conditions. For a vertex model whose partition functions are the Schur polynomials, the two sides of each instance of the Yang-Baxter equation admit exactly one weight-preserving matching of their summands. Carried across a grid, this forced matching is the classical RSK correspondence in the form of Fomin's growth diagrams. In natural coordinates the local matching becomes the combinatorial three-dimensional R, a set-theoretic solution of the Zamolodchikov tetrahedron equation. Its periodic closure returns the combinatorial R-matrices of one-row crystals. The forced matching is special to the Schur weights. For the Hall-Littlewood and q-Whittaker deformations and their spin versions, at generic parameter values no deterministic matching works for all boundary data. Reading each instance of the Yang-Baxter equation probabilistically, we replace the matching by a coupling of the two sides - a bijectivization, or probabilistic bijection - and obtain Markov operators that transport probability measures attached to vertex models. Iterated over the grid, these operators produce randomized RSK-type dynamics and interacting particle systems, including q-PushTASEP and the stochastic six-vertex model.

A qKZ-type discrete flat connection with multiplicative step $p$ has monodromy operators when $p$ is a root of unity. The monodromy operators are commuting endomorphisms of the connection. We construct their eigensections and eigenvalues. The construction is based on the presentation of the discrete flat connection as a discrete Gauss-Manin connection.

We show that every classifiable simple C*-algebra, in particular the Jiang-Su algebra, admits outer actions of several discrete quantum groups, such as finite quantum groups and duals of certain free compact quantum groups. As a consequence of the construction, we provide new examples of hyperfinite subfactors with the Temperley-Lieb-Jones standard invariant and show that $\operatorname{SU}_q(2)$ can act minimally on some Araki-Woods factors for infinitely many $q$.

This paper provides a complex-analytic interpretation of certain topologies on the tensor algebra, which are a central tool within strict deformation quantization. We prove that the symmetric tensor algebra $\operatorname{S}(V)$ of a barelled nuclear DF-space $V$ endowed with any of these topologies may be understood as an algebra of bounded Fréchet holomorphic functions on the strong dual space $V'_\beta$. The coarsest of the topologies then corresponds to the topology of uniform convergence on bounded subsets. The finer ones provide an infinite-dimensional generalization of the order of an entire holomorphic function. To facilitate these results, we study the interplay between boundedness and continuity of holomorphic mappings between locally convex spaces. As a byproduct, we establish a simple sufficient criterion for Fréchet holomorphy as well as several completeness results for spaces of bounded functions.

We construct a complex spherical fusion category with cyclic Haagerup-Izumi fusion rules for every odd n >= 3, and deduce pseudo-unitary existence. The proof has four parts: explicit real coefficients and their quadratic identities; a contour calculation for a range of cubic Fourier coefficients; algebraic completion of all cubics; and categorical reconstruction. Matrix inversion supplies reflection, and an extension of the dimension-field automorphism supplies the positive-dimensional category.

We construct the KLR algebra (in type $A$) as affine sections of a coherent sheaf $\mathcal{E}$ on a product of projective spaces $\mathbb{P}^n$. It was shown by Elias-Qi that the Lie algebra $\mathfrak{sl}_2$ acts on such KLR algebras; our construction geometrically explains this action as stemming from the $\mathfrak{sl}_2$-action on the twisting sheaves $\mathcal{O}(k)$ of $\mathbb{P}^1$ and gives the maximal finite-dimensional $\mathfrak{sl}_2$-submodule of KLR as global sections of $\mathcal{E}$. We furthermore extend the Elias-Qi action to a Witt action. Kleshchev-Loubert-Miemietz showed that the KLR of type $A$ has a graded affine cellular structure; our construction also gives a geometric incarnation of this.

After proving the moonshine conjecture, Borcherds asked in 1995 whether there exist only finitely many holomorphic automorphic products of singular weight. We show that, up to conjugation, there are exactly 27 such products on lattices of the form $2U\oplus L$. The classification proceeds by identifying each product at a standard $1$-dimensional cusp with the superdenominator of a BKM superalgebra that carries a graded module structure over an affine Lie superalgebra $\hat{\mathfrak{g}}$. The auxiliary algebras $\hat{\mathfrak{g}}$ are shown to satisfy strong conditions that reduce the possibilities to an initial list of 1372 candidates. An elimination argument leaves exactly 85 algebras $\hat{\mathfrak{g}}$ that correspond to the 27 products. This classification is closely connected with certain distinguished families of vertex operator superalgebras. Our results provide a systematic framework for extending affine Lie superalgebras to hyperbolic ones with modularity, generalizing the work of Feingold and Frenkel (1983).

