INTEGRABLE MODULES OVER QUANTUM SYMMETRIC PAIR COIDEAL SUBALGEBRAS

Verma and simple modules for quantum groups at non-abelian groups

Given a quantum group, we prove that the canonical bases of the tensor products of its integrable highest weight modules can be obtained from the canonical bases of the integrable highest weight modules of a bigger quantum group. As a result, based on the positivity of the canonical bases of the integrable highest weight modules due to Lusztig, we prove that the canonical bases of the tensor products have the positivity.

Let $(\bf U, \bf U^\imath)$ be a quantum symmetric pair of Kac-Moody type. The $\imath$quantum groups $\bf U^\imath$ and the universal $\imath$quantum groups $\widetilde{\bf U}^\imath$ can be viewed as a generalization of quantum groups and Drinfeld doubles $\widetilde{\bf U}$. In this paper we formulate and establish Serre-Lusztig relations for $\imath$quantum groups in terms of $\imath$divided powers, which are an $\imath$-analog of Lusztig's higher order Serre relations for quantum groups. This has applications to braid group symmetries on $\imath$quantum groups.

The free-fermion point refers to a GL ( 2 ) × GL ( 1 ) parametrized Yang-Baxter equation within the six-vertex model. It has been known for a long time that this is connected with the quantum group U q ( gl ( 1 | 1 ) ) . We demonstrate that R -matrices from the finite quantum superalgebra U q ( gl ( 1 | 1 ) ) produce a dense subset of the free-fermionic Yang-Baxter equations of the six-vertex model, matching those of the prime, simple modules in the affine quantum superalgebra U q ( gl ˆ ( 1 | 1 ) ) . Either of these quantum groups can be used to generate the full free-fermion point, and we discuss them both. Our discussion includes 6 families of six-vertex models used by Brubaker, Bump, and Friedberg in connection with Tokuyama's theorem, a deformation of the Weyl character formula. Thus our work gives quantum group interpretations for those models, known informally as Tokuyama ice.

For quantum symmetric pairs $(\textbf {U}, \textbf {U}^\imath )$ of Kac–Moody type, we construct $\imath$-canonical bases for the highest weight integrable $\textbf U$-modules and their tensor products regarded as $\textbf {U}^\imath$-modules, as well as an $\imath$-canonical basis for the modified form of the $\imath$-quantum group $\textbf {U}^\imath$. A key new ingredient is a family of explicit elements called $\imath$-divided powers, which are shown to generate the integral form of $\dot {\textbf {U}}^\imath$. We prove a conjecture of Balagovic–Kolb, removing a major technical assumption in the theory of quantum symmetric pairs. Even for quantum symmetric pairs of finite type, our new approach simplifies and strengthens the integrality of quasi-$K$-matrix and the constructions of $\imath$-canonical bases, by avoiding a case-by-case rank-one analysis and removing the strong constraints on the parameters in a previous work.

Let $$\widetilde{{{\mathbf {U}}}}^\imath $$ be the universal $$\imath $$ quantum group arising from quantum symmetric pairs. Recently, Lu and Wang formulated a Drinfeld-type presentation for $$\widetilde{{{\mathbf {U}}}}^\imath $$ of split affine ADE type. In this paper, we generalize their results by establishing a Drinfeld-type presentation for $$\widetilde{{{\mathbf {U}}}}^\imath $$ of arbitrary split affine type.

It is natural to define a Coxeter transformation in quantum groups by the product of Lusztig's symmetries in some order. In this note, we show that the Ringel–Hall algebra approach enables us to determine the behavior of its action in case of finite type.

We construct equivariant vector bundles over quantum projective spaces making use of parabolic Verma modules over the quantum general linear group. Using an alternative realization of the quantized coordinate ring of projective space as a subalgebra in the algebra of functions on the quantum group, we reformulate quantum vector bundles in terms of quantum symmetric pairs. In this way, we prove complete reducibility of modules over the corresponding coideal stabilizer subalgebras, via the quantum Frobenius reciprocity.

The $q$-deformed associative algebra $U_q'(\mathfrak{so}_n)$ is an example of an $\imath$quantum group. Finite-dimensional simple modules over $U_q'(\mathfrak{so}_n)$ were constructed by Gavrilik and Klimyk which limit to finite-dimensional simple modules over the universal enveloping algebra $U(\mathfrak{so}_n)$ as $q \to 1$. These modules are examples of Gelfand-Tsetlin modules since they are equal to the direct sum of common finite-dimensional eigenspaces under the action of a commutative subalgebra called the Gelfand-Tsetlin subalgebra $\Gamma \subset U_q'(\mathfrak{so}_n)$. In this thesis we begin to study the category of Gelfand-Tsetlin modules over $U_q'(\mathfrak{so}_n)$, which contains the category of finite-dimensional modules. Following the example of Drozd, Futorny and Ovsienko for $U(\mathfrak{gl}_n)$; Mazorchuk and Turowska for $U_q(\mathfrak{gl}_n)$; and Mazorchuk for $U(\mathfrak{so}_n)$; we construct infinite-dimensional analogs to the Gavrilik-Klimyk classical $U_q'(\mathfrak{so}_n)$-modules by removing the conditions on the combinatorial patterns parameterizing the bases. We call these generic Gelfand-Tsetlin modules. We find upper bounds for lengths of these modules for $n=3$ and $n=4$. Taking the limit as $q \to 1$, we obtain generic Gelfand-Tsetlin modules over $U(\mathfrak{so}_n)$ whose matrix coefficients are rational functions, thus extending the work of Mazorchuk. We use these large new families of modules to embed $U_q'(\mathfrak{so}_n)$ and $U(\mathfrak{so}_n)$ into skew group algebras of shift operators. These embeddings map the Gelfand-Tsetlin subalgebras to certain invariant algebras under the actions of the direct products of ($q$-)Weyl groups. Lastly, we use this fact to show the Gelfand-Tsetlin subalgebras are Harish-Chandra subalgebras, as defined by Drozd-Futorny-Ovsienko.

