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.

Given a quiver with potential $(Q,W)$, Kontsevich–Soibelman constructed a cohomological Hall algebra (CoHA) on the critical cohomology of the stack of representations of $(Q,W)$. Special cases of this construction are related to work of Nakajima, Varagnolo, Schiffmann–Vasserot, Maulik–Okounkov, Yang–Zhao, etc. about geometric constructions of Yangians and their representations; indeed, given a quiver Q, there exists an associated pair $(\widetilde{Q}, \widetilde{W})$ whose CoHA is conjecturally the positive half of the Maulik–Okounkov Yangian $Y_{\text {MO}}(\mathfrak {g}_{Q})$.

For a quiver with potential $(Q,W)$, we follow a suggestion of Kontsevich–Soibelman and study a categorification of the above algebra constructed using categories of singularities. Its Grothendieck group is a K-theoretic Hall algebra (KHA) for quivers with potential. We construct representations using framed quivers, and we prove a wall-crossing theorem for KHAs. We expect the KHA for $(\widetilde{Q}, \widetilde{W})$ to recover the positive part of quantum affine algebra $U_{q}(\widehat {\mathfrak {g}_{Q}})$ defined by Okounkov–Smirnov.

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.

The notion of quantized characters was introduced in our previous paper as a natural quantization of characters in the context of asymptotic representation theory for quantum groups. As in the case of ordinary groups, the representation associated with any extreme quantized character generates a von Neumann factor. From the viewpoint of operator algebras (and measurable dynamical systems), it is natural to ask what is the Murray–von Neumann–Connes type of the resulting factor. In this paper, we give a complete solution to this question when the inductive system is of quantum unitary groups $U_{q}(N)$.

The geometric Satake correspondence gives an equivalence of categories between the representations of a semisimple group $G$ and the spherical perverse sheaves on the affine Grassmannian $Gr$ of its Langlands dual group. Bezrukavnikov and Finkelberg developed a derived version of this equivalence which relates the derived category of $G^{\vee }$-equivariant constructible sheaves on $Gr$ with the category of $G$-equivariant ${\mathcal{O}}(\mathfrak{g})$-modules. In this paper, we develop a K-theoretic version of the derived geometric Satake which involves the quantum group $U_{q}\mathfrak{g}$. We define a convolution category $K\operatorname{Conv}(Gr)$ whose morphism spaces are given by the $G^{\vee }\times \mathbb{C}^{\times }$-equivariant algebraic K-theory of certain fibre products. We conjecture that $K\operatorname{Conv}(Gr)$ is equivalent to a full subcategory of the category of $U_{q}\mathfrak{g}$-equivariant ${\mathcal{O}}_{q}(G)$-modules. We prove this conjecture when $G=\operatorname{SL}_{n}$. A key tool in our proof is the $\operatorname{SL}_{n}$ spider, which is a combinatorial description of the category of $U_{q}\mathfrak{sl}_{n}$ representations. By applying horizontal trace, we show that the annular $\operatorname{SL}_{n}$ spider describes the category of $U_{q}\mathfrak{sl}_{n}$-equivariant ${\mathcal{O}}_{q}(\operatorname{SL}_{n})$-modules. Then we use quantum loop algebras to relate the annular $\operatorname{SL}_{n}$ spider to $K\operatorname{Conv}(Gr)$. This gives a combinatorial/diagrammatic description of both categories and proves our conjecture.

We formulate and study Howe–Moore type properties in the setting of quantum groups and in the setting of rigid $C^{\ast }$-tensor categories. We say that a rigid $C^{\ast }$-tensor category ${\mathcal{C}}$ has the Howe–Moore property if every completely positive multiplier on ${\mathcal{C}}$ has a limit at infinity. We prove that the representation categories of $q$-deformations of connected compact simple Lie groups with trivial center satisfy the Howe–Moore property. As an immediate consequence, we deduce the Howe–Moore property for Temperley–Lieb–Jones standard invariants with principal graph $A_{\infty }$. These results form a special case of a more general result on the convergence of completely bounded multipliers on the aforementioned categories. This more general result also holds for the representation categories of the free orthogonal quantum groups and for the Kazhdan–Wenzl categories. Additionally, in the specific case of the quantum groups $\text{SU}_{q}(N)$, we are able, using a result of the first-named author, to give an explicit characterization of the central states on the quantum coordinate algebra of $\text{SU}_{q}(N)$, which coincide with the completely positive multipliers on the representation category of $\text{SU}_{q}(N)$.

We show how to use Jantzen’s sum formula for Weyl modules to prove semisimplicity criteria for endomorphism algebras of $\mathbf{U}_{q}$-tilting modules (for any field $\mathbb{K}$ and any parameter $q\in \mathbb{K}-\{0,-1\}$). As an application, we recover the semisimplicity criteria for the Hecke algebras of types $\mathbf{A}$ and $\mathbf{B}$, the walled Brauer algebras and the Brauer algebras from our more general approach.

In the context of varieties of representations of arbitrary quivers, possibly carrying loops, we define a generalization of Lusztig Lagrangian subvarieties. From the combinatorial study of their irreducible components arises a structure richer than the usual Kashiwara crystals. Along with the geometric study of Nakajima quiver varieties, in the same context, this yields a notion of generalized crystals, coming with a tensor product. As an application, we define the semicanonical basis of the Hopf algebra generalizing quantum groups, which was already equipped with a canonical basis. The irreducible components of the Nakajima varieties provide the family of highest weight crystals associated to dominant weights, as in the classical case.

