Papers

Let $\hbar$ be a non-zero real number and put $q= e^{-\pi i \hbar}$. Let $G$ be a semisimple split real Lie group of classical type, with Lie algebra $\mathfrak{g}$ and associated Borel subalgebra $\mathfrak{b} \subseteq \mathfrak{g}$. We show that there exists a locally compact quantum group $(L_{\hbar}(B),\Delta)$ which integrates the Borel part $U_q(\mathfrak{b})$ of the Drinfeld–Jimbo quantized enveloping algebra $U_q(\mathfrak{g})$, viewed as a Hopf $*$-algebra. We further compute the associated Haar weights and modular data for $(L_{\hbar}(B),\Delta)$. Up to an excess Cartan part, the Drinfeld double of $(L_{\hbar}(B),\Delta)$ can be seen as an analytic quantization of the function algebra of the totally positive semigroup $G^+\subseteq G$.

We construct an action of the positive half of the Witt algebra on colored $\mathfrak{gl}_N$-Khovanov-Rozansky homology and prove that this action is compatible with the functoriality of colored link homology. To establish this result, we use Bar-Natan's framework of canopolies to formulate and prove the relevant statements at the level of tangles.

We give an explicit formula for the Lusztig parameters of determinantial modules over symmetric quiver Hecke algebras. The formula applies to every Bruhat pair and every fixed reduced expression of the upper Weyl group element. In finite types $A$, $D$, and $E$, we identify the parameters of the modules in Ménard's seed using Demazure weaves. We then factor prefix determinantial modules into these seed modules, with multiplicities given by Lusztig cycles. We also give a criterion for recognizing monomials in this fixed seed and an explicit formula recovering their multiplicities from Lusztig parameters.

We prove that there is no uniform description of $\mathrm{LG}^{(n)}$ in terms of colored Jones polynomials for all knots, even when allowing infinitely many colors and arbitrary operations. We prove the $n=1$ case, using an existing cabling formula and the separation of the Conway and Kinoshita–Terasaka knots by $\mathrm{LG}^{(2)}$. We develop an odd cabling formula and use a Vandermonde argument to show that for each $n\geq 2$ some satellite of each of these knots is distinguished by $\mathrm{LG}^{(n)}$.

We study the Lax operator for Macdonald symmetric functions introduced by Nazarov and Sklyanin in 2015, extending the spectral theory developed by Mickler-Moll in the Jack case. We extend the NS operator to a Lax operator acting on symmetric functions with coefficients in polynomials in an auxiliary variable, and prove it has an orthogonal eigenbasis indexed by partitions and their addable corners, with the corner contents as eigenvalues, and leading terms given by Macdonald symmetric functions. We obtain the Macdonald analogue of the author's earlier Jack box-sum identity for Littlewood-Richardson coefficients. We then strengthen this box-sum identity to a two-parameter formula, proved using a finite-rank product formula of Warnaar. Its residues give explicit linear relations among Macdonald Littlewood-Richardson coefficients.

We revisit the relation between the Kontsevich commutative graph complex and the homology of $\GL(n,\Z)$ found by Brown–Chan–Galatius–Payne using the sharbly complex of Ash–Miller–Patzt.

In this work, we propose a mathematical description of the critical point of boundary phase transitions in 2+1D topological orders. Given two boundary phases, $\mathcal{C}_A$ and $\mathcal{C}_B$, with corresponding Lagrangian algebras $A$ and $B$ in a 2+1D topological order $\mathcal{C}$, respectively, we propose that the critical point of the phase transition between these two phases, if it exists, corresponds to $D = A \times_{M} B$, which is the fiber product of $A$ and $B$ over $A$-$B$-algebra $M$. We prove that $D$ is also a condensable algebra. Additionally, we present an alternative classification of condensable algebras in $Z(Vec_G^\omega)$ and reveal splitting phenomenon that arises during the condensation of certain algebras. We also perform explicit computations for the condensable algebras within several examples of $Z(Vec_G^\omega)$, covering both Abelian and non-Abelian groups $G$.

