arXiv subjects

Ngau Lam

Publications and source records attributed to Ngau Lam.

18 recordsLinked to original sources

arXiv2025

Dualities of Gaudin models with irregular singularities for general linear Lie (super)algebras

We prove an equivalence between the actions of the Gaudin algebras with irregular singularities for $\mathfrak{gl}_d$ and $\mathfrak{gl}_{p+m|q+n}$ on the Fock space of $d(p+m)$ bosonic and $d(q+n)$ fermionic oscillators. This establishes a duality of $(\mathfrak{gl}_d, \mathfrak{gl}_{p+m|q+n})$ for Gaudin models. As an application, we show that the Gaudin algebra with irregular singularities for $\mathfrak{gl}_{p+m|q+n}$ acts cyclically on each weight space of a certain class of infinite-dimensional modules over a direct sum of Takiff superalgebras over $\mathfrak{gl}_{p+m|q+n}$ and that the action is diagonalizable with a simple spectrum under a generic condition. We also study the classical versions of Gaudin algebras with irregular singularities and demonstrate a duality of $(\mathfrak{gl}_d, \mathfrak{gl}_{p+m|q+n})$ for classical Gaudin models.

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arXiv2025

Bethe algebras for unitarizable modules over classical Lie (super)algebras and a duality

Let $\mathfrak{g}$ denote the classical Lie algebra $\mathfrak{gl}_d$, $\mathfrak{sp}_{2d}$, or $\mathfrak{so}_{2d}$ with a fixed $*$-structure $\sigma$. Let $M_1, \ldots, M_\ell$ be unitarizable $\mathfrak{g}$-modules (with respect to $\sigma$), and let ${\bf z}=(z_1, \ldots, z_\ell) \in \mathbb{C}^\ell$. We investigate the action of the Bethe algebra $\mathcal{B}_{\mathfrak{g}}^\mu$ for $\mathfrak{g}$ with respect to $\mu \in \mathfrak{g}^*$ on the tensor product $\underline{M}({\bf z}):=M_1(z_1) \otimes \cdots \otimes M_\ell(z_\ell)$ of evaluation $\mathfrak{g}[t]$-modules. We show that if $\mu \circ \sigma$ equals the complex conjugation of $\mu$, then $\mathcal{B}_{\mathfrak{g}}^\mu$ is diagonalizable on any finite-dimensional $\mathcal{B}_{\mathfrak{g}}^\mu$-submodule of $\underline{M}({\bf z})$ for ${\bf z} \in \mathbb{R}^\ell$. This, together with the result derived from the duality of Bethe algebras (see below), suggests that a simple spectrum conjecture for $\mathcal{B}_{\mathfrak{g}}^\mu$ should hold. We establish a duality of Bethe algebras for the general linear Lie (super)algebras $\mathfrak{gl}_d$ and $\mathfrak{gl}_{p+m|q+n}$. As an application, we show that under a generic condition, the Bethe algebra for $\mathfrak{gl}_{p+m|q+n}$ with respect to ${\bf z} \in \mathbb{C}^{p+q+m+n}$ is diagonalizable with a simple spectrum on any weight space of $L_1(w_1) \otimes \cdots \otimes L_d(w_d)$, where the $L_i$ are (infinite-dimensional) unitarizable highest weight $\mathfrak{gl}_{p+m|q+n}$-modules corresponding to generalized partitions of depth 1, and $w_1, \ldots, w_d \in \mathbb{C}$. We also obtain the corresponding result for $\mathfrak{gl}_{p+m}$ by setting $q=n=0$.

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arXiv2024

The Gaudin model for the general linear Lie superalgebra and the completeness of the Bethe ansatz

Let $\mathfrak{B}_{m|n}(\underline{\boldsymbol{z}})$ be the Gaudin algebra of the general linear Lie superalgebra $\mathfrak{gl}_{m|n}$ with respect to a sequence $\underline{\boldsymbol{z}} \in \mathbb{C}^\ell$ of pairwise distinct complex numbers, and let $M$ be any $\ell$-fold tensor product of irreducible polynomial modules over $\mathfrak{gl}_{m|n}$. We show that the singular space $M^{\rm sing}$ of $M$ is a cyclic $\mathfrak{B}_{m|n}(\underline{\boldsymbol{z}})$-module and the Gaudin algebra $\mathfrak{B}_{m|n}(\underline{\boldsymbol{z}})_{M^{\rm sing}}$ of $M^{\rm sing}$ is a Frobenius algebra. We also show that $\mathfrak{B}_{m|n}(\underline{\boldsymbol{z}})_{M^{\rm sing}}$ is diagonalizable with a simple spectrum for a generic $\underline{\boldsymbol{z}}$ and give a description of an eigenbasis and its corresponding eigenvalues in terms of the Fuchsian differential operators with polynomial kernels. This may be interpreted as the completeness of a reformulation of the Bethe ansatz for $\mathfrak{B}_{m|n}(\underline{\boldsymbol{z}})_{M^{\rm sing}}$.

