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We construct a unique braid group action on modified $q$-Weyl algebra $\mathbf A_q(S)$. Under this action, we give a realization of the braid group action on quasi-split $\imath$quantum groups $^{\imath}\mathbf U(S)$ of type $\mathrm{AIII}$. Furthermore, we directly construct a unique braid group action on polynomial ring $\mathbb P$ which is compatible with the braid group action on $\mathbf A_q(S)$ and $^{\imath}\mathbf U(S)$.

Entanglement, a fundamental feature of quantum mechanics, has long been recognized as a valuable resource in enabling secure communications and surpassing classical limits. However, previous research has primarily concentrated on static entangled states generated at a single point in time, overlooking the crucial role of the quantum dynamics responsible for creating such states. Here, we propose a framework for investigating entanglement across multiple time points, termed temporal entanglement, and demonstrate that the performance of a quantum network in transmitting information is inherently dependent on its temporal entanglement. Through case studies, we showcase the capabilities of our framework in enhancing conventional quantum teleportation and achieving exponential performance growth in the protocol of quantum repeaters. Additionally, our framework effectively doubles the communication distance in certain noise models. Our results address the longstanding question surrounding temporal entanglement within non-Markovian processes and its impact on quantum communication, thereby pushing the frontiers of quantum information science., 14 pages (main text) + 64 pages (supplemental material), 6 figures (main text) + 30 figures (supplemental material). Comments welcome!

With the help of path realization and affine energy function, we give a Young wall construction of level-1 highest weight crystals $B(\lambda)$ over $U_{q}(G_{2}^{(1)})$ and $U_{q}(D_{4}^{(3)})$. Our construction is based on four different shapes of colored blocks, $\mathbf O$-block, $\mathbf I$-block, $\mathbf L$-block and $\mathbf{LL}$-block, obtained by cutting the unit cube in three different ways.

In this paper, we develop the perfect basis theory for quantum Borcherds-Bozec algebras $U_{q}(\mathfrak g)$ and their irreducible highest weight modules $V(\lambda)$. We show that the perfect graph (resp. dual perfect graph) of every perfect basis (resp. dual perfect basis) of $U_{q}^{-}(\mathfrak g)$ (resp. $V(\lambda)$) is isomorphic to $B(\infty)$ (resp. $B(\lambda)$). For this purpose, we define a new class of Kashiwara operators which is different from the one given by Bozec and prove all the interlocking inductive statements in Kashiwara's grand loop argument, which shows the existence and the uniqueness of crystal bases for quantum Borcherds-Bozec algebras.

For a quasi-split Satake diagram, we define a modified $q$-Weyl algebra, and show that there is an algebra homomorphism between it and the corresponding $\imath$quantum group. In other words, we provide a differential operator approach to $\imath$quantum groups. Meanwhile, the Oscillator representations are obtained. The crystal basis of the irreducible subrepresentations of these Oscillator representations are constructed.

We provide a construction of global bases for quantum Borcherds-Bozec algebras and their integrable highest weight representations., All authors contribute equally

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 K-theory approach to $\imath$-quantum groups associated to the Satake diagram in \eqref{eq1}, which is the Langlands dual picture of that constructed in \cite{BKLW14}, where a geometric realization of the $\imath$-quantum group is provided by using perverse sheaves. As an application of the main results, we prove Li's conjecture \cite{L18} for the special cases with the satake diagram in \eqref{eq1}., Comment: 28 pages

We introduce an affine Schur algebra via the affine Hecke algebra associated to Weyl group of affine type C. We establish multiplication formulas on the affine Hecke algebra and affine Schur algebra. Then we construct monomial bases and canonical bases for the affine Schur algebra. The multiplication formula allows us to establish a stabilization property of the family of affine Schur algebras that leads to the modified version of an algebra ${\mathbf K}^{\mathfrak c}_n$. We show that ${\mathbf K}^{\mathfrak c}_n$ is a coideal subalgebra of quantum affine algebra ${\bf U}(\hat{\mathfrak{gl}}_n)$, and $\big({\mathbf U}(\hat{ \mathfrak{gl}}_n), {\mathbf K}^{\mathfrak c}_n)$ forms a quantum symmetric pair. The modified coideal subalgebra is shown to admit monomial and stably canonical bases. We also formulate several variants of the affine Schur algebra and the (modified) coideal subalgebra above, as well as their monomial and canonical bases. This work provides a new and algebraic approach which complements and sheds new light on our previous geometric approach on the subject. In the appendix by four of the authors, new length formulas for the Weyl groups of affine classical types are obtained in a symmetrized fashion., v2: added an appendix "Length formulas in symmetrized forms". 87 pages, 4 figures v3: minor revisions, reference updated. To appear in the Memoirs of the American Mathematical Society

We establish algebraically and geometrically a duality between the Iwahori-Hecke algebra of type D and two new quantum algebras arising from the geometry of N-step isotropic flag varieties of type D. This duality is a type D counterpart of the Schur-Jimbo duality of type A and the Schur-like duality of type B/C discovered by Bao-Wang. The new algebras play a role in the type D duality similar to the modified quantum gl(N) in type A, and the modified coideal subalgebras of quantum gl(N) in type B/C. We construct canonical bases for these two algebras., 40 pages

We introduce a multi-parameter twisted Hopf algebra associated with a root datum and show that its modified form is isomorphic to Lusztig's modified quantum algebra under certain restrictions on the parameters. By taking various specializations of the parameters, we obtain similar results for two/multi-parameter quantum algebras and Kang-Kashiwara-Oh's multi-parameter quantum superalgebras. As a consequence, the categories of weight modules of these algebras are identical.

A geometric categorification is given for arbitrary-large-finite-dimensional quotients of quantum osp(1|2) and the tensor product of its simple modules. The modified quantum osp(1|2) of Clark-Wang, a new version in this paper and the modified quantum sl(2) are isomorphic to each other over a field containing Q(v) and a root of -1., Comment: Final version. To appear in Transactions of AMS

By using perverse sheaves on representation spaces of quivers over $k[t]/(t^n)$ and jet schemes over flag varieties, we construct a geometric composition algebra $\mathbf K$ under Lusztig's framework on geometric realizations of the negative part of quantum algebras. Simple perverse sheaves in $\mathbf K$ form the canonical basis of $\mathbf K$. The relationships among the algebra $\mathbf K$, the composition algebra of locally projective representations of quivers over $k[t]/(t^n)$ and quantum generalized Kac-Moody algebra are provided., Simplify some proofs and some other revisions on the earlier version

We establish a three-parameter Schur duality between the affine Hecke algebra of type C and a coideal subalgebra of quantum affine $\mathfrak{sl}_n$. At the equal parameter specializations, we obtain Schur dualities of types BCD., 25 pages. Minor corrections, to appear in Math. Res. Letters