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Kang Lu

Publications and source records attributed to Kang Lu.

At least 19 recordsLinked to original sources
arXiv2026

Quasi-split iYangians: minimalistic presentations and coideal structures

We study iYangians, namely twisted Yangians in the Drinfeld current presentation, associated with quasi-split symmetric pairs of type $\mathsf{ADE}$ with nontrivial diagram involution. We establish minimalistic presentations in terms of degree-zero and degree-one generators. Type $\mathsf A_{2n}$ is treated separately: the isolated rank-two case requires two additional relations, whereas in higher rank these relations are forced by the neighboring noncentral orbit. As a by-product, we strengthen Theorem 3.1 in arxiv:2511.07136 by proving that the extra relation in the minimalistic presentation of split iYangian is redundant whenever $\mathfrak g$ has rank at least two. We also construct explicit injective homomorphisms from quasi-split iYangians into Yangians, identify their images as right coideal subalgebras, and obtain isomorphisms between the Drinfeld and $J$ presentations. Finally, we derive triangular estimates for the Drinfeld currents and their coproducts and apply them to the ${}^\imath\ell$-weights arising by restriction from finite-dimensional Yangian modules.

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arXiv2026

Twisted super Yangians of quasi-split type A

We introduce the twisted super Yangian $Y^{\mathfrak{s}}_{\imath}$ of quasi-split type A in the Drinfeld current presentation for an arbitrary symmetric parity sequence $\mathfrak{s}$. We prove via Gauss decomposition that $Y^{\mathfrak{s}}_{\imath}$ is isomorphic to the special twisted super Yangian $\mathscr{SY}^{\mathfrak{s}}$, a subalgebra of the twisted super Yangian $\mathscr{Y}^{\mathfrak{s}}$ previously defined via the R-matrix presentation. As a corollary, we show that the algebras $Y^{\mathfrak{s}}_{\imath}$ corresponding to different symmetric parity sequences with the same $\mathfrak{m}$ and $\mathfrak{n}$ are isomorphic. We also establish a PBW theorem for $Y^{\mathfrak{s}}_{\imath}$ and describe the center of $\mathscr{Y}^{\mathfrak{s}}$ in terms of Gaussian generators, thereby generalizing known results for the nonsuper quasi-split type A case.

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arXiv2026

Shifted affine iquantum groups of quasi-split ADE types

We formulate shifted affine iquantum groups of arbitrary quasi-split ADE types via Drinfeld presentations. We construct GKLO-type representations of shifted affine iquantum groups via algebras of difference operators, which allow us to construct truncated shifted affine iquantum groups. This provides a q-deformation of truncated shifted iYangians in our prior work arising as a quantization of affine Grassmannian islices.

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arXiv2026

Representations of shifted super Yangians and finite $W$-superalgebras of type A

In this article, we study the representation theory of shifted super Yangians and finite $W$-superalgebras of type A. A criterion for the finite dimensionality of irreducible modules is obtained in the standard parity case. Furthermore, we provide an explicit Gelfand-Tsetlin character formula for Verma modules of finite $W$-superalgebras. As an application, we show that the centers of the finite $W$-superalgebras associated to any even nilpotent elements belonging to the same general linear Lie superalgebra are all isomorphic to the center of the universal enveloping superalgebra.

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arXiv2025

Shifted twisted Yangians of quasi-split ADE types

Associated to all quasi-split Satake diagrams of type ADE and even spherical coweights $\mu$, we introduce the shifted iYangians ${}^\imath Y_\mu$ and establish their PBW bases. We construct the iGKLO representations of ${}^\imath Y_\mu$, which factor through quotients called truncated shifted iYangians ${}^\imath Y_\mu^\lambda$. In type AI with $\mu$ dominant, a variant of ${}^\imath Y_\mu^{N\varpi_1^\vee}$ is identified with the truncated shifted iYangians in another definition, which are isomorphic to finite W-algebras of type BCD. These new family of algebras has connections and applications to fixed point loci of affine Grassmannian slices which will be developed in a sequel.