Given a pair of complementary subgroups $P$ and $N$ of a finite group $G$, we first discuss how the same type of functions $\mathbb E\colon P\times N\to \mathbb T$, which we call bi-$1$-cocycles, appear naturally in the study of projective representations of $G$ of Heisenberg type and second cohomology of $G$ and of its dual quantum group $\widehat G$. We show then that such representations are irreducible and the corresponding ordinary and dual $2$-cocycles are nondegenerate if and only if the matrix $(\mathbb E(p,n))_{p,n}$ is invertible, if and only if it is Hadamard. We illustrate the result with examples arising from symplectic nilpotent Lie algebras over finite fields with two complementary Lagrangian subalgebras.

We derive the triangular prism equations (TPE) for near-group fusion rules of type $G+|G|$, where $G$ is a finite abelian group. We prove that spherical categorification with trivial Frobenius–Schur indicator over any field in which $|G|$ is invertible is equivalent to the solvability of the TPE system. For every cyclic group, we construct an explicit solution to the TPE system using hyperbolic gamma functions, yielding both a spherical category with negative simple-object dimensions and, after Galois conjugation, a pseudo-unitary category with positive dimensions. For scalar solutions with positive dimensions, we establish an exact singular-value formula and a uniform gap, which provide explicit sufficient criteria for unitarity. Finally, we show that the unitarity of the cyclic Galois-conjugated coefficients for arbitrary orders corresponds to a specialization of a real-multiplication conjecture implied by the order-one abelian Stark conjecture, revealing a deep connection between categorical unitarity and Hilbert's twelfth problem for real quadratic fields.

We establish a quantum super Schur–Weyl duality for the mirabolic Hecke algebra \(H_n^{\mathrm{mir}}(q)\) with \(q\) indeterminate and determine the corresponding bimodule decomposition. The centralizer of the mirabolic action is shown as a direct sum of homogeneous quantum Schur superalgebras. We describe the tensor-space annihilator and the resulting faithful quotient. When nonzero, the annihilator is generated by an explicit spectral idempotent attached to the smallest rectangle excluded by the hook condition. From the quantum super Schur–Weyl duality, we derive a super Frobenius formula for mirabolic Hecke algebras. This formula then yields a Murnaghan–Nakayama rule and Regev-type formulas for the irreducible characters. We also construct a super mirabolic RSK bijection from words in a super alphabet with an additional even letter to hook semistandard insertion tableaux together with recording pairs that distinguish the positions of the additional letter. This correspondence yields a Roichman formula for irreducible characters of \(H_n^{\mathrm{mir}}(q)\). Finally, we establish a parameter-inversion isomorphism between \(H_n^{\mathrm{mir}}(q^{-1})\) and the \(q\)-rook monoid algebra \(R_n(q)\) and use it to transport the duality, annihilators, and character formulas in both directions. The transported tensor decomposition provides a representation-theoretic interpretation of hook and two-row character sums for \(R_n(q)\).

In this paper, we introduce the crystallization $C(SO_0(2n+1)/SO_0(2n-1))$ of the $C^*$-algebra $C(SO_q(2n+1)/SO_q(2n-1))$. We show that every irreducible representation of $C(SO_0(2n+1)/SO_0(2n-1))$ arises as the limit, as $q \to 0^+$, of a family of irreducible representations of $C(SO_q(2n+1)/SO_q(2n-1))$. As a consequence, we prove that $C(SO_0(2n+1)/SO_0(2n-1))$ is the universal $C^*$-algebra generated by a finite set of generators and relations. We then construct a chain of unital essential short exact sequences of $C^*$-algebras, enabling us to compute the $K$-groups of $C(SO_0(2n+1)/SO_0(2n-1))$ and determine explicit generators for them. Using the theory of homogeneous $C^*$-extensions, we prove that $C(SO_0(2n+1)/SO_0(2n-1))$ is isomorphic to $C(SO_q(2n+1)/SO_q(2n-1))$ for every $q \in (0,1)$.