Let $Q$ be a finite acyclic valued quiver. We establish an alternative cluster multiplication formula in the quantum cluster algebra associated with $Q$, which exhibits certain symmetries similar to 2-Calabi–Yau properties of cluster categories. Motivated by this formula, we introduce a Hall algebra $\mathcal {D}\mathcal {H}_\Lambda^{{cl}}(\widetilde{\mathcal{A}})$ via a certain quotient of a derived Hall subalgebra associated with $Q$, serving as an algebra counterpart to cluster categories of hereditary algebras. As our main result, we prove that the quantum cluster algebra associated with $Q$ is isomorphic to the exceptional subalgebra of $\mathcal{D}\mathcal{H}_\Lambda^{cl}(\widetilde{\mathcal{A}})$, namely, the subalgebra generated by exceptional objects. Furthermore, when $Q$ is of Dynkin type, the quantum cluster algebra is naturally isomorphic to $\mathcal{D}\mathcal{H}_\Lambda^{{cl}}(\widetilde{\mathcal{A}})$ itself. This establishes a Hall algebra realization for all acyclic quantum cluster algebras. As an application, we introduce the Hall subalgebra of $\mathcal{D}\mathcal{H}_\Lambda^{cl}(\widetilde{\mathcal{A}})$ generated by projectives, yielding a Hall algebra analogue of the lower bound quantum cluster algebras. Then we show that the standard monomials of quantum projective cluster variables form a basis of the quantum cluster algebras via Hall algebra approach. Moreover, we show that the Auslander–Reiten translation of cluster categories induces an algebra automorphism of $\mathcal {D}\mathcal {H}_\Lambda^{{cl}}(\widetilde{\mathcal{A}})$. Finally, we apply the BGP-reflections to $\mathcal {D}\mathcal {H}_\Lambda^{{cl}}(\widetilde{\mathcal{A}})$, and prove that the Hall algebra $\mathcal {D}\mathcal {H}_\Lambda^{{cl}}(\widetilde{\mathcal{A}})$ is invariant up to isomorphism under the mutations at sink vertices of $Q$.

We construct a categorification of the functor of grouplike elements of a Hopf algebra in the setting of braided monoidal categories. Given a braided bistrong monoidal functor admitting a right adjoint, we define a functor assigning to each Hopf monoid a group object obtained as a subobject of its underlying object by means of finite limits. We prove that this construction is functorial and provides a right adjoint to the induced free Hopf monoid functor, thereby extending the classical correspondence between groups and grouplike elements of Hopf algebras to a categorical setting.

Let $G$ be a standard reductive group over an algebraically closed field of positive characteristic and let $\mathfrak{g} = \operatorname{Lie}(G)$. The category of Harish-Chandra bimodules consists of $G$-equivariant $U(\mathfrak{g})$-bimodules such that the adjoint $\mathfrak{g}$-action integrates to a $G$-action. In this paper we classify the simple Harish-Chandra bimodules and determine their central characters.

We study iHopf algebras associated with quantum supergroups of basic type. For a quantum supersymmetric pair $(\mathbf{U}, \mathbf{U}^\imath)$, we realize $\mathbf{U}^\imath$ and $\mathbf{U}$ as the iHopf algebras of the Borel quantum group ${\hat{\mathbf{U}}}_q^{\geq0}$ and the tensor product algebra ${\hat{\mathbf{U}}}_q^{\geq0}\otimes {\hat{\mathbf{U}}}_q^{\geq0}$, respectively, while the coideal subalgebra structure is encoded by an embedding between the iHopf algebras. Moreover, we derive an explicit conversion formula between the iHopf multiplication and the original multiplication, which provides a general mechanism transforming the defining relations of quantum (super)groups into relations of iquantum (super)groups. In particular, all the Serre relations that occur in quasi-split iquantum supergroups of basic type are characterized.

