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Wednesday, 4 November 2026: 5 new papers, 5 cross-lists and 4 replacements in Quantum Algebra (math.QA), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
The Askey-Wilson function $\phi(x, \gamma)$ is a meromorphic continuation of the Askey-Wilson polynomials in their spectral variable off the lattice $\gamma = \tilde{a} q^m, m \in \mathbb{Z}_+$. We show that the analytic part of the Askey-Wilson function $\phi^{an}(x, \tilde{a} q^\gamma)$ is the Laplace transform of a positive measure of total measure $\phi^{an}(x, s_0) < \infty$, under certain conditions on the parameters governing the function. We deduce log-convexity of this function, as well as positive-definiteness and an integral formula for a certain intertwining operator associated with the Askey-Wilson operator $L$. This offers a $q$-analogue of several results, in the trigonometric limit due to Sahi-McSwiggen, and in the rational (i.e. Dunkl) limit due to R\"osler.
In this article we establish new obstructions to the categorification of near-group fusion rings by combining the values of irreducible characters of the Grothendieck ring with the Galois symmetry of the induced unit object $I(1)$ in the Drinfeld center. As a result, we show that a near-group ring of type $G+m$, where $G$ is an abelian group and both $|G|$ and $m$ are odd, can only categorify if $m=|G|$.
The reflective algebra $(R_H(A),K_{\mathrm{ref}}(A))$, introduced by Laugwitz, Walton and Yakimov, is a quasitriangular left $H$-comodule algebra constructed from a quasitriangular Hopf algebra $(H,R)$ and a left $H$-comodule algebra $A$. We study the structure and factorizability of finite-dimensional reflective algebras. When $(H,R)$ is factorizable, we show that $(R_H(A),K_{\mathrm{ref}}(A))$ is isomorphic, as a quasitriangular left $H$-comodule algebra, to $A\otimes H$ equipped with the trivial left $H$-coaction on $A$ and the universal $K$-matrix induced by $R_{21}R$. In this case, we also classify all universal $K$-matrices on $R_H(A)$. Finally, we prove that $(R_H(A),K_{\mathrm{ref}}(A))$ is factorizable if and only if $(H,R)$ is factorizable and $A$ is simple as an algebra.
Three constructions of vertex operator algebras realizing the Haagerup center were announced independently and almost simultaneously. The relations between these constructions are studied in this paper. The generators are expressed in a common lattice basis for $E_6\oplus A_2$. It is proved that the constructions of Riesen and Huang give isomorphic vertex operator algebras, and that this algebra and the algebra constructed by Jiao have isomorphic fixed point subalgebras under automorphisms of order two. The isomorphisms are given explicitly on generators. The common fixed point subalgebra contains the parafermion vertex operator algebras $K(\mathfrak{sl}_2,3)$ and $K(\mathfrak{sl}_2,13)$. A general rationality criterion for unitary vertex operator algebras is given in terms of positive linear combinations of conformal vectors and is applied to these constructions. The two vertex operator algebras are simple current extensions of the common fixed point subalgebra and have equivalent module categories as ribbon categories. The equivalence follows from decomposition with respect to a nondegenerate pointed subcategory with four simple objects in the module category of the common fixed point subalgebra. It is also proved that the conformal nets associated with these unitary vertex operator algebras are completely rational.
In a weakly integral braided fusion category, a prime divides none of the conjugacy-class dimensions if and only if it does not divide the Frobenius-Perron dimension of the adjoint subcategory. We prove that these conditions hold precisely when the corresponding Sylow subgroup of the group of invertible objects generates a braided Deligne factor whose dimension is the largest power of that prime dividing the Frobenius-Perron dimension of the category. We also prove that the prime graph of the conjugacy-class dimensions has at most two connected components. If the graph is disconnected, the category is solvable. The bound is attained by the representation category of the Drinfeld double of the symmetric group on three letters.
We encode certain sets of partitions by regular languages of boundary words, so that their generating functions become rational series in the quantum plane $RU=qUR$. An equality between two partition counts is certified by writing the difference of their series as $(1-U)a+b(1-R)$, with $a,b$ converging $q$-adically. We show that $a,b$ may be chosen as solutions of finite holonomic systems of $q$-difference equations computed from the automata, and describe finite certificates of their convergence, together with bounded searches for such certificates; the searches need not succeed, but a false identity is always detected. As an application, we prove the three-row, level-two case of Primc's conjecture, which equates counts of coloured partitions subject to local multiplicity conditions with counts of coloured partitions whose parts lie in prescribed residue classes modulo eight.
In this note, we begin with considering the simple one-dimensional field theory describing a single free particle on manifold, and apply the BRST formalism and BV quantization to this model following the standard procedures in physics. Through calculations that, while not fully rigorous, have a more transparent physical interpretation, then subsequent geometric identifications, we recover the deformation quantization and algebraic index theorem on symplectic manifold. This note thus elucidates the physical origins and mechanisms of these mathematical results.
We construct a universal quantum computational encoding via braiding in Links--Gould fusion spaces associated to $U_q^H(\mathfrak{sl}(2|1))$ where $q$ is on the unit circle and is not a root of unity. The construction uses only neglectons and is realized in the morphism spaces $\mathrm{Hom}_{U_q^H}(V(0,\alpha),V(0,-\frac{1}{2})^{\otimes 2N}\otimes V(0,\alpha)).$ Under explicit positivity and arithmetic constraints these spaces admit a $2^N$-dimensional computational subspace with a positive definite Hermitian form. We prove single-qubit density in $\mathrm{U}(2,1)$ and approximate arbitrary unitary operators on $N$-qubit computational subspaces with arbitrarily small leakage. The calculations use explicit inclusions, projections, and Hermitian pairings without requiring $F$-matrices.
Almost 30 years ago Littelmann conjectured that there is a unitriangular base change between the dual canonical basis and the path vector basis for all symmetrizable Kac-Moody Lie algebras. In this paper we give a positive answer to the conjecture for the case of classical Lie algebras and type $G_2$, when the path model is given by standard Young tableaux.
Using Mac Lane's PROPs, one can introduce multilinear polynomial identities that involve all the operations of a Hopf algebra: the (co)multiplication, the (co)unit and the antipode. We find bases for polynomial identities and calculate their codimensions for the Sweedler algebra, treating it first as an ordinary associative algebra over a field and second as a Hopf algebra.