Kotelskiy, Watson and Zibrowius show that the Bar-Natan invariant $\unicode{x0414}(T)$ associated to a $4$-ended oriented tangle $T$ extends to a twisted complex $\unicode{x0414}^\infty(T)$ over an $A_\infty$-enhancement of the Bar-Natan algebra. We prove that $\unicode{x0414}^\infty(T)$ itself is a tangle invariant, settling a conjecture from their work. We deduce this from an invariance statement for a matrix multifactorization $\mathcal M(T)$, generalizing a construction of Ballinger to the coefficient ring $\mathbb Z[G]$. This multifactorization is well defined up to a notion of 1-homotopy equivalence. The proof rests on explicit formulas for special deformation retracts of Koszul matrix factorizations, which realize delooping in this setting and may be of independent interest.

We construct chiral extensions of the cotangent Yangian using vertex algebroids. At zero level, the construction yields a vertex algebra with a compatible coproduct, whose Zhu reduction recovers the cotangent Yangian of Abedin and Niu. We also construct vertex algebra extensions at non-zero levels and study level-additive coproducts. The zero-level vertex algebra admits two natural quasi-conformal structures. We compute the corresponding sphere one-point conformal blocks and equip each block space with an associative product induced by the coproduct. For the first quasi-conformal structure, the blocks space can be identified with the dual of a certain Lie algebroid homology; for the second, we describe the block algebra and its classical limit in terms of mini-twistor geometry.

To quantize the reduction of a coisotropic submanifold of a Poisson manifold, Cattaneo and Felder constructed a homotopy Poisson algebra together with a deformation quantization. Its degree-zero cohomology is the reduced Poisson algebra, while the full homotopy structure retains information about the formal embedding even when the reduced space is singular or trivial. On the integration side, Cattaneo showed that a coisotropic submanifold integrates to a Lagrangian subgroupoid of any symplectic groupoid integrating the ambient Poisson manifold. It is now recognized that both constructions fit into shifted symplectic geometry: the Lagrangian subgroupoid presents a $1$-shifted Lagrangian morphism of differentiable stacks, and the homotopy Poisson algebra describes the associated $0$-shifted Poisson structure. However, such morphisms form only a special class of $1$-shifted Lagrangians. We extend this picture to $1$-shifted Lagrangians and, more generally, $1$-shifted coisotropics in $1$-shifted symplectic differentiable stacks. We establish an integration theorem for their infinitesimal description in terms of twisted Dirac structures and construct a natural Poisson bracket on invariant functions. Under regularity hypotheses, we equip the associated Lie algebroid complex with a flat $P_\infty$-structure and a curved $A_\infty$-quantization, independent of the auxiliary choices up to isomorphism. The induced bracket in degree zero recovers the reduced Poisson bracket, and suitable cohomological vanishing conditions yield a deformation quantization of invariant functions. This extends the Cattaneo–Felder picture to more general reduction procedures, with the same infinitesimal data governing integration and quantization.

We show that the Fomin–Kirillov algebras $FK_n$ are infinite-dimensional for $n\geq6$. This answers a question posed by Fomin and Kirillov in 1997. To prove this, we demonstrate that the Nichols algebras supported on the conjugacy class of transpositions of the symmetric group are infinite-dimensional for both relevant 2-cocycles. For this, we develop a new method of partially semisimplifying a Nichols algebra along an automorphism of order $p$ in characteristic $p$.