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arXiv2025

Minimalistic Presentation and Coideal Structure of Twisted Yangians

We introduce a minimalistic presentation for the twisted Yangian ${}^\imath\mathscr Y$ associated with split symmetric pairs (or Satake diagrams) introduced in arXiv:2406.05067 via a Drinfeld type presentation. As applications, we establish an injective algebra homomorphism from ${}^\imath\mathscr Y$ to the Yangian $\mathscr Y$, thereby identifying ${}^\imath\mathscr Y$ as a right coideal subalgebra of $\mathscr Y$ and proving its isomorphism with the twisted Yangian in the $J$ presentation. Furthermore, we provide estimates for the Drinfeld generators of ${}^\imath\mathscr Y$ and describe their coproduct images in terms of the Drinfeld generators of $\mathscr Y$ under this identification.

math.QA↗
arXiv2025

Shifted twisted Yangians and affine Grassmannian islices

In a prequel we introduced the shifted iYangians ${}^\imath Y_\mu$ associated to quasi-split Satake diagrams of type ADE and even spherical coweights $\mu$, and constructed the iGKLO representations of ${}^\imath Y_\mu$, which factor through truncated shifted iYangians ${}^\imath Y_\mu^\lambda$. In this paper, we show that ${}^\imath Y_\mu$ quantizes the involutive fixed point locus ${}^\imath W_\mu$ arising from affine Grassmannians of type ADE, and supply strong evidence toward the expectation that ${}^\imath Y_\mu^\lambda$ quantizes a top-dimensional component of the affine Grassmannian islice ${}^\imath\overline{W}_\mu^\lambda$. We identify the islices ${}^\imath\overline{W}_\mu^\lambda$ in type AI with suitable nilpotent Slodowy slices of type BCD, building on the work of Lusztig and Mirkovi\'c-Vybornov in type A. We propose a framework for producing ortho-symplectic (and hybrid) Coulomb branches from split (and nonsplit) Satake framed double quivers, which are conjectured to relate closely to the islices ${}^\imath\overline{W}_\mu^\lambda$ and the algebras ${}^\imath Y_\mu^\lambda$.

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arXiv2025

Shifted twisted Yangians and finite $W$-algebras of classical type

We introduce parabolic presentations of twisted Yangians of types AI and AII, interpolating between the R-matrix presentation and the Drinfeld presentation. Then we formulate and provide parabolic presentations for the shifted twisted Yangians. We define quotient algebras known as truncated shifted twisted Yangians and equip them with baby comultiplications, generalizing the work of Brundan and Kleshchev. PBW bases for all (truncated) shifted twisted Yangians of type AI and AII are established along the way. Applying the theory of universal equivariant quantizations of conic symplectic singularities we show that the truncated twisted shifted Yangian is isomorphic to the finite $W$-algebra which quantizes a suitable Slodowy slice. This provides a presentation of the finite $W$-algebra associated with every even nilpotent element in type {\sf B} and {\sf C}, as well as every nilpotent element with two Jordan blocks in type {\sf D}. Finally we make a conjecture which would supply presentations in the remaining even cases in type {\sf D}.

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arXiv2024

A Drinfeld type presentation of twisted Yangians of quasi-split type

We formulate a family of algebras, twisted Yangians (of simply-laced quasi-split type) in Drinfeld type current generators and defining relations. These new algebras admit PBW type bases and are shown to be a deformation of twisted current algebras. For all quasi-split type excluding the even rank case in type AIII, we show that the twisted Yangians can be realized via a degeneration on the Drinfeld type presentation of affine $\imath$quantum groups. For both even and odd rank cases in type AIII, we use the Gauss decomposition method to show that these new algebras are isomorphic to Molev-Ragoucy's reflection algebras defined in the R-matrix presentation.

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arXiv2024

Affine $\imath$quantum groups and twisted Yangians in Drinfeld presentations

We formulate a family of algebras, twisted Yangians (of split type) in current generators and relations, via a degeneration of the Drinfeld presentation of affine $\imath$quantum groups (associated with split Satake diagrams). These new algebras admit PBW type bases and are shown to be a deformation of twisted current algebras; presentations for twisted current algebras are also provided. For type AI, it matches with the Drinfeld presentation of twisted Yangian obtained via Gauss decomposition. We conjecture that our split twisted Yangians are isomorphic to the corresponding ones in RTT presentation.

