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Yu-Feng Yao

Publications and source records attributed to Yu-Feng Yao.

16 recordsLinked to original sources
arXiv2024

Tensor product weight modules over the affine-Virasoro algebra

In this paper, we study the tensor products of irreducible highest weight modules with irreducible loop modules over the affine-Virasoro algebra with aid of the ``shifting technique" established for the Virasoro algebra in [H. Chen, X. Guo, K. Zhao, Tensor product weight modules over the Virasoro algebra, J. Lond. Math. Soc. 88(2013), 829-844.]. All such tensor product modules are indecomposable modules with infinite-dimensional weight spaces. Moreover, we obtain the necessary and sufficient conditions for such tensor product modules to be irreducible. Therefore, we obtain a class of new irreducible weight modules over the affine-Virasoro algebra. Finally, the necessary and sufficient conditions for any two such tensor product modules to be isomorphic are also determined.

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arXiv2024

Enabling Collaborative Clinical Diagnosis of Infectious Keratitis by Integrating Expert Knowledge and Interpretable Data-driven Intelligence

Although data-driven artificial intelligence (AI) in medical image diagnosis has shown impressive performance in silico, the lack of interpretability makes it difficult to incorporate the "black box" into clinicians' workflows. To make the diagnostic patterns learned from data understandable by clinicians, we develop an interpretable model, knowledge-guided diagnosis model (KGDM), that provides a visualized reasoning process containing AI-based biomarkers and retrieved cases that with the same diagnostic patterns. It embraces clinicians' prompts into the interpreted reasoning through human-AI interaction, leading to potentially enhanced safety and more accurate predictions. This study investigates the performance, interpretability, and clinical utility of KGDM in the diagnosis of infectious keratitis (IK), which is the leading cause of corneal blindness. The classification performance of KGDM is evaluated on a prospective validation dataset, an external testing dataset, and an publicly available testing dataset. The diagnostic odds ratios (DOR) of the interpreted AI-based biomarkers are effective, ranging from 3.011 to 35.233 and exhibit consistent diagnostic patterns with clinic experience. Moreover, a human-AI collaborative diagnosis test is conducted and the participants with collaboration achieved a performance exceeding that of both humans and AI. By synergistically integrating interpretability and interaction, this study facilitates the convergence of clinicians' expertise and data-driven intelligence. The promotion of inexperienced ophthalmologists with the aid of AI-based biomarkers, as well as increased AI prediction by intervention from experienced ones, demonstrate a promising diagnostic paradigm for infectious keratitis using KGDM, which holds the potential for extension to other diseases where experienced medical practitioners are limited and the safety of AI is concerned.

cs.AI↗
arXiv2023

Parabolic BGG categories and their block decomposition for Lie superalgebras of Cartan type

In this paper, we study the parabolic BGG categories for graded Lie superalgebras of Cartan type over complex numbers. The gradation of such a Lie superalgebra $\ggg$ naturally arises, with the zero component $\ggg_0$ being a reductive Lie algebra. We first show that there are only two proper parabolic subalgebras containing Levi subalgebra $\ggg_0$: the ``maximal one" $\sfp_\max$ and the ``minimal one" $\sfp_\min$. Furthermore, the parabolic BGG category arising from $\sfp_\max$, essentially turns out to be a subcategory of the one arising from $\sfp_\min$. Such a priority of $\sfp_\min$ in the sense of representation theory reduces the question to the study of the ``minimal parabolic" BGG category $\comi$ associated with $\sfp_\min$. We prove the existence of projective covers of simple objects in these categories, which enables us to establish a satisfactory block theory. Most notably, our main results are as follows: (1) We classify and obtain a precise description of the blocks of $\comi$. (2) We investigate indecomposable tilting and indecomposable projective modules in $\comi$, and compute their character formulas.

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arXiv2022

Quasi-simple modules and Loewy lengths in modular representations of reductive Lie algebras

Let $\frak g$ be a reductive Lie algebra over an algebraically closed field of characteristic $p>0$. In this paper, we study the representations of $\frak g$ with a $p$-character $χ$ of standard Levi form associated with a given subset $I$ of the simple root system $Π$ of $\frak g$. Let $U_χ({\frak g})$ be the reduced enveloping algebra of $\frak g$. A notion "quasi-simple module" (denoted by $\mathcal L_χ(λ)$) is introduced. The properties of such a module turn out to be better than those of the corresponding simple module $\widehat L_χ(λ)$. It enables us to investigate the $U_χ({\frak g})$-modules from a new point of view, and correspondingly gives rise new consequences. First, we show that the first self extension of $\mathcal L_χ(λ)$ is zero, and the projective dimension of $\mathcal L_χ(λ)$ is finite when $λ$ is $p$-regular. These properties make it significant to rewrite the formula of Lusztig's Hope (Lusztig's conjecture on the irreducible characters in the category of $U_χ({\frak g})$-modules) by replacing $\widehat L_χ(λ)$ by $\mathcal L_χ(λ)$. Second, with the aid of quasi-simple modules, we get a formula on the Loewy lengths of standard modules and proper standard modules over $U_χ({\frak g})$. And by studying some examples, we formulate some conjectures on the Loewy lengths of indecomposable projective $\frak g$-modules, standard modules and proper standard modules.