We develop Delsarte theory for quantum association schemes, allowing both composition and the quantum Schur product to be noncommutative. A MacWilliams identity and positivity of the inner and dual distributions yield conic upper bounds for codes and lower bounds for designs in terms of effective size, which extends subset cardinality. These bounds recover classical linear programs for symmetric association schemes and semidefinite programs for homogeneous coherent configurations. We also quantize Schurian schemes using finite-group representations and characterize quantum block designs in quantum Johnson schemes arising from irreducible stabilizer representations by averaging over subgroups fixing prescribed points individually. In a quantum Johnson scheme on $M_2(\mathbb{C})^{\oplus10}$, we prove that the minimum effective size of a nonzero positive element satisfying the strength-two design condition is $20$, attained by complementary projections outside the center of the underlying algebra. Every such element in the center has effective size $40$, while the conic lower bound is $40/3$, strictly below the actual minimum.

We prove that the Laurent polynomial in $\mathbb{Z}[q^{\pm 1}]$ that is the top coefficient of the Links-Gould invariant of the boundary of a Seifert surface is multiplicative under arbitrary Murasugi sums of minimal genus Seifert surfaces. This extends a previous result of the second and third authors which only holds for plumbing, i.e. 4-Murasugi sum. Our technique relies on quantum supergroup themselves and PBW basis rather than the skein relations used in the previous work.

We unify Jones' representations of braid groups and the oriented Thompson's group by introducing the oriented dyadic braid group. This finitely generated group contains every braid group and the oriented Thompson's group. We construct an embedding of the oriented Thompson's group into this group such that taking braid closures recovers Jones' construction of oriented links. We also introduce the dyadic Temperley-Lieb algebra and construct a unitary representation of the oriented dyadic braid group extending the classical Jones representations of braid groups. Along the Thompson group's embedding, its distinguished matrix coefficient agrees with the corresponding coefficient of Jones' representation of the oriented Thompson's group.

The Akutsu-Deguchi-Ohtsuki (ADO) invariants of links are a non-semisimple version of the colored Jones polynomials. We prove that the top coefficients of the ADO invariants of links in $S^3$ are multiplicative under Murasugi-sum along connected minimal genus Seifert surfaces. More generally, for any simple complex Lie algebra $\mathfrak{g}$ of type ADE and any $r\geq 3$, we prove that the top coefficient of the $r$-th $\mathfrak{g}$-ADO invariant is multiplicative under Murasugi-sum. As a corollary, we get that all the $\mathfrak{g}$-ADO invariants of fibred links are $q$-monic, generalizing a previous result of the authors for $\mathfrak{sl}_2$. Moreover, whenever $\mathfrak{u}_q(\mathfrak{g})$ is ribbon, we find an expression for the top coefficient of the $r$-th $\mathfrak{g}$-ADO invariant in terms of Hopf algebra integrals of universal $\mathfrak{u}_q(\mathfrak{g})$-invariants of bottom tangles.

The alternating composition of $N = 2p$ weighted differential operators $w_j(x)\cdot\partial_x^{\,p}$ of strict order $p$ on the line $\mathbb{R} \ni x$ is again an operator of order $p$; its coefficient is the universal constant $c(p)$ times the Wronskian of the weights $w_1,\ldots,w_N$. Lie brackets of vector fields fix $c(p=1)=1$; we want to find $c(p \geqslant 2)$: e.g., $c(2) = 2$ or $c(3) = 90$. Direct symbolic expansion (over $|S_{2p}| =(2p)!$ permutations) fails for $p \geqslant 4$. Taking the monomials $w_j = x^{j-1}$ reduces the summation to the much smaller set $\Phi_p \subseteq S_{2p-1} \subsetneq S_{2p}$ of late-growing permutations. Expressing $c(p)$ as a signed sum of products of falling factorials, we implement and speed up the algorithm that gains all the integer values up to $c(18) = 4.881\ldots \cdot 10^{462}$. The resulting sequence is new, now registered as OEIS A392714; its (sub)leading-order growth rate is $\log c(p) \simeq 2p^2\log p -b p^2 + \overline{o}(p^2)$ for $p\gg 1$, with $b\geqslant 2.6744$.