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arXiv2024

Gaudin Hamiltonians on unitarizable modules over classical Lie (super)algebras

Let $M$ be a tensor product of unitarizable irreducible highest weight modules over the Lie (super)algebra $\mathcal{G}$, where $\mathcal{G}$ is $\mathfrak{gl}(m|n)$, $\mathfrak{osp}(2m|2n)$ or $\mathfrak{spo}(2m|2n)$. We show, using super duality, that the singular eigenvectors of the (super) Gaudin Hamiltonians for $\mathcal{G}$ on $M$ can be obtained from the singular eigenvectors of the Gaudin Hamiltonians for the corresponding Lie algebras on some tensor products of finite-dimensional irreducible modules. As a consequence, the (super) Gaudin Hamiltonians for $\mathcal{G}$ are diagonalizable on the space spanned by singular vectors of $M$ and hence on $M$. In particular, we establish the diagonalization of the Gaudin Hamiltonians, associated to any of the orthogonal Lie algebra $\mathfrak{so}(2n)$ and the symplectic Lie algebra $\mathfrak{sp}(2n)$, on the tensor product of infinite-dimensional unitarizable irreducible highest weight modules.

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arXiv2023

Quadratic and cubic Gaudin Hamiltonians and super Knizhnik-Zamolodchikov equations for general linear Lie superalgebras

We show that under a generic condition, the quadratic Gaudin Hamiltonians associated to $\mathfrak{gl}(p+m|q+n)$ are diagonalizable on any singular weight space in any tensor product of unitarizable highest weight $\mathfrak{gl}(p+m|q+n)$-modules. Moreover, every joint eigenbasis of the Hamiltonians can be obtained from some joint eigenbasis of the quadratic Gaudin Hamiltonians for the general linear Lie algebra $\mathfrak{gl}(r+k)$ on the corresponding singular weight space in the tensor product of some finite-dimensional irreducible $\mathfrak{gl}(r+ k)$-modules for $r$ and $k$ sufficiently large. After specializing to $p=q=0$, we show that similar results hold as well for the cubic Gaudin Hamiltonians associated to $\mathfrak{gl}(m|n)$. We also relate the set of singular solutions of the (super) Knizhnik-Zamolodchikov equations for $\mathfrak{gl}(p+m|q+n)$ to the set of singular solutions of the Knizhnik-Zamolodchikov equations for $\mathfrak{gl}(r+k)$ for $r$ and $k$ sufficiently large.

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arXiv2020

Solutions of super Knizhnik-Zamolodchikov equations

We establish an explicit bijection between the sets of singular solutions of the (super) KZ equations associated to the Lie superalgebra, of infinite rank, of type $\mf{a, b,c,d}$ and to the corresponding Lie algebra. As a consequence, the singular solutions of the super KZ equations associated to the classical Lie superalgebra, of finite rank, of type $\mf{a, b,c,d}$ for the tensor product of certain parabolic Verma modules (resp., irreducible modules) are obtained from the singular solutions of the KZ equations for the tensor product of the corresponding parabolic Verma modules (resp., irreducible modules) over the corresponding Lie algebra of sufficiently large rank, and vice versa. The analogous results for some special kinds of trigonometric (super) KZ equations are obtained.

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arXiv2019

Projective modules over classical Lie algebras of infinite rank in the parabolic category

We study the truncation functors and show the existence of projective cover of each irreducible module in parabolic BGG category $\mathcal O$ over infinite rank Lie algebra of types $\mathfrak{a,b,c,d}$. Moreover, $\mathcal O$ is a Koszul category. As a consequence, the corresponding parabolic BGG category $\overline{\mathcal O}$ over infinite rank Lie superalgebra of types $\mathfrak{a,b,c,d}$ through the super duality is also a Koszul category.

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arXiv2018

Symmetric structure for the endomorphism algebra of projective-injective module in parabolic category

We show that for any singular dominant integral weight $λ$ of a complex semisimple Lie algebra $\mathfrak{g}$, the endomorphism algebra $B$ of any projective-injective module of the parabolic BGG category $\mathcal{O}_λ^{\mathfrak{p}}$ is a symmetric algebra (as conjectured by Khovanov) extending the results of Mazorchuk and Stroppel for the regular dominant integral weight. Moreover, the endomorphism algebra $B$ is equipped with a homogeneous (non-degenerate) symmetrizing form. In the appendix, there is a short proof due to K. Coulembier and V. Mazorchuk showing that the endomorphism algebra $B_λ^{\mathfrak{p}}$ of the basic projective-injective module of $\mathcal{O}_λ^{\mathfrak{p}}$ is a symmetric algebra.

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arXiv2016

An inversion formula for some Fock spaces

A symmetric bilinear form on a certain subspace $\widehat{\mathbb T}^{\bf b}$ of a completion of the Fock space $\mathbb T^{\bf b}$ is defined. The canonical and dual canonical bases of $\widehat{\mathbb T}^{\bf b}$ are dual with respect to the bilinear form. As a consequence, the inversion formula connecting the coefficients of the canonical basis and that of the dual canonical basis of $\widehat{\mathbb T}^{\bf b}$ expanded in terms of the standard monomial basis of $\mathbb T^{\bf b}$ is obtained. Combining with the Brundan's algorithm for computing the elements in the canonical basis of $\widehat{\mathbb{T}}^{{\bf b}_{\mathrm{st}}}$, we have an algorithm computing the elements in the canonical basis of $\widehat{\mathbb{T}}^{\bf b}$ for arbitrary ${\bf b}$.

