Quantum Algebra Papers of the Week

We study the passage from ghosts to SICs.

We give an explicit formula for elements $H_{2k}$ of conformal weight $2k$, $k\ge1$, in the universal Virasoro vertex operator algebra $V$ at every central charge, such that $H_{2k}(0)H_{2l}\in TV$ for all $k,l\ge1$.

We present LeanVOA, a Lean 4 library for the theory of vertex operator algebras (VOAs), together with a companion book that sets each formalized result next to its textbook statement.

We construct simple coherent classes and compute their normalized difference operators for the two-node quiver Coulomb branch with gauge group $GL_1\times GL_2$ and $m-1$ framing columns at the second node.

We prove that the bottom degree of Huang-Lepowsky's $P(w)$-tensor product module $M^1\boxtimes_{P(w)}M^2$ is isomorphic to the left $A(V)$-module $M^1\odot_{w,0}M^2$ arising from the restricted conformal blocks on $(¶^1,\infty,w,0)$.

For a quiver $Q$, we introduce and study $Q$-operads (or operads over $Q$), defined as a special sort of $\mathbf{fc}$-multicategories or virtual double categories, in which the directions of the arrows of $Q$ encode the permitted input orderings.

We characterize the weights of minimal singular vectors of $V^\kappa(\mathfrak{g})$ with respect to the generalized Cartan matrix, when $V^\kappa(\mathfrak{g})$ is not simple and $\kappa$ is not critical.

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.

We establish a finite system of consistency equations for triangle relation planar 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.

We obtain classification results for rational vertex operator algebras generated in low conformal weights.

We construct an explicit algebra homomorphism from the whole quantum group ${U_{q^{1/2}}}( {\widehat {\mathfrak{sl}_2}} )$ to $\mathcal A_q$, whose images together with either of the two frozen variables generate $\mathcal A_q$.

Following the knot-quiver correspondence, we investigate quiver counterparts of the HOMFLYPT skein relations by interpreting link invariants in terms of holomorphic worldsheets and their multicover contributions. Extending this framework to more general worldsheet geometries, we conjecture that, up to quiver equivalence, the adjacency matrix of the quiver associated with any link in a skein triple admits a block decomposition whose diagonal blocks are quiver adjacency matrices associated with the other two links.

These invariants, originally constructed by means of Fourier-Mukai transform functors between derived categories of coherent sheaves on certain iterated flag varieties, can be given an alternative algebraic description by using tilting bundles. We use this method and provide several examples of its application, in the hope of showing its utility to explicit pen-and-paper computations.

We study the space of highest-weight vectors in $L_n\otimes L_m$ for the diagonal $\mathfrak{sl}_2$-action.

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.

We classify finite composites of the positive Fibonacci category with arbitrary pointed fusion categories, retaining the entire source functor and making no unitarity assumption on the target.

We propose a program to classify tensor categories generated by prescribed seed tensor categories.

We give invariants of pairs $(M,\Gamma)$ consisting of a closed connected oriented three-manifold and an oriented framed ribbon graph embedded in $M$ colored by a the Drinfeld center of a unimodular chromatic category $\mathcal C$.

We study the chiralization of Poisson--Lie theory through graded symplectic geometry.