In this paper, we show that for any coquasi-Hopf algebra $H$ and any $H$-Majid bimodule $M$, the cotensor coalgebra $\operatorname{CoT}_H(M)$ admits a graded coquasi-Hopf algebra structure. We then define the standard filtration of a coquasi-Hopf algebra and show that the associated graded coquasi-Hopf algebra admits an embedding into a cotensor coquasi-Hopf algebra that preserves degrees zero and one. If, in addition, this coquasi-Hopf algebra is coquasitriangular (respectively, coribbon), then the cotensor coquasi-Hopf algebra is also coquasitriangular (respectively, coribbon), and the above embedding is a morphism of coquasitriangular (respectively, coribbon) coquasi-Hopf algebras. As a byproduct, we prove that the coquasi-antipode of a coquasi-Hopf algebra with the dual Chevalley property is bijective. Moreover, we establish graded quasi-coquasi-Hopf pairings between tensor quasi-Hopf algebras and cotensor coquasi-Hopf algebras.

We study the quantum Cayley graphs associated with the Kac-Paljutkin quantum group. We determine which projections give rise to undirected, irreflexive and connected quantum graphs and classify them up to isomorphism. Their spectra and chromatic numbers are computed. For the special case of the projection $p = I_2$ we also compute the associated quantum Cuntz-Krieger algebra.

We obtain recurrence relations for formal characters of Feigin-Stoyanovsky type subspaces of standard $B_2^{(1)}$-modules at fixed level. The recurrences are derived directly from the combinatorial bases described by difference and initial conditions. Unlike in type $A_\ell^{(1)}$, the right side in general involves several shifted characters. We give explicit polynomial coefficients and prove that the resulting system, together with the constant term normalization, determines all formal characters at fixed level uniquely. At level $1$ we solve the recurrence system explicitly and obtain fermionic formulas for all three characters. We also relate these characters to the multigraded generating functions for exponent triples parametrizing the Meurman-Primc monomial basis of standard $A_1^{(1)}$-modules.

We study the Iwahori Gaiotto conjecture for cotangent types and its application to the representation theory of the (mixed, small, and Lusztig) quantum supergroups. Using local and global models and SW-Zastava spaces, we construct factorization functors and compare the Iwahori Gaiotto category with the category $O$ of the mixed quantum supergroup. We then study the images of intersection cohomology sheaves and the metaplectic Hecke action. In type A, we describe the finite and affine monodromic categories, the corresponding Yokonuma–Hecke actions, and their role in the Kazhdan–Lusztig algorithm. The last section treats the pointwise purity.

We define an intrinsic unital completely positive (ucp) map from the Cuntz algebra to the Pythagorean with finitely many generators. This permits to lift states and cyclic representations. This process restricts to the Cuntz-Toeplitz algebras and the contraction algebras. Taking inductive limits we obtain an effective machinery for constructing states on the infinite Cuntz algebra $\mathcal O_\infty$. Combining this approach with previous techniques of the first author we obtain large families of pairwise inequivalent irreducible representations of $\mathcal O_\infty$: one family being in an obvious bijection with the unit sphere of Hilbert's space.

We consider the type AIII (quasi split) quantum symmetric pair subalgebras $U_q'$. We define a new Cartan subalgebra in one of the two subfamilies. Together with Letzter's Cartan subalgebra in the other, we construct a triangular decomposition of $U_q'$ via the Letzter map. Then we define Verma modules as induced modules and prove that finite dimensional simple $U_q'$-modules are quotients of our Verma modules. These treatments are uniform in both subfamilies. In the second part we define a diagrammatic category $Web^B$ that controls polynomial representations of $U_q'$. We prove a multiplicity-free decomposition of the quantum exterior powers $\bigwedge^k\mathbb{V}$ as a $U_q'$-module, using a explicit eigenspace decomposition of certain dot morphisms in $Web^B$. As applications, we deduce the multiplicity-freeness of the involved anti-spherical Hecke module. For $\bigwedge^k\mathbb{V}$, we explicitly determine the highest weight vectors and their weights, thereby determining the isomorphism classes of their irreducible summands. Finally, we determine the kernel of the $Web^B$-action on certain finite dimensional representation category of $U_q'$.