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arXiv2023

Twisted super Yangians of type AIII and their representations

We study the super analogue of the Molev-Ragoucy reflection algebras, which we call twisted super Yangians of type AIII, and classify their finite-dimensional irreducible representations under certain conditions. These superalgebras are coideal subalgebras of the super Yangian $\mathscr{Y}(\mathfrak{gl}_{m|n})$ and are associated with symmetric pairs of type AIII in Cartan's classification. We establish the Schur-Weyl type duality between degenerate affine Hecke algebras of type BC and twisted super Yangians.

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arXiv2023

Isomorphism between twisted $q$-Yangians and affine $\imath$quantum groups: type AI

By employing Gauss decomposition, we establish a direct and explicit isomorphism between the twisted $q$-Yangians (in R-matrix presentation) and affine $\imath$quantum groups (in current presentation) associated to symmetric pair of type AI introduced by Molev-Ragoucy-Sorba and Lu-Wang, respectively. As a corollary, we obtain a PBW type basis for affine $\imath$quantum groups of type AI.

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arXiv2023

A Drinfeld type presentation of twisted Yangians

We develop a Gauss decomposition approach to establish a Drinfeld type current presentation for Olshanski's twisted Yangians associated to the orthogonal Lie algebras (also called twisted Yangians of type AI), settling a longstanding open problem. We expect that this will open the door for finding current presentations for other twisted Yangians.

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arXiv2022

On Bethe eigenvectors and higher transfer matrices for supersymmetric spin chains

We study the $\mathfrak{gl}_{m|n}$ XXX spin chains defined on tensor products of highest $\mathfrak{gl}_{m|n}$-modules. We show that the on-shell Bethe vectors are eigenvectors of higher transfer matrices and compute the corresponding eigenvalues, confirming Conjecture 5.15 of arXiv:2007.15573 and extending the main results of arXiv:0605015 to supersymmetric case. We then take the classical limits and obtain the corresponding results for the $\mathfrak{gl}_{m|n}$ Gaudin models.

math-ph↗
arXiv2022

Completeness of Bethe ansatz for Gaudin models associated with gl(1|1)

We study the Gaudin models associated with $\mathfrak{gl}(1|1)$. We give an explicit description of the algebra of Hamiltonians (Gaudin Hamiltonians) acting on tensor products of polynomial evaluation $\mathfrak{gl}(1|1)[t]$-modules. It follows that there exists a bijection between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the highest weights and evaluation parameters. In particular, our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one. Therefore, we confirm Conjecture 8.3 from arXiv:1809.01279. We also give dimensions of the generalized eigenspaces. Moreover, we express the generating pseudo-differential operator of Gaudin transfer matrices associated to antisymmetrizers in terms of the quadratic Gaudin transfer matrix and the center of $\mathrm{U}(\mathfrak{gl}(1|1)[t])$.

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arXiv2021

A note on odd reflections of super Yangian and Bethe ansatz

In a recent paper arXiv:2109.09462, Molev introduced analogues of the odd reflections for the super Yangian $\mathrm{Y}(\mathfrak{gl}_{m|n})$ and obtained a transition rule for the change of highest weights when the parity sequence is altered. In this note, we reproduce the results from a different point of view and discuss their relations with the fermionic reproduction procedure of the XXX-type Bethe ansatz equations introduced in arXiv:1811.11225. We give an algorithm that how the $q$-characters change under the odd reflections. We also take the chance to compute explicitly the $q$-characters of skew representations of $\mathrm{Y}(\mathfrak{gl}_{m|n})$ for arbitrary parity sequences.

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arXiv2021

Bethe ansatz equations for orthosymplectic Lie superalgebras and self-dual superspaces

We study solutions of the Bethe ansatz equations associated to the orthosymplectic Lie superalgebras $\mathfrak{osp}_{2m+1|2n}$ and $\mathfrak{osp}_{2m|2n}$. Given a solution, we define a reproduction procedure and use it to construct a family of new solutions which we call a population. To each population we associate a symmetric rational pseudo-differential operator $\mathcal R$. Under some technical assumptions, we show that the superkernel $W$ of $\mathcal R$ is a self-dual superspace of rational functions, and the population is in a canonical bijection with the variety of isotropic full superflags in $W$ and with the set of symmetric complete factorizations of $\mathcal R$. In particular, our results apply to the case of even Lie algebras of type D${}_m$ corresponding to $\mathfrak{osp}_{2m|0}=\mathfrak{so}_{2m}$.

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