arXiv subjects

Chih-Whi Chen

Publications and source records attributed to Chih-Whi Chen.

At least 19 recordsLinked to original sources

arXiv2026

Categorical Equivalences of Finite W-Superalgebras and Clifford Twists

Associated with an even nilpotent element $e$ in a basic classical Lie superalgebra $\mathfrak{g}$, we study, in full generality, two constructions of finite $W$-superalgebras, defined via Whittaker models and isotropic subspaces, respectively. We prove that both formulations are independent of the various choices made in their constructions, thereby yielding, for a fixed good grading for $e$, at most two isomorphism classes of $W$-superalgebras. In the case when there are two non-isomorphic versions, we establish that they differ precisely by a Clifford extension. Consequently, when the two $W$-superalgebras are non-isomorphic, their module categories are equivalent up to a Clifford twist. Building on this equivalence and utilizing the Skryabin equivalence, we classify their irreducible representations in terms of generalized Whittaker modules over $\mathfrak{g}$

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arXiv2025

Whittaker modules and representations of finite $W$-algebras of queer Lie superalgebras

We study various categories of Whittaker modules over the queer Lie superalgebras $\mathfrak q(n)$. We formulate standard Whittaker modules and reduce the problem of composition factors of these standard Whittaker modules to that of Verma modules in the BGG categories $\mathcal O$ of $\mathfrak q(n)$. We also obtain an analogue of Losev-Shu-Xiao decomposition for the finite $W$-superalgebras $U(\mathfrak q(n), E)$ of $\mathfrak q(n)$ associated to an odd nilpotent element $E\in \mathfrak q(n)_{\bar{1}}$. As an application, we establish several equivalences of categories of Whittaker $\mathfrak q(n)$-modules and analogues of BGG category of $U(\mathfrak q(n), E)$-modules. In particular, we reduce the multiplicity problem of Verma modules over $U(\mathfrak q(n), E)$ to that of the Verma modules in the BGG categories $\mathcal O$ of $\mathfrak q(n)$.

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arXiv2025

Whittaker categories, properly stratified categories and Fock space categorification for Lie superalgebras

We study various categories of Whittaker modules over a type I Lie superalgebra realized as cokernel categories that fit into the framework of properly stratified categories. These categories are the target of the Backelin functor $Γ_ζ$. We show that these categories can be described, up to equivalence, as Serre quotients of the BGG category $\mathcal O$ and of certain singular categories of Harish-Chandra $(\mathfrak g,\mathfrak g_{\bar 0})$-bimodules. We also show that $Γ_ζ$ is a realization of the Serre quotient functor. We further investigate a $q$-symmetrized Fock space over a quantum group of type A and prove that, for general linear Lie superalgebras our Whittaker categories, the functor $Γ_ζ$ and various realizations of Serre quotients and Serre quotient functors categorify this $q$-symmetrized Fock space and its $q$-symmetrizer. In this picture, the canonical and dual canonical bases in this $q$-symmetrized Fock space correspond to tilting and simple objects in these Whittaker categories, respectively.

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arXiv2025

Serre functors for Lie superalgebras and tensoring with $S^{\mathrm{top}}(\mathfrak{g}_{\overline{1}})$

We show that the action of the Serre functor on the subcategory of projective-injective modules in a parabolic BGG category $\mathcal O$ of a quasi-reductive finite dimensional Lie superalgebra is given by tensoring with the top component of the symmetric power of the odd part of our superalgebra. As an application, we determine, for all strange Lie suepralgebras, when the subcategory of projective injective modules in the parabolic category $\mathcal O$ is symmetric.

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arXiv2024

Typical representations of Takiff superalgebras

We investigate representations of the $\ell$-th Takiff superalgebras $\widetilde{\mathfrak g}_\ell := \widetilde{\mathfrak g}\otimes \mathbb C[θ]/(θ^{\ell+1})$, for $\ell>0$, associated with a basic classical and a periplectic Lie superalgebras $\widetilde{\mathfrak g}$. We introduce the odd reflections and formulate a general notion of typical representations of the Takiff superalgebras $\widetilde{\mathfrak g}_\ell$. As a consequence, we provide a complete description of the characters of the finite-dimensional modules over type I Takiff superalgebras. For the Lie superalgebras $\widetilde{\mathfrak g}= \mathfrak{gl}(m|n)$ and $\mathfrak{osp}(2|2n)$, we prove that the Kac induction functor of $\widetilde{\mathfrak g}_\ell$ leads to an equivalence from an arbitrary typical Jordan block of the category $\mathcal O$ for $\widetilde{\mathfrak g}_\ell$ to a Jordan block of the category $\mathcal O$ for the even subalgebra of $\widetilde{\mathfrak g}_\ell$. We also obtain a classification of non-singular simple Whittaker modules over the Takiff superalgebras.