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arXiv2014

Brundan-Kazhdan-Lusztig conjecture for general linear Lie superalgebras

In the framework of canonical and dual canonical bases of Fock spaces, Brundan in 2003 formulated a Kazhdan-Lusztig type conjecture for the characters of the irreducible and tilting modules in the BGG category for the general linear Lie superalgebra for the first time. In this paper, we prove Brundan's conjecture and its variants associated to all Borel subalgebras in full generality.

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arXiv2011

Super duality for general linear Lie superalgebras and applications

We apply the super duality formalism recently developed by the authors to obtain new equivalences of various module categories of general linear Lie superalgebras. We establish the correspondence of standard, tilting, and simple modules, as well as the identification of the u-homology groups, under these category equivalences. As an application, we obtain a complete solution of the irreducible character problem for some new parabolic BGG categories of gl(m|n)-modules, including the full BGG category of gl(m|2)-modules, in terms of type A Kazhdan-Lusztig polynomials.

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arXiv2010

Super duality and homology of unitarizable modules of Lie algebras

The u-homology formulas for unitarizable modules at negative levels over classical Lie algebras of infinite rank of types gl(n), sp(2n) and so(2n) are obtained. As a consequence, we recover the Enright's formulas for three Hermitian symmetric pairs of classical types (SU(p; q); SU(p) X SU(q)), (Sp(2n);U(n)) and (SO*(2n);U(n)).

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arXiv2010

Super duality and irreducible characters of ortho-symplectic Lie superalgebras

We formulate and establish a super duality which connects parabolic categories $O$ between the ortho-symplectic Lie superalgebras and classical Lie algebras of $BCD$ types. This provides a complete and conceptual solution of the irreducible character problem for the ortho-symplectic Lie superalgebras in a parabolic category $O$, which includes all finite-dimensional irreducible modules, in terms of classical Kazhdan-Lusztig polynomials.

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arXiv2010

Irreducible Characters of General Linear Superalgebra and Super Duality

We develop a new method to solve the irreducible character problem for a wide class of modules over the general linear superalgebra, including all the finite-dimensional modules, by directly relating the problem to the classical Kazhdan-Lusztig theory. We further verify a parabolic version of a conjecture of Brundan on the irreducible characters in the BGG category $\mc{O}$ of the general linear superalgebra. We also prove the super duality conjecture.

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arXiv2008

A BGG-type resolution for tensor modules over general linear superalgebra

We construct a Bernstein-Gelfand-Gelfand type resolution in terms of direct sums of Kac modules for the finite-dimensional irreducible tensor representations of the general linear superalgebra. As a consequence it follows that the unique maximal submodule of a corresponding reducible Kac module is generated by its proper singular vector.

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arXiv2003

Character Formula for Infinite Dimensional Unitarizable Modules of the General Linear Superalgebra

The Fock space of $m+p$ bosonic and $n+q$ fermionic quantum oscillators forms a unitarizable module of the general linear superalgebra $gl_{m+p|n+q}$. Its tensor powers decompose into direct sums of infinite dimensional irreducible highest weight $gl_{m+p|n+q}$-modules. We obtain an explicit decomposition of any tensor power of this Fock space into irreducibles, and develop a character formula for the irreducible $gl_{m+p|n+q}$-modules arising in this way.

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arXiv2003

Infinite-dimensional Lie superalgebras and hook Schur functions

Making use of a Howe duality involving the infinite-dimensional Lie superalgebra $\hgltwo$ and the finite-dimensional group $GL_l$ we derive a character formula for a certain class of irreducible quasi-finite representations of $\hgltwo$ in terms of hook Schur functions. We use the reduction procedure of $\hgltwo$ to $\hat{gl}_{n|n}$ to derive a character formula for a certain class of level 1 highest weight irreducible representations of $\hat{gl}_{n|n}$, the affine Lie superalgebra associated to the finite-dimensional Lie superalgebra $gl_{n|n}$. These modules turn out to form the complete set of integrable $\hat{gl}_{n|n}$-modules of level 1. We also show that the characters of all integrable level 1 highest weight irreducible $\hat{gl}_{m|n}$-modules may be written as a sum of products of hook Schur functions.

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arXiv2000

Finite conformal modules over the N=2,3,4 superconformal algebras

In this paper we continue the study of representation theory of formal distribution Lie superalgebras initiated in q-alg/9706030. We study finite Verma-type conformal modules over the N=2, N=3 and the two N=4 superconformal algebras and also find explicitly all singular vectors in these modules. From our analysis of these modules we obtain a complete list of finite irreducible conformal modules over the N=2, N=3 and the two N=4 superconformal algebras.

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