Let $\mathsf{u}_q(\mathfrak{g})$ be the small quantum group associated with a complex semisimple Lie algebra $\mathfrak{g}$ and a primitive root of unity q, satisfying certain restrictions. We establish the equivalence between three different actions of $\mathfrak{g}$ on the center of $\mathsf{u}_q(\mathfrak{g})$ and on the higher derived center of $\mathsf{u}_q(\mathfrak{g})$. Based on the triviality of this action for $\mathfrak{g} = \mathfrak{sl}_2, \mathfrak{sl}_3, \mathfrak{sl}_4$, we conjecture that, in finite type A, central elements of the small quantum group $\mathsf{u}_q(\mathfrak{sl}_n)$ arise as the restriction of central elements in the big quantum group $\mathsf{U}_q(\mathfrak{sl}_n)$. We also study the role of an ideal $\mathsf{z}_{\mathrm{Hig}}$ known as the Higman ideal in the center of $\mathsf{u}_q(\mathfrak{g})$. We show that it coincides with the intersection of the Harish-Chandra center and its Fourier transform, and compute the dimension of $\mathsf{z}_{\mathrm{Hig}}$ in type A. As an illustration we provide a detailed explicit description of the derived center of $\mathsf{u}_q(\mathfrak{sl}_2)$ and its various symmetries.

We obtain the explicit generators of the annihilator ideal for the tensor product of any finite dimensional simple module over quantum group [Formula: see text], by using the weight property of ideals in [Formula: see text] when q is not a root of unity. As an application, we give a presentation of quantum generalized Schur algebra.

Let $\Sigma$ be a finite type surface, and $G$ a complex algebraic simple Lie group with Lie algebra $\mathfrak{g}$. The quantum moduli algebra of $(\Sigma,G)$ is a quantization of the ring of functions of $X_G(\Sigma)$, the variety of $G$-characters of $\pi_1(\Sigma)$, introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the mid '90s. It can be realized as the invariant subalgebra of so-called graph algebras, which are $U_q(\mathfrak{g})$-module-algebras associated to graphs on $\Sigma$, where $U_q(\mathfrak{g})$ is the quantum group corresponding to $G$. We study the structure of the quantum moduli algebra in the case where $\Sigma$ is a sphere with $n+1$ open disks removed, $n\geq 1$, using the graph algebra of the "daisy" graph on $\Sigma$ to make computations easier. We provide new results that hold for arbitrary $G$ and generic $q$, and develop the theory in the case where $q=\epsilon$, a primitive root of unity of odd order, and $G={\rm SL}(2,{\mathbb C})$. In such a situation we introduce a Frobenius morphism that provides a natural identification of the center of the daisy graph algebra with a finite extension of the coordinate ring $\mathcal{O}(G^n)$. We extend the quantum coadjoint action of De-Concini-Kac-Procesi to the daisy graph algebra, and show that the associated Poisson structure on the center corresponds by the Frobenius morphism to the Fock-Rosly Poisson structure on $\mathcal{O}(G^n)$. We show that the set of fixed elements of the center under the quantum coadjoint action is a finite extension of ${\mathbb C}[X_G(\Sigma)]$ endowed with the Atiyah-Bott-Goldman Poisson structure. Finally, by using Wilson loop operators we identify the Kauffman bracket skein algebra $K_{\zeta}(\Sigma)$ at $\zeta:={\rm i}\epsilon^{1/2}$ with this quantum moduli algebra specialized at $q=\epsilon$.

On the Vassiliev knot invariants

In this paper we study the BGG-categories \mathcal O_q associated to quantum groups. We prove that many properties of the ordinary BGG-category \mathcal O for a semisimple complex Lie algebra carry over to the quantum case. Of particular interest is the case when q is a complex root of unity. Here we prove a tensor decomposition for both simple modules, projective modules, and indecomposable tilting modules. Using the known Kazhdan-Lusztig conjectures for \mathcal O and for finite dimensional U_q -modules we are able to determine all irreducible characters as well as the characters of all indecomposable tilting modules in \mathcal O_q . As a consequence of these results we are able to recover also a known result, namely that the generic quantum case behaves like the classical category \mathcal O .

We introduce the notion of affinizations and R-matrices for arbitrary quiver Hecke algebras. It is shown that they enjoy similar properties to those for symmetric quiver Hecke algebras. We next define a duality datum \mathcal{D} and construct a tensor functor \mathfrak F^{\mathcal{D}}\colon \mathrm {Mod}_gr(R^{\mathcal D}) \to \mathrm {Mod}_gr(R) between graded module categories of quiver Hecke algebras R and R^{\mathcal{D}} arising from \mathcal{D} . The functor \mathfrak F^{\mathcal{D}} sends finite-dimensional modules to finite-dimensional modules, and is exact when R^{\mathcal{D}} is of finite type. It is proved that affinizations of real simple modules and their R-matrices give a duality datum. Moreover, the corresponding duality functor sends a simple module to a simple module or zero when R^{\mathcal{D}} is of finite type. We give several examples of the functors \mathfrak F^{\mathcal{D}} from the graded module category of the quiver Hecke algebra of type D_\ell , C_\ell , B_{\ell-1} , A_{\ell-1} to that of type A_\ell , A_\ell , B_\ell , B_\ell , respectively.