We introduce the notions of quasi-Laurent and Laurent families of simple modules over quiver Hecke algebras of arbitrary symmetrizable types. We prove that such a family plays a similar role of a cluster in quantum cluster algebra theory and exhibits a quantum Laurent positivity phenomenon similar to the basis of the quantum unipotent coordinate ring $\mathcal {A}_q(\mathfrak {n}(w))$, coming from the categorification. Then we show that the families of simple modules categorifying Geiß–Leclerc–Schröer (GLS) clusters are Laurent families by using the Poincaré–Birkhoff–Witt (PBW) decomposition vector of a simple module $X$ and categorical interpretation of (co)degree of $[X]$. As applications of such $\mathbb {Z}\mspace {1mu}$-vectors, we define several skew-symmetric pairings on arbitrary pairs of simple modules, and investigate the relationships among the pairings and $\Lambda$-invariants of $R$-matrices in the quiver Hecke algebra theory.

Let $G$ be a reductive group, and let $\check {G}$ be its Langlands dual group. Arkhipov and Bezrukavnikov proved that the Whittaker category on the affine flags ${\operatorname {Fl}}_G$ is equivalent to the category of $\check {G}$-equivariant quasi-coherent sheaves on the Springer resolution of the nilpotent cone. This paper proves this theorem in the quantum case. We show that the twisted Whittaker category on ${\operatorname {Fl}}_G$ and the category of representations of the mixed quantum group are equivalent. In particular, we prove that the quantum category $\mathsf {O}$ is equivalent to the twisted Whittaker category on ${\operatorname {Fl}}_G$ in the generic case. The strong version of our main theorem claims a motivic equivalence between the Whittaker category on ${\operatorname {Fl}}_G$ and a factorization module category, which holds in the de Rham setting, the Betti setting, and the $\ell$-adic setting.

We apply Takesaki’s and Connes’s ideas on structure analysis for type III factors to the study of links (a short term of Markov kernels) appearing in asymptotic representation theory.

Let $\mathfrak {g}$ be a complex semisimple Lie algebra with associated Yangian $Y_{\hbar }\mathfrak {g}$. In the mid-1990s, Khoroshkin and Tolstoy formulated a conjecture which asserts that the algebra $\mathrm {D}Y_{\hbar }\mathfrak {g}$ obtained by doubling the generators of $Y_{\hbar }\mathfrak {g}$, called the Yangian double, provides a realization of the quantum double of the Yangian. We provide a uniform proof of this conjecture over $\mathbb {C}[\kern-1.2pt\![{\hbar }]\!\kern-1.2pt]$ which is compatible with the theory of quantized enveloping algebras. As a by-product, we identify the universal R-matrix of the Yangian with the canonical element defined by the pairing between the Yangian and its restricted dual.

We use the dual functional realization of loop algebras to study the prime irreducible objects in the Hernandez–Leclerc (HL) category for the quantum affine algebra associated with $\mathfrak {sl}_{n+1}$. When the HL category is realized as a monoidal categorification of a cluster algebra (Hernandez and Leclerc (2010, Duke Mathematical Journal 154, 265–341); Hernandez and Leclerc (2013, Symmetries, integrable systems and representations, 175–193)), these representations correspond precisely to the cluster variables and the frozen variables are minimal affinizations. For any height function, we determine the classical decomposition of these representations with respect to the Hopf subalgebra $\mathbf {U}_q(\mathfrak {sl}_{n+1})$ and describe the graded multiplicities of their graded limits in terms of lattice points of convex polytopes. Combined with Brito, Chari, and Moura (2018, Journal of the Institute of Mathematics of Jussieu 17, 75–105), we obtain the graded decomposition of stable prime Demazure modules in level two integrable highest weight representations of the corresponding affine Lie algebra.

We establish an equivalence between two approaches to quantization of irreducible symmetric spaces of compact type within the framework of quasi-coactions, one based on the Enriquez–Etingof cyclotomic Knizhnik–Zamolodchikov (KZ) equations and the other on the Letzter–Kolb coideals. This equivalence can be upgraded to that of ribbon braided quasi-coactions, and then the associated reflection operators (K-matrices) become a tangible invariant of the quantization. As an application we obtain a Kohno–Drinfeld type theorem on type $\mathrm {B}$ braid group representations defined by the monodromy of KZ-equations and by the Balagović–Kolb universal K-matrices. The cases of Hermitian and non-Hermitian symmetric spaces are significantly different. In particular, in the latter case a quasi-coaction is essentially unique, while in the former we show that there is a one-parameter family of mutually nonequivalent quasi-coactions.

We introduce the quantum isomeric supercategory and the quantum affine isomeric supercategory. These diagrammatically defined supercategories, which can be viewed as isomeric analogues of the HOMFLYPT skein category and its affinisation, provide powerful categorical tools for studying the representation theory of the quantum isomeric superalgebras (commonly known as quantum queer superalgebras).