We propose a new formula relating the rationalized Drinfeld associators obtained from the Knizhnik–Zamolodchikov equation to the quantum $6j$-symbols of the algebra $\mathcal{U}_q(\mathfrak{g})$: $ U_{q} = U \cdot \Phi^{KZ}_{\mathbb{Q}} $. We prove this relation in the case of the $\mathfrak{sl}_2$ Lie algebra for the tensor cube of the fundamental and the first symmetric representation. We also check it perturbatively for the $\mathfrak{sl}_N$ algebra for representations $[2]\otimes [1] \otimes [1]$, $[2]\otimes [2] \otimes [1]$, the non-trivial cases with multiplicities — $[2,1]^{\otimes 3}$, $[2,2]^{\otimes 3}$, and in the case of the $\mathfrak{so}_5$ Lie algebra for the tensor cube of the fundamental representation.

We construct an explicit rational \(\mathrm{SL}_2(\mathbb C)\)-action on the underlying vector space of the rank-two \(\beta\gamma\)-system. The construction is motivated by Creutzig's branching decomposition with respect to \(L_{-1}(\mathfrak{sl}_2)\) and a rank-one Heisenberg vertex algebra. We realize the raising and lowering operators as residues of lattice-type fields, prove that they preserve the original \(\beta\gamma\)-system despite being defined on localizations, and establish their commutation relations and integrability. The resulting action commutes with the affine \(L_{-1}(\mathfrak{sl}_2)\)-action and all nonzero Heisenberg modes, while the Heisenberg zero mode serves as its Cartan operator. Each affine multiplicity space is thereby identified with \(V_n\otimes\mathcal F_{\mathrm{osc}}\), where \(V_n\) is the irreducible \(\mathrm{SL}_2(\mathbb C)\)-module of highest weight \(n\) and \(\mathcal F_{\mathrm{osc}}\) is the Heisenberg oscillator Fock space. Thus the Heisenberg charge decomposition is realized as the weight decomposition of \(V_n\). The infinitesimal action is not by vertex algebra derivations and does not preserve the symmetric conformal grading.

Explicit R-matrices of quantum groups are routinely transcribed from the literature, yet whether they satisfy an identity depends on basis order, tensor placement, the coproduct and, for superalgebras, the Koszul sign rule. quantum-group is a Python package that builds such matrices exactly with SymPy and returns every identity as an exact residual matrix, whose nonzero entries locate the convention that disagrees. It covers U_q(sl_2) modules, tensor products and the fundamental R-matrix, and the graded Yang–Baxter equation for GL_q(2|1). Applied to the thesis it was written for, it exposed an R-matrix that satisfies the Yang–Baxter equation but intertwines the opposite coproduct.

For a non-degenerate braided fusion category $\mathcal B$, we study the Grothendieck $\mathbb Z_+$-ring of $\mathcal B$-module categories. We define a convolution of étale algebras in the Drinfeld center $\mathcal Z(\mathcal B)$ and show that the full center of a relative tensor product of module categories is canonically isomorphic to the convolution of their full centers. The module Grothendieck ring has a canonical upper-triangular grading by the poset of localizations of $\mathcal B$. We describe each localization stratum by double-coset endpoint blocks and a group-ring-valued transition matrix, obtaining an explicit sandwich multiplication formula. When $\mathcal B$ is the center of a fusion category, the transition matrix of the maximal stratum is the Gram matrix of the Lagrangian algebras, so linear relations among their classes give rise to radicals. We compute stratum rings and the radicals of their complexifications in examples from classical finite geometry, Drinfeld doubles of finite groups, exceptional WZW categories, and the Haagerup center. The examples exhibit noncommutative module Grothendieck rings with semisimple complexifications, as well as nonzero radicals in both intermediate and maximal complexified strata.