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arXiv2024

Whittaker modules of central extensions of Takiff superalgebras and finite supersymmetric $W$-algebras

For a basic classical Lie superalgebra $\mathfrak s$, let $\mathfrak g$ be the central extension of the Takiff superalgebra $\mathfrak s\otimesΛ(θ)$, where $θ$ is an odd indeterminate. We study the category of $\mathfrak g$-Whittaker modules associated with a nilcharacter $χ$ of $\mathfrak g$ and show that it is equivalent to the category of $\mathfrak s$-Whittaker modules associated with a nilcharacter of $\mathfrak s$ determined by $χ$. In the case when $χ$ is regular, we obtain, as an application, an equivalence between the categories of modules over the supersymmetric finite $W$-algebras associated to the odd principal nilpotent element at non-critical levels and the category of the modules over the principal finite $W$-superalgebra associated to $\mathfrak s$. Here, a supersymmetric finite $W$-algebra is conjecturally the Zhu algebra of a supersymmetric affine $W$-algebra. This allows us to classify and construct irreducible representations of a principal finite supersymmetric $W$-algebra.

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arXiv2023

Kostant's problem for Whittaker modules

We study the classical problem of Kostant for Whittaker modules over Lie algebras and Lie superalgebras. We give a sufficient condition for a positive answer to Kostant's problem for the standard Whittaker modules over reductive Lie algebras. Under the same condition, the positivity of the answer for simple Whittaker modules is reduced to that for simple highest weight modules. We develop several reduction results to reduce the Kostant's problem for standard and simple Whittaker modules over a type I Lie superalgebra to that for the corresponding Whittaker modules over the even part of this Lie superalgebra.

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arXiv2023

Whittaker categories of quasi-reductive Lie superalgebras and principal finite W-superalgebras

We study the Whittaker category $\mathcal N(ζ)$ of the Lie superalgebra $\mathfrak g$ for an arbitrary character $ζ$ of the even subalgebra of the nilpotent radical associated with a triangular decomposition of $\mathfrak g$. We prove that the Backelin functor from either the integral subcategory or any strongly typical block of the BGG category to the Whittaker category sends irreducible modules to irreducible modules or zero. The category $\mathcal N(ζ)$ provides a suitable framework for studying finite $W$-superalgebras associated with an even principal nilpotent element. For the periplectic Lie superalgebras $\mathfrak{p}(n)$, we formulate the principal finite $W$-superalgebras $W_ζ$ and establish a Skryabin-type equivalence. For a basic classical and a strange Lie superalgebras, we prove that the category of finite-dimensional modules over a given principal finite $W$-superalgebra $W_ζ$ is equivalent to $\mathcal N(ζ)$ under the Skryabin equivalence, for a non-singular character $ζ$. As a consequence, we give a super analogue of Soergel's Struktursatz for a certain Whittaker functor from the integral BGG category $\mathcal O$ to the category of finite-dimensional modules over $W_ζ$.

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arXiv2022

Whittaker categories of quasi-reductive Lie superalgebras and quantum symmetric pairs

We show that, for an arbitrary quasi-reductive Lie superalgebra with a triangular decomposition and a character $ζ$ of the nilpotent radical, the associated Backelin functor $Γ_ζ$ sends Verma modules to standard Whittaker modules provided the latter exist. As a consequence, this gives a complete solution to the problem of determining the composition factors of the standard Whittaker modules in terms of composition factors of Verma modules in the category $\mathcal O$. In the case of the ortho-symplectic Lie superalgebras, we show that the Backelin functor $Γ_ζ$ and its target category, respectively, categorify a $q$-symmetrizing map and the corresponding $q$-symmetrized Fock space associated with a quasi-split quantum symmetric pair of type $AIII$.

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arXiv2021

Annihilator ideals and blocks of Whittaker modules over quasireductive Lie superalgebras

We extend Kostant's result on annihilator ideals of non-singular simple Whittaker modules over Lie algebras to (possibly singular) simple Whittaker modules over Lie superalgebras. We describe these annihilator ideals in terms of certain primitive ideals coming from the category $\mathcal O$ for quasireductive Lie superalgebras. To determine these annihilator ideals, we develop annihilator-preserving equivalences between certain full subcategories of the Whittaker category $\tilde{\mathcal N}$ and the categories of certain projectively presentable modules in the category $\mathcal O$. These equivalences lead to a classification of simple Whittaker modules that lie in the integral central blocks when restricted to the even subalgebra. We make a connection between the linkage classes of integral blocks of $\mathcal O$ and of $\tilde{\mathcal N}$. In particular, they can be computed via Kazhdan-Lusztig combinatorics for Lie superalgebras of type $\mathfrak{gl}$ and $\mathfrak{osp}$. We then give a description of the integral blocks of the category $\tilde{\mathcal N}$ of Whittaker modules for Lie superalgebras $\mathfrak{gl}(m|n)$, $\mathfrak{osp}(2|2n)$ and $\mathfrak{pe}(n)$.