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arXiv2021

A new class of irreducible modules over the affine-Virasoro algebra of type $A_1$

In this paper, we construct a class of non-weight modules over the affine-Virasoro algebra of type $A_1$ by taking tensor products of a finite number of irreducible modules $M(λ, α, β, γ)$ with irreducible highest weight modules $V(η, ε, θ)$. We obtain the necessary and sufficient conditions for such tensor product modules to be irreducible, and determine the necessary and sufficient conditions for such two modules to be isomorphic. We also compare these modules with other known non-weight modules, showing that these irreducible modules are new.

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arXiv2021

Tensor products of the defining representations over the Witt algebra in positive characteristic

Let $A(1):=k[X]/(X^p)$ be the natural representation of the Witt algebra $W(1)$ over an algebraically closed field of prime characteristic $p>3$. In this note, we decompose the $W(1)$-module $A(1)\otimes A(1)$ into two invariant subspaces, and precisely construct their Jordan-Hölder composition series. As a consequence, we obtain all decomposition factors of the tensor product of the simple restricted $W(1)$-module with "highest" weight $p-1$.

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arXiv2021

On structure of graded restricted simple Lie algebras of Cartan type as modules over the Witt algebra

Any graded restricted simple Lie algebra of Cartan type contains a subalgebra isomorphic to the Witt algebra over a field of prime characteristic. As some analogue of study on branching rules for restricted non-classical Lie algebras, it is shown that each graded restricted simple Lie algebra of Cartan type can be decomposed into a direct sum of restricted baby Verma modules and simple modules as an adjoint module over the Witt algebra. In particular, the composition factors are precisely determined.

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arXiv2021

Irreducible tensor product modules over the affine-Virasoro algebra of type $A_1$

In this paper, we construct a class of non-weight modules over the affine-Virasoro algebra of type $A_1$ by taking tensor products of irreducibles defined in [Q. Chen, J. Han, Non-weight modules over the affine-Virasoro algebra of type $A_1$, J. Math. Phys. 60, 071707 (2019)] with irreducible highest weight modules. The irreducibility and the isomorphism classes of these modules are determined. Moreover, we show that these tensor product modules are different from the known non-weight modules. Finally, we realize some tensor product modules as induced modules from modules over certain subalgebras of the affine-Virasoro algebra of type $A_1$, and give sufficient and necessary conditions for these induced modules to be reducible.

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arXiv2021

On Chevalley restriction theorem for semi-reductive algebraic groups and its applications

An algebraic group is called semi-reductive if it is a semi-direct product of a reductive subgroup and the unipotent radical. Such a semi-reductive algebraic group naturally arises and also plays a key role in the study of modular representations of non-classical finite-dimensional simple Lie algebras in positive characteristic, and some other cases. Let $G$ ba a connected semi-reductive algebraic group over an algebraically closed field $\mathbb{F}$ and $\mathfrak{g}=Lie(G)$. It turns out that $G$ has many same properties as reductive groups, such as the Bruhat decomposition. In this note, we obtain an analogue of classical Chevalley restriction theorem for $\mathfrak{g}$, which says that the $G$-invariant ring $\mathbb{F}[\mathfrak{g}]^G$ is a polynomial ring if $\mathfrak{g}$ satisfies a certain "posivity" condition suited for lots of cases we are interested in. As applications, we further investigate the nilpotent cones and resolutions of singularities for semi-reductive Lie algebras.

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arXiv2020

Deep Sequential Feature Learning in Clinical Image Classification of Infectious Keratitis

Infectious keratitis is the most common entities of corneal diseases, in which pathogen grows in the cornea leading to inflammation and destruction of the corneal tissues. Infectious keratitis is a medical emergency, for which a rapid and accurate diagnosis is needed for speedy initiation of prompt and precise treatment to halt the disease progress and to limit the extent of corneal damage; otherwise it may develop sight-threatening and even eye-globe-threatening condition. In this paper, we propose a sequential-level deep learning model to effectively discriminate the distinction and subtlety of infectious corneal disease via the classification of clinical images. In this approach, we devise an appropriate mechanism to preserve the spatial structures of clinical images and disentangle the informative features for clinical image classification of infectious keratitis. In competition with 421 