The deformed Noumi-Sano operator $H_{n, m}^d(x, y)$ of order $d$ is a simultaneous extension of the Macdonald operator and the Noumi-Sano operator. In this paper, we construct an integral transform associated with eigenvalue problem of $H_{n, m}^d(x, y)$. Using the integral transform, we obtain an integral representation of eigenfunctions for $H_{n, m}^d(x, y)$.

Habiro's cyclotomic expansion of the colored Jones polynomial has a higher rank analogue conjectured by Chen–Liu–Zhu for colored $SU(n)$ invariants. We prove this conjecture for torus knots $T(2,2p+1)$ and give a Bailey theoretic realization of the cyclotomic coefficients. For $n\geq2$ and $K_p=T(2,2p+1)$, the colored $SU(n)$ invariants admit an expansion $$ J_N^{SU(n)}(K_p;q) = \sum_{m=0}^{N} \left(\prod_{j=0}^{m-1}\{N-j\}\{N+n+j\}\right) H_m^{(n,p)}(q), $$ where $H_m^{(n,p)}(q)\in\mathbb Z[q^{\pm1}]$ is independent of the color $N$. The proof identifies the cyclotomic basis with a Newton basis and rewrites the resulting Newton coefficients as an ordinary Bailey transform. The Lin–Zheng formula for $T(2,2p+1)$ then gives a well-poised Bailey kernel. A terminating very-well-poised ${}_6\phi_5$ summation diagonalizes the kernel and reduces the integrality problem to an ordinary Bailey transition. We prove a uniform integrality theorem for these transitions in a formal integral $q$-difference operator algebra. As a direct corollary, the expansion yields the corresponding congruence relations and proves part(i) of the Chen–Liu–Zhu $SU(n)$ volume conjecture for $T(2,2p+1)$.

In this work, we complete the description of the one-dimensional partial actions of 8-dimensional Hopf algebras by computing the remaining cases: the Kac-Paljutkin algebra $\mathcal{A}$ and the unique non-semisimple non-pointed Hopf algebra $\mathcal{K}$. We prove that all these partial actions are symmetric, study their associated partial smash products and we determine all their partial coactions of dimension one. Beyond the 8-dimensional setting, we investigate which right coideal subalgebras of a Hopf algebra $H$ can be realized as partial smash products $\underline{ \Bbbk \# H}$ over the base field. In particular, we show that every right coideal subalgebra of a finite-dimensional cosemisimple Hopf algebra arises in this way.

Evidence suggests a tight connection between the existence of antipode-like maps on bialgebra-like structures, and the Frobenius property for the associated free Hopf module functor. In this paper, we prove that in any braided monoidal category satisfying few mild assumptions, a bimonoid is a one-sided Hopf monoid and the antipode is a bimonoid anti-homomorphism, if and only if the free Hopf module functor is Frobenius. Our interest in these structures stems from having found genuine examples of one-sided Hopf monoids in contexts where the braiding is non-trivial. In fact, we provide a construction of the free one-sided Hopf monoid over a comonoid in any symmetric monoidal category with some non-restrictive additional assumptions. We present several examples of this construction, and describe the resulting one-sided (sometimes two-sided) Hopf monoids.

We prove that all Morita classes $\mu_k \in H_{4k}(Aut(F_{2k+2});\mathbb{Q})$ are nonzero using a novel description of the Hopf algebra structure on Steinberg homology. We prove analogous results with twisted coefficients. Our results disprove a case of a finiteness conjecture of Kontsevich.

We prove braided tensor equivalences between the categories of weight modules of the affine vertex algebra of $\mathfrak{sl}_2$ at any admissible level and categories associated to partially semisimplified versions of quantum $\mathfrak{sl}_{2|1}$ at roots of unity.

We determine the defining ideal of the simple affine vertex algebra $L_{2-\ell}(\mathfrak{so}_{2\ell})$ for every odd $\ell\ge5$. Perše's quadratic singular vector alone generates the maximal ideal of the universal affine vertex algebra at this level. Together with the established even-rank presentation, this gives a complete parity-dependent description of this type-$D$ collapsing family at $k=2-\ell$: one quadratic generator in odd rank, and a quadratic generator together with two Pfaffian generators in even rank. The proof establishes a rank reduction for the quadratic quotients under minimal Drinfeld–Sokolov reduction, valid in both parities. Nonvanishing of reduction on nonzero graded subquotients then lifts simplicity along the odd-rank chain from the known base case $D_3\cong A_3$ at level $-1$.