We introduce the combinatorial notion of a q-factorization graph intended as a tool to study and express results related to the classification of prime simple modules for quantum affine algebras. These are directed graphs equipped with three decorations: a coloring and a weight map on vertices, and an exponent map on arrows (the exponent map can be seen as a weight map on arrows). Such graphs do not contain oriented cycles and, hence, the set of arrows induces a partial order on the set of vertices. In this first paper on the topic, beside setting the theoretical base of the concept, we establish several criteria for deciding whether or not a tensor product of two simple modules is a highest-$\ell $-weight module and use such criteria to prove, for type A, that a simple module whose q-factorization graph has a totally ordered vertex set is prime.

We present a categorical point of view on dynamical quantum groups in terms of categories of Harish-Chandra bimodules. We prove Tannaka duality theorems for forgetful functors into the monoidal category of Harish-Chandra bimodules in terms of a slight modification of the notion of a bialgebroid. Moreover, we show that the standard dynamical quantum groups $F(G)$ and $F_q(G)$ are related to parabolic restriction functors for classical and quantum Harish-Chandra bimodules. Finally, we exhibit a natural Weyl symmetry of the parabolic restriction functor using Zhelobenko operators and show that it gives rise to the action of the dynamical Weyl group.

We introduce and study a fermionisation procedure for the cohomological Hall algebra $\mathcal{H}_{\Pi_Q}$ of representations of a preprojective algebra, that selectively switches the cohomological parity of the BPS Lie algebra from even to odd. We do so by determining the cohomological Donaldson–Thomas invariants of central extensions of preprojective algebras studied in the work of Etingof and Rains, via deformed dimensional reduction. Via the same techniques, we determine the Borel–Moore homology of the stack of representations of the $\unicode{x03BC}$ -deformed preprojective algebra introduced by Crawley–Boevey and Holland, for all dimension vectors. This provides a common generalisation of the results of Crawley-Boevey and Van den Bergh on the cohomology of smooth moduli schemes of representations of deformed preprojective algebras and my earlier results on the Borel–Moore homology of the stack of representations of the undeformed preprojective algebra.

We define slope subalgebras in the shuffle algebra associated to a (doubled) quiver, thus yielding a factorization of the universal R-matrix of the double of the shuffle algebra in question. We conjecture that this factorization matches the one defined by [1, 18, 32, 33, 34] using Nakajima quiver varieties.

Let $U_q'({\mathfrak {g}})$ be a quantum affine algebra with an indeterminate $q$, and let $\mathscr {C}_{\mathfrak {g}}$ be the category of finite-dimensional integrable $U_q'({\mathfrak {g}})$-modules. We write $\mathscr {C}_{\mathfrak {g}}^0$ for the monoidal subcategory of $\mathscr {C}_{\mathfrak {g}}$ introduced by Hernandez and Leclerc. In this paper, we associate a simply laced finite-type root system to each quantum affine algebra $U_q'({\mathfrak {g}})$ in a natural way and show that the block decompositions of $\mathscr {C}_{\mathfrak {g}}$ and $\mathscr {C}_{\mathfrak {g}}^0$ are parameterized by the lattices associated with the root system. We first define a certain abelian group $\mathcal {W}$ (respectively $\mathcal {W} _0$) arising from simple modules of $\mathscr {C}_{\mathfrak {g}}$ (respectively $\mathscr {C}_{\mathfrak {g}}^0$) by using the invariant $\Lambda ^\infty$ introduced in previous work by the authors. The groups $\mathcal {W}$ and $\mathcal {W} _0$ have subsets $\Delta$ and $\Delta _0$ determined by the fundamental representations in $\mathscr {C}_{\mathfrak {g}}$ and $\mathscr {C}_{\mathfrak {g}}^0$, respectively. We prove that the pair $( \mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} _0, \Delta _0)$ is an irreducible simply laced root system of finite type and that the pair $( \mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} , \Delta )$ is isomorphic to the direct sum of infinite copies of $( \mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} _0, \Delta _0)$ as a root system.

We explicitly determine the defining relations of all quantum symmetric pair coideal subalgebras of quantised enveloping algebras of Kac–Moody type. Our methods are based on star products on noncommutative ${\mathbb N}$-graded algebras. The resulting defining relations are expressed in terms of continuous q-Hermite polynomials and a new family of deformed Chebyshev polynomials.

We compare crystal combinatorics of the level $2$ Fock space with the classification of unitary irreducible representations of type B rational Cherednik algebras to study how unitarity behaves under parabolic restriction. We show that the crystal operators that remove boxes preserve the combinatorial conditions for unitarity, and that the parabolic restriction functors categorifying the crystals send irreducible unitary representations to unitary representations. Furthermore, we find the supports of the unitary representations.

We prove an explicit inverse Chevalley formula in the equivariant K-theory of semi-infinite flag manifolds of simply laced type. By an ‘inverse Chevalley formula’ we mean a formula for the product of an equivariant scalar with a Schubert class, expressed as a $\mathbb {Z}\left [q^{\pm 1}\right ]$ -linear combination of Schubert classes twisted by equivariant line bundles. Our formula applies to arbitrary Schubert classes in semi-infinite flag manifolds of simply laced type and equivariant scalars $e^{\lambda }$ , where $\lambda $ is an arbitrary minuscule weight. By a result of Stembridge, our formula completely determines the inverse Chevalley formula for arbitrary weights in simply laced type except for type $E_8$ . The combinatorics of our formula is governed by the quantum Bruhat graph, and the proof is based on a limit from the double affine Hecke algebra. Thus our formula also provides an explicit determination of all nonsymmetric q-Toda operators for minuscule weights in ADE type.

We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided multi-fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively two-, three- and four-dimensional framed local topological field theories. In particular, we produce a framed three-dimensional local topological field theory attached to the category of representations of a quantum group at any value of $q$.

We study webs in quantum type C, focusing on the rank three case. We define a linear pivotal category $\textbf {Web}(\mathfrak {sp}_{6})$ diagrammatically by generators and relations, and conjecture that it is equivalent to the category $\textbf {FundRep}(U_q(\mathfrak {sp}_{6}))$ of quantum $\mathfrak {sp}_{6}$ representations generated by the fundamental representations, for generic values of the parameter q. We prove a number of results in support of this conjecture, most notably that there is a full, essentially surjective functor $\textbf {Web}(\mathfrak {sp}_{6}) \rightarrow \textbf {FundRep}(U_q(\mathfrak {sp}_{6}))$ , that all $\textrm {Hom}$ -spaces in $\textbf {Web}(\mathfrak {sp}_{6})$ are finite-dimensional, and that the endomorphism algebra of the monoidal unit in $\textbf {Web}(\mathfrak {sp}_{6})$ is one-dimensional. The latter corresponds to the statement that all closed webs can be evaluated to scalars using local relations; as such, we obtain a new approach to the quantum $\mathfrak {sp}_{6}$ link invariants, akin to the Kauffman bracket description of the Jones polynomial.

We present new examples of finite-dimensional Nichols algebras over fields of characteristic 2 from braided vector spaces that are not of diagonal type, admit realizations as Yetter–Drinfeld modules over finite abelian groups, and are analogous to Nichols algebras of finite Gelfand–Kirillov dimension in characteristic 0. New finite-dimensional pointed Hopf algebras over fields of characteristic 2 are obtained by bosonization with group algebras of suitable finite abelian groups.

We settle several fundamental questions about the theory of universal deformation quantization of Lie bialgebras by giving their complete classification up to homotopy equivalence. Moreover, we settle these questions in a greater generality: we give a complete classification of the associated universal formality maps. An important new technical ingredient introduced in this paper is a polydifferential endofunctor ${\mathcal {D}}$ in the category of augmented props with the property that for any representation of a prop ${\mathcal {P}}$ in a vector space $V$ the associated prop ${\mathcal {D}}{\mathcal {P}}$ admits an induced representation on the graded commutative algebra $\odot ^\bullet V$ given in terms of polydifferential operators. Applying this functor to the minimal resolution $\widehat {\mathcal {L}\textit{ieb}}_\infty$ of the genus completed prop $\widehat {\mathcal {L}\textit{ieb}}$ of Lie bialgebras we show that universal formality maps for quantizations of Lie bialgebras are in one-to-one correspondence with morphisms of dg props

\[F: \mathcal{A}\textit{ssb}_\infty \longrightarrow {\mathcal{D}}\widehat{\mathcal{L}\textit{ieb}}_\infty \]

$\mathcal {A}\textit{ssb}_\infty$

$\mathfrak{A}$

${\mathcal {L}} ie_\infty$

${\mathcal {L}} ie_\infty$

$\mathsf {Def}({\mathcal {A}} ss{\mathcal {B}}_\infty \rightarrow {\mathcal {E}} nd_{\odot ^\bullet V})$

$\mathsf {Def}({\mathcal {L}} ie{\mathcal {B}}\rightarrow {\mathcal {E}} nd_V)$

$\odot V$

$V$

$\mathsf {Def}(\mathcal {A}\textit{ssb}_\infty \stackrel {F}{\rightarrow } {\mathcal {D}}\widehat {\mathcal {L}\textit{ieb}}_\infty )$

$\{F_\mathfrak{A}\}$

$GRT=GRT_1\rtimes {\mathbb {K}}^*$

$\{\mathfrak{A}\}$

Generalised quantum determinantal rings are the analogue in quantum matrices of Schubert varieties. Maximal orders are the noncommutative version of integrally closed rings. In this paper, we show that generalised quantum determinantal rings are maximal orders. The cornerstone of the proof is a description of generalised quantum determinantal rings, up to a localisation, as skew polynomial extensions.