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These notes are an introduction to the theory of quantum symmetries of finite and infinite sets, graphs, and locally compact spaces.

We prove that the star product for quantum symmetric pair coideal subalgebras is short. We apply this result to obtain new conceptual proofs, from first principles, of several fundamental facts about quantum symmetric pairs. In particular,…

It has recently been observed that, in contrast to the classical case, holomorphic structures on line bundles over the quantum projective line are not uniquely determined by degree. We formulate a fixed-point-theoretic framework for the…

We study the structure and representation theory of the principal W-algebra $\mathsf{W}^{\mathsf{k}}_{\mathrm{pr}}$ of $\mathsf{V}^{\mathsf{k}}(\mathfrak{psl}_{2|2})$. The defining operator product expansions are computed, as is the Zhu…

We study the shuffle algebra realization of the positive subalgebra $Y_n^{>}(\mathbb{k})$ of the Yangian associated to $\mathfrak{sl}_n$ over an algebraically closed field $\mathbb{k}$ of characteristic $p>2$. In contrast to the…

We describe and compute various families of commuting elements of the matrix shuffle algebra of type $\mathfrak{gl}_{n|m}$, which is expected to be isomorphic to quantum toroidal $\mathfrak{gl}_{n|m}$. Our formulas are given in terms of…

For G a group, we present a G-graded version of chromatic maps and skein modules and use them to define a 2+1-G-HQFT out of a G-chromatic category. The construction applies to the representations of unrestricted quantum groups at root of…

Let $H$ be a generalized Liu algebra over an algebraically closed field $k$ of characteristic zero. We prove that all simple Yetter-Drinfeld modules over $H$ are finite-dimensional and present an explicit classification of these modules.…

We construct a quasi-particle basis of the integrable highest weight module of highest weight $3\Lambda_0$ for the twisted affine Lie algebra of type $A_2^{(2)}$ in the principal realization. More specifically, by introducing the concept of…

We give a combinatorial model structure to the category of, not necessarily conilpotent, differential graded (dg) cocommutative coalgebras and an $\infty$-category structure to the category of curved Lie algebras over an algebraically…

The aim of this short note is to establish a 2-equivalence between a certain 2-category of foams and that of singular Soergel bimodules of type A.

This is a continuation of the previous paper (arXiv:2508.01944) in this series. We recontextualise Cirio and Martins' work to motivate our fundamental conjecture that the Drinfeld-Kohno (Lie) 2-algebra has trivial cohomology. It is then…

Given a pivotal module category over a spherical fusion category, we introduce the encircling module, a module over the fusion algebra defined using the pivotal structure, and prove that it is isomorphic to the NIM-rep as a fusion algebra…

Let $\g$ be a simple complex Lie algebra of type $G_2$, $F_4$, or $E_8$, and let $G$ be the unique connected simply connected Lie group with $\mathrm{Lie}(G)=\g$ with compact real form $K$. We prove a triangular decomposition theorem for…

Progress on the conjecture of Banica and Bichon that the classical permutation group is a maximal quantum subgroup of the quantum permutation group remains limited to a handful of small-parameter results. By Tannaka--Krein duality, any…

The overarching goal of this thesis was to develop categorical methods that connect enumerative geometry, as studied in mirror symmetry, with large $N$ gauge theories. In the first part, we established a relation between graph complexes,…

We introduce a functor $\Psi$ that associates to a dioperad $P$ acting on a vector space $V$ a two-colored operad $\Psi(P)$ acting on the pair $(V, V^*)$. The construction is based on a simple pictorial idea: by selecting one input or…

For $q$ generic, Jimbo showed that $q$-tensor space $V_q^{\otimes r}$ (where $V_q$ is the $n$-dimensional vector representation) satisfies Schur--Weyl duality with respect to the commuting actions of the quantized enveloping algebra…

This paper is devoted to studying the centre of the multi-parameter quantum group $U_{q,G}(\mathfrak{g})$ introduced by Okado and Yamane, where $\mathfrak{g}$ is a complex simple Lie algebra, and all parameters lie in general position. We…

We construct a new class of quantum vertex algebras associated with the normalized Yang $R$-matrix. They are obtained as Yangian deformations of certain $\mathcal{S}$-commutative quantum vertex algebras and their $\mathcal{S}$-locality…