Related papers: Invariant theory of $\imath$quantum groups of type…
We construct quantum supersymmetric pairs $({\bold U},{\bold U}^\imath)$ of type AIII and elucidate their fundamental properties. An $\imath$Schur duality between the $\imath$quantum supergroup ${\bold U}^\imath$ and the Hecke algebra of…
We provide a geometric realization of the quasi-split affine $\imath$quantum group of type AIII$_{2n-1}^{(\tau)}$ in terms of equivariant K-groups of non-connected Steinberg varieties of type C. This uses a new Drinfeld type presentation of…
We survey some recent developments on the theory of dual canonical bases for quantum groups and $\imath$quantum groups. The $\imath$quiver algebras were introduced by Wang and the first author, which are used to give two realizations of…
We develop the non-commutative polynomial version of the invariant theory for the quantum general linear supergroup ${\rm{ U}}_q(\mathfrak{gl}_{m|n})$. A non-commutative ${\rm{ U}}_q(\mathfrak{gl}_{m|n})$-module superalgebra…
Certain quantum topological invariants of three manifolds can be written in the form of the Gaussian sum. It is shown that such topological invariants can be approximated efficiently by a quantum computer. The invariants discussed here are…
We show how to construct, starting from a quasi-Hopf algebra, or quasi-quantum group, invariants of knots and links. In some cases, these invariants give rise to invariants of the three-manifolds obtained by surgery along these links. This…
The classical invariant theory for the queer Lie superalgebra is an investigation of the $\mathrm{U}(\mathfrak{q}_n)$-invariant sub-superalgebra of the symmetric superalgebra $\mathrm{Sym}(V^{\oplus r}\oplus V^{*\oplus s})$ for…
We develop a geometric approach toward an interplay between a pair of quantum Schur algebras of arbitrary finite type. Then by Beilinson-Lusztig-MacPherson's stabilization procedure in the setting of partial flag varieties of type A (resp.…
We develop invariant theory for the quantum group ${\rm U}_q$ of $G_2$ at generic $q$ in the setting of braided symmetric algebras. Let ${\mathcal A}_m$ be the braided symmetric algebra over $m$-copies of the $7$-dimensional simple ${\rm…
The (universal) $\imath$quantum groups are as a vast generalization of (Drinfeld double) quantum groups. We establish an algebra homomorphism from universal $\imath$quantum group of split type to a certain quantum torus, which can be viewed…
let $\widetilde{\bf U}^\imath$ be a quasi-split universal $\imath$quantum group associated to a quantum symmetric pair $(\widetilde{\bf U}, \widetilde{\bf U}^\imath)$ of Kac-Moody type with a diagram involution $\tau$. We establish the…
The Reshetikhin - Turaeve approach to topological invariants of three - manifolds is generalized to quantum supergroups. A general method for constructing three - manifold invariants is developed, which requires only the study of the…
Expanding the classic works of Kazhdan-Lusztig and Deodhar, we establish bar involutions and canonical (i.e., quasi-parabolic KL) bases on quasi-permutation modules over the type B Hecke algebra, where the bases are parameterized by cosets…
A quantum symmetric pair consists of a quantum group $\mathbf U$ and its coideal subalgebra ${\mathbf U}^{\imath}_{\boldsymbol{\varsigma}}$ with parameters $\boldsymbol{\varsigma}$ (called an $\imath$quantum group). We initiate a Hall…
Various constructions for quantum groups have been generalized to $\imath$-quantum groups. Such generalization is called $\imath$-program. In this paper, we fill one of parts in the $\imath$-program. Namely, we provide an equivariant…
This is the first of two papers on quasi-split affine quantum symmetric pairs $\big(\widetilde{\mathbf U}(\widehat{\mathfrak g}), \widetilde{{\mathbf U}}^\imath \big)$, focusing on the real rank one case, i.e., $\mathfrak{g}=…
We consider the quantum-group self-duality equation in the framework of the gauge theory on a deformed twistor space. Quantum deformation of the Atiyah-Drinfel'd-Hitchin-Manin and t'Hooft multi-instanton solutions are constructed.
We survey some recent development on the theory of $\imath$Hall algebras. Starting from $\imath$quivers (aka quivers with involutions), we construct a class of 1-Gorenstein algebras called $\imath$quiver algebras, whose semi-derived Hall…
We develop a general theory of canonical bases for quantum symmetric pairs $(\mathbf{U}, \mathbf{U}^\imath)$ with parameters of arbitrary finite type. We construct new canonical bases for the simple integrable $\mathbf{U}$-modules and their…
By choosing an unconventional polarization of the connection phase space in (2+1)-gravity on the torus, a modular invariant quantum theory is constructed. Unitary equivalence to the ADM-quantization is shown.