arXiv subjects

Hongda Lin

Publications and source records attributed to Hongda Lin.

9 recordsLinked to original sources

arXiv2026

The double super Yangians in type A for arbitrary $0^m 1^n$-sequences and their bosonic representations

In this paper, we introduce the double super Yangian $\mathrm{DY}_{h}(\mathfrak{gl}_{m|n}^{\mathfrak{s}})$ and $\mathrm{DY}_{h}(\mathfrak{sl}_{m|n}^{\mathfrak{s}})$ associated with any fixed $0^{m}1^{n}$--sequence $\mathfrak{s}$. First, we establish an explicit isomorphism between the Drinfeld and R--matrix presentations of $\mathrm{DY}_{h}(\mathfrak{gl}^{\mathfrak{s}}_{m|n})$. We then generalize the notion of the quantum Berezinian to $\mathrm{DY}_{h}(\mathfrak{gl}_{m|n}^{\mathfrak{s}})$, and employ it to construct the R--matrix presentation of $\mathrm{DY}_{h}(\mathfrak{sl}_{m|n}^{\mathfrak{s}})$ and prove that it is isomorphic to the Drinfeld presentation. As an application, we present level--1 bosonic representations for $\mathrm{DY}_{h}(\mathfrak{gl}_{m|n}^{\mathfrak{s}})$ and $\mathrm{DY}_{h}(\mathfrak{sl}_{m|n}^{\mathfrak{s}})$ in terms of their Drinfeld current generators.

math.RT↗

arXiv2026

Affine Yangians as Limits of Quantum Toroidal Algebras

We establish a degeneration isomorphism between quantum toroidal algebras and untwisted affine Yangians, valid for all untwisted affine Kac-Moody Lie algebras. Specifically, we prove that the affine Yangian $Y_\hbar(\mathfrak{g})$ is isomorphic, as a $\mathbb{C}[\hbar]$-algebra, to the associated graded algebra of the quantum toroidal algebra $U_\hbar(\mathfrak{g}^{\mathrm{tor}})$ with respect to a canonical filtration. This result constitutes the affine analogue of Drinfeld's conjecture on the relationship between Yangians and quantum loop algebras, previously established in the finite-dimensional setting by Gautam--Toledano Laredo and by Guay--Ma. As principal applications of this isomorphism, we derive two fundamental structural properties of affine Yangians: a Poincar\'e--Birkhoff--Witt (PBW) basis for $Y_\hbar(\mathfrak{g})$ in all untwisted affine types, and the identification of its classical limit as the universal enveloping algebra $U(\mathfrak{g}[u])$ of the polynomial current Lie algebra. A key ingredient of independent interest is our construction of a PBW basis for $U_\hbar(\mathfrak{g}^{\mathrm{tor}})$ itself, which relies on a new torsion-freeness argument for the quantum toroidal algebra and the topological Nakayama lemma.

math.QA↗

arXiv2025

Representations of Quantum Affine General Linear Superalgebras at Arbitrary 01-Sequences

In this paper, we investigate finite-dimensional irreducible representations of the quantum affine general linear superalgebra $\mathrm{U}_q\big(\widehat{\mathfrak{gl}}_{m|n,\mathbf{s}}\big)$ for arbitrary 01-sequences $\mathbf{s}$, using the RTT presentation. We systematically construct the RTT presentation for quantum general linear superalgebra $\mathrm{U}_q\big(\mathfrak{gl}_{m|n,\mathbf{s}}\big)$, and derive a PBW basis induced by the action of the braid group, compatible with non-standard parities. We determine the necessary and sufficient conditions for the finite-dimensionality of irreducible representations of $\mathrm{U}_q\big(\mathfrak{gl}_{m|n,\mathbf{s}}\big)$ and extend the results to the affine case via the evaluation homomorphism. Specific cases such as $(m,n)=(1,1)$ demonstrate that all finite-dimensional representations are tensor products of typical evaluation representations. This work extends existing representation frameworks and classification methods to encompass arbitrary 01-sequences, establishing the foundation for subsequent research on representations of quantum affine superalgebras.

math.QA↗

arXiv2025

Drinfeld super Yangian of the exceptional Lie superalgebra $D(2,1;\lambda)$

In this paper, we establish the first rigorous framework for the Drinfeld super Yangian associated with an exceptional Lie superalgebra, which lacks a classical Lie algebraic counterpart. Specifically, we systematically investigate the Drinfeld presentation and structural properties of the super Yangian associated with the exceptional Lie superalgebra $D(2,1;\lambda)$. First, we introduce a Drinfeld presentation for the super Yangian associated with the exceptional Lie superalgebra $D(2,1;\lambda)$, explicitly constructing its current generators and defining relations. A key innovation is the construction of a Poincar\'e-Birkhoff-Witt (PBW) basis using degeneration techniques from the corresponding quantum loop superalgebra. Furthermore, we demonstrate that the super Yangian possesses a Hopf superalgebra structure, explicitly providing the coproduct, counit, and antipode.