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arXiv2021

On semisimplicity of Jantzen middles for the periplectic Lie superalgebra

We prove that an integral block of the category $\mathcal O$ of the periplectic Lie superalgebra contains a non-semisimple Jantzen middle if and only if it contains a simple module of atypical highest weight. As a consequence, every atypical integral block of $\mathcal O$ does not admit a Kazhdan-Lusztig theory in the sense of Cline, Parshall and Scott.

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arXiv2021

Whittaker modules for classical Lie superalgebras

We classify simple Whittaker modules for classical Lie superalgebras in terms of their parabolic decompositions. We establish a type of Miličić-Soergel equivalence of a category of Whittaker modules and a category of Harish-Chandra bimodules. For classical Lie superalgebras of type I, we reduce the problem of composition factors of standard Whittaker modules to that of Verma modules in their BGG categories $\mathcal O$. As a consequence, the composition series of standard Whittaker modules over the general linear Lie superalgebras $\mathfrak{gl}(m|n)$ and the ortho-symplectic Lie superalgebras $\mathfrak{osp}(2|2n)$ can be computed via the Kazhdan-Lusztig combinatorics.

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arXiv2020

Blocks and characters of $G(3)$-modules of non-integral weights

We classify blocks in the BGG category $\mathcal O$ of modules of non-integral weights for the exceptional Lie superalgebra $G(3)$. We compute the characters for tilting modules of non-integral weights in $\mathcal O$. Reduction methods are established to connect non-integral blocks of $G(3)$ with blocks of the special linear Lie algebra $\mathfrak{sl}(2)$, the exceptional Lie algebra $G_2$, the general linear Lie superalgebras $\mathfrak{gl}(1|1)$, $\mathfrak{gl}(2|1)$ and the ortho-symplectic Lie superalgebra $\mathfrak{osp}(3|2)$.

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arXiv2020

Some homological properties of category $\mathcal O$ for Lie superalgebras

For classical Lie superalgebras of type I, we provide necessary and sufficient conditions for a Verma supermodule $Δ(λ)$ to be such that every non-zero homomorphism from another Verma supermodule to $Δ(λ)$ is injective. This is applied to describe the socle of the cokernel of an inclusion of Verma supermodules over the periplectic Lie superalgebras $\mathfrak{pe}(n)$ and, furthermore, to reduce the problem of description of $\mathrm{Ext}^1_{\mathcal O}(L(μ),Δ(λ))$ for $\mathfrak{pe}(n)$ to the similar problem for the Lie algebra $\mathfrak{gl}(n)$. Additionally, we study the projective and injective dimensions of structural supermodules in parabolic category $\mathcal O^{\mathfrak p}$ for classical Lie superalgebras. In particular, we completely determine these dimensions for structural supermodules over the periplectic Lie superalgebra $\mathfrak{pe}(n)$ and the ortho-symplectic Lie superalgebra $\mathfrak{osp}(2|2n)$.

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arXiv2020

Parabolic category $\mathcal O^{\mathfrak p}$ for periplectic Lie superalgebras $\mathfrak{pe}(n)$

We provide a linkage principle in an arbitrary parabolic category $\mathcal O^{\mathfrak p}$ for the periplectic Lie superalgebras $\mathfrak{pe}(n)$. As an application, we classify indecomposable blocks in $\mathcal O^{\mathfrak p}$. We classify indecomposable tilting modules in $\mathcal O^{\mathfrak p}$ whose characters are controlled by the Kazhdan-Lusztig polynomials of type $\bf A$ Lie algebras. We establish the complete list of characters of indecomposable tilting modules in $\mathcal O^{\mathfrak p}$ for $\mathfrak{pe}(3)$.

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arXiv2020

Blocks and characters of $D(2|1;ζ)$-modules of non-integral weights

We classify blocks in the BGG category $\mathcal O$ of modules of non-integral weights for the exceptional Lie superalgebra $D(2|1;ζ)$. We establish various reduction methods, which connect some types of non-integral blocks of $D(2|1;ζ)$ with integral blocks of general linear Lie superalgebras $\mathfrak{gl}{(1|1)}$ and $\mathfrak{gl}{(2|1)}$. We compute the characters for irreducible $D(2|1;ζ)$-modules of non-integral weights in $\mathcal O$.

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arXiv2019

Tilting modules for classical Lie superalgebras

We study tilting and projective-injective modules in a parabolic BGG category $\mathcal O$ for an arbitrary classical Lie superalgebra. We establish a version of Ringel duality for this type of Lie superalgebras which allows to express the characters of tilting modules in terms of those of simple modules in that category. We also obtain a classification of projective-injective modules in the full BGG category $\mathcal O$ for all simple classical Lie superalgebras. We then classify and give an explicit combinatorial description of parabolic subalgebras of the periplectic Lie superalgebras and apply our results to study their tilting modules in more detail.

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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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