We study the Gukov–Manolescu quantum invariant for oriented links. While this invariant is a single series in the knot case, we find that links admit multiple such series, each one consistent with the Melvin–Morton–Rozansky expansion of the colored Jones polynomials. We prove that convergent inverted state sums yield a family of multivariable Gukov–Manolescu series, indexed by monomials of the Alexander polynomial. We conjecture that these monomials correspond precisely to the fibered faces of the Thurston norm ball and provide extensive computational evidence for this correspondence. Further results concerning the leading term, the corresponding single-variable invariant, and the effect of partial Dehn surgery are also established.

Racinet's double shuffle Lie algebra $\mathfrak{dm}_0$ encodes the double shuffle relations of multiple zeta values. We study two analogues of $\mathfrak{dm}_0$ for multiple Eisenstein series and $q$-analogues of multiple zeta values, namely the space $\operatorname{BARI}_{\operatorname{swap},\mathrm{il}}$ of swap-invariant alternil bimoulds with the uri bracket of Kühn and Schneps, and Burmester's balanced double shuffle space $\mathfrak{bm}_0$. Both were conjectured to be Lie algebras. We prove both conjectures and show that $\mathfrak{bm}_0$ is isomorphic to the Lie algebra of finite-depth polynomial elements of $\operatorname{BARI}_{\operatorname{swap},\mathrm{il}}$.

The Clifford hierarchy is a nested sequence of sets of quantum gates that can be fault-tolerantly performed using gate teleportation within standard quantum error correction schemes. The importance of these gates has motivated numerous studies of their structure. Zeng-Chen-Chuang conjectured in 2007 that all hierarchy gates are generalised semi-Clifford, i.e. take the form $C_1 \Pi D C_2$ for Clifford gates $C_1, C_2$, a permutation gate $\Pi$, and a diagonal gate $D$; Beigi-Shor proved in 2008 that this holds for all third-level gates. We construct a five-qubit gate that is in the fifth level of the Clifford hierarchy but is not generalised semi-Clifford. Rather than simply present and verify our counterexample to the generalised semi-Clifford conjecture, we show how its form can be deduced. Our counterexample also demonstrates that the Clifford hierarchy is not closed under inverses.

We prove the Chen–Yang volume conjecture for all sufficiently long integral Dehn fillings of any fixed marked fundamental shadow-link exterior. For each fixed filling, the exponential growth of its $SO(3)$ Turaev–Viro invariants recovers its hyperbolic volume along the full sequence of odd levels. The filling coefficients may have mixed signs and unrelated magnitudes. We also establish a complete asymptotic expansion of the signed $SO(3)$ Witten–Reshetikhin–Turaev invariant and identify the absolute leading coefficient explicitly in terms of adjoint Reidemeister torsion. Fixed even colors on the filling cores recover characters of the geometric holonomy. The key difficulty is cancellation in the signed surgery sum. Our main analytic tool transfers an exact reflection symmetry from a continuous model to the finite quantum sums. We control the error below the exponential scale of the surviving contribution. We also apply the method to a one-edge state sum restricted to central colors. The dominant contributions cancel, and for each sufficiently large fixed number of blocks we determine the smaller surviving exponential rate and its nonzero leading coefficient.

Let $L$ be a finite-dimensional pointed Hopf algebra over an algebraically closed field. We establish an intrinsic characterization of index-two extensions: every Hopf algebra $H$ containing $L$ as a Hopf subalgebra and satisfying $H_0=L_0$ and $\dim H=2\dim L$ is a normalized quadratic extension of $L$. For a fixed support $(g,h)$, such extensions are described by a coalgebra Hochschild $2$-cocycle together with Ore-type data $(\sigma,\delta,u,v)$ satisfying explicit compatibility conditions. Their isomorphism classes are parameterized by the orbits of gauge transformations and Hopf automorphisms of $L$. As an application, we classify non-connected pointed Hopf algebras of dimension $16$ in characteristic $2$ with one-dimensional infinitesimal braiding whose diagram is not a Nichols algebra.