math.QA↗

arXiv2025

RTT presentation of coideal subalgebra of quantized enveloping algebra of type CI

The pair consisting of a quantum group and its corresponding coideal subalgebra, known as a quantum symmetric pair, was developed independently by M. Noumi and G. Letzter through different approaches. The purpose of this paper is threefold. First, for symmetric pairs $(\mathfrak{sp}_{2n},\mathfrak{gl}_n)$, we construct a coideal subalgebra $U_q^{tw}(\mathfrak{gl}_n)$ of the quantized enveloping algebra of type CI using the $R$-matrix presentation, based on the work of Noumi. Second, we derive a Poincar\'e-Birkhoff-Witt(PBW) basis for $U_q^{tw}(\mathfrak{gl}_n)$ by the $\mathbb{A}$-form approach. As a consequence of the isomorphism btween $U_q^{tw}(\mathfrak{gl}_n)$ and the $\imath$quantum group $\mathcal{U}^{\imath}$, our method also yields the PBW basis for the $\imath$quantum group of type CI. Finally, as an application of the $R$-matrix presentation, we construct a Poisson algebra $\mathcal{P}_n$ associated with $U_q^{tw}(\mathfrak{gl}_n)$, and explicitly describe the action of the braid group $\mathcal{B}_n$ on the elements of $\mathcal{P}_n$.

math.QA↗

arXiv2025

Quantum Berezinian for the Twisted Super Yangian

Motivated by an open problem proposed in Molev's book \cite[Section 2.16, Example 16]{Mo07}, we investigate the quantum Berezinian $\mathfrak{B}^{tw}(u)$ associated with the twisted super Yangian, which is a coideal sub-superalgebra of the super Yangian of the general linear Lie superalgebra. We provide an explicit formulation of $\mathfrak{B}^{tw}(u)$, and we also construct the center of the twisted super Yangian. This construction enables us to define the special twisted super Yangian, which is isomorphic to the quotient of the twisted super Yangian by its center. Moreover, we demonstrate the quantum Sylvester theorem for both the generator matrix and the quantum Berezinian.

math.QA↗

arXiv2024

Braid group action and quantum affine superalgebra for type $\mathfrak{osp}(2m+1|2n)$

In this paper, we investigate the structure of the quantum affine superalgebra associated with the orthosymplectic Lie superalgebra $\mathfrak{osp}(2m+1|2n)$ for $m\geqslant 1$. The Drinfeld-Jimbo presentation for this algebra, denoted as $U_q[\mathfrak{osp}(2m+1|2n)^{(1)}]$, was originally introduced by H. Yamane. We provide the definition of the Drinfeld presentation $\mathcal{U}_q[\mathfrak{osp}(2m+1|2n)^{(1)}]$. To establish the isomorphism between the Drinfeld-Jimbo presentation and the Drinfeld presentation of the quantum affine superalgebra for type $\mathfrak{osp}(2m+1|2n)$, we introduce a braid group action to define quantum root vectors of the quantum superalgebra. Specifically, we present an efficient method for verifying the isomorphism between two presentations of the quantum affine superalgebra associated with the type $\mathfrak{osp}(2m+1|2n)$.

math.QA↗

arXiv2024

$R$-Matrix Presentation of Quantum Affine Superalgebra for Type $\mathfrak{osp}(2m+1|2n)$

In our preceding research, we introduced the Drinfeld presentation of the quantum affine superalgebra associated to the orthosymplectic Lie superalgebra $\mathfrak{osp}(2m+1|2n)$ for $m>0$. We provided the isomorphism between its Drinfeld-Jimbo presentation and Drinfeld presentation using braid group actions as a fundamental method. Based on this work, our current study delves into its $R$-matrix presentation, wherein we establish a clear isomorphism between the $R$-matrix presentation and the Drinfeld presentation. In particular, our contribution extends the investigations of Jing, Liu and Molev concerning quantum affine algebra in type BCD to the realm of supersymmetry.

math.QA↗

arXiv2023

From quantum loop superalgebras to super Yangians

The goal of this paper is to generalize a statement by Drinfeld, asserting that Yangians can be constructed as limit forms of the quantum loop algebras, to the super case. We establish a connection between quantum loop superalgebra and super Yangian of the general linear Lie superalgebra $\mathfrak{gl}_{M|N}$ in RTT type presentation. In particular, we derive the Poincar\'e-Birkhoff-Witt(PBW) theorem for the quantum loop superalgebra $\mathrm{U}_q\big(\mathfrak{Lgl}_{M|N}\big)$. Additionally, we investigate the application of the same argument to twisted super Yangian of the ortho-symplectic Lie superalgebra. For this purpose, we introduce the twisted quantum loop superalgebra as a one-sided coideal of $\mathrm{U}_q\big(\mathfrak{Lgl}_{M|2n}\big)$ with respect to the comultiplication.

math.QA↗