It is known that the normalized standard generators of the free orthogonal quantum group $O_{N}^{+}$ converge in distribution to a free semicircular system as $N\,\to \,\infty$. In this note, we substantially improve this convergence result by proving that, in addition to distributional convergence, the operator normof any non-commutative polynomial in the normalized standard generators of $O_{N}^{+}$ converges as $N\,\to \,\infty$ to the operator norm of the corresponding non-commutative polynomial in a standard free semicircular system. Analogous strong convergence results are obtained for the generators of free unitary quantum groups. As applications of these results, we obtain a matrix-coefficient version of our strong convergence theorem, and we recover a well-known ${{\mathcal{L}}^{2}}\,-\,{{\mathcal{L}}^{\infty }}$ norm equivalence for noncommutative polynomials in free semicircular systems.

We construct a duality isomorphism in equivariant periodic cyclic homology analogous to Baaj-Skandalis duality in equivariant Kasparov theory. As a consequence we obtain general versions of the Green-Julg theorem and the dual Green-Julg theorem in periodic cyclic theory.

Throughout we work within the framework of bornological quantum groups, thus in particular incorporating at the same time actions of arbitrary classical Lie groups as well as actions of compact or discrete quantum groups. An important ingredient in the construction of our duality isomorphism is the notion of a modular pair for a bornological quantum group, closely related to the concept introduced by Connes and Moscovici in their work on cyclic cohomology for Hopf algebras.

We complete the calculation of the twisted cyclic homology of the quantised coordinate ring = ℂq [SL(2)] of SL(2) that we began in [14]. In particular, a nontrivial cyclic 3-cocycle is constructed which also has a nontrivial class in Hochschild cohomology and thus should be viewed as a noncommutative geometry analogue of a volume form.

Let G be a connected, simply connected, simple complex algebraic group and let ϵ be a primitive ℓth root of one, ℓ odd and 3∤ℓ if G is of type G2. We determine all Hopf algebra quotients of the quantized coordinate algebra 𝒪ϵ(G).

We construct families of irreducible representations for a class of quantum groups Uq(fm(K,H). First, we realize these quantum groups as hyperbolic algebras. Such a realization yields natural families of irreducible weight representations for Uq(fm(K,H)). Second, we study the relationship between Uq(fm(K,H)) and Uq(fm(K)). As a result, any finite-dimensional weight representation of Uq(fm(K,H)) is proved to be completely reducible. Finally, we study the Whittaker model for the center of Uq(fm(K,H)), and a classification of all irreducible Whittaker representations of Uq(fm(K,H)) is obtained.

We study braided Hochschild and cyclic homology of ribbon algebras in braided monoidal categories, as introduced by Baez and by Akrami and Majid. We compute this invariant for several examples coming from quantum groups and braided groups.

We provide an alternative approach to the Faddeev–Reshetikhin–Takhtajan presentation of the quantum group $\uqg$, with $L$-operators as generators and relations ruled by an $R$-matrix. We look at $\uqg$ as being generated by the quantum Borel subalgebras $U_q(\mathfrak{b}_+)$ and $U_q(\mathfrak{b}_-)$, and use the standard presentation of the latter as quantum function algebras. When $\mathfrak{g}=\mathfrak{gl}_n$, these Borel quantum function algebras are generated by the entries of a triangular $q$-matrix. Thus, eventually, $U_q(\mathfrak{gl}_n)$ is generated by the entries of an upper triangular and a lower triangular $q$-matrix, which share the same diagonal. The same elements generate over $\Bbbk[q,q^{-1}]$ the unrestricted $\Bbbk [q,q^{-1}]$-integral form of $U_q(\mathfrak{gl}_n)$ of De Concini and Procesi, which we present explicitly, together with a neat description of the associated quantum Frobenius morphisms at roots of 1. All this holds, mutatis mutandis, for $\mathfrak{g}=\mathfrak{sl}_n$ too.

we interpret the five-parameter family of macdonald–koornwinder polynomials as vector valued spherical functions on quantum grassmannians.

For quantized irreducible flag manifolds the locally finite part of the dual coalgebra is shown to coincide with a natural quotient coalgebra $\overline{U}$ of $U_q ( \mathfrak{g} )$. On the way the coradical filtration of $\overline{U}$ is determined. A graded version of the duality between $\overline{U}$ and the quantized coordinate ring is established. This leads to a natural construction of several examples of quantized vector spaces.

As an application, covariant first order differential calculi on quantized irreducible flag manifolds are classified.

This paper continues the study of quantised function algebras $\mathcal{O}_{\epsilon}[G]$ of a semisimple group $G$ at an $\ell$th root of unity $\epsilon$. These algebras were introduced by De Concini and Lyubashenko in 1994, and studied further by De Concini and Procesi and by Gordon, amongst others. Our main purpose here is to increase understanding of the finite-dimensional factor algebras $\mathcal{O}_{\epsilon}[G](g)$, for $g \in G$. We determine the representation type and block structure of these factors, and (for many $g$) describe them up to isomorphism. A series of parallel results is obtained for the quantised Borel algebras $U_{\epsilon}^{\geq 0}$ and $U_{\epsilon}^{\leq 0}$.

2000 Mathematical Subject Classification: 16W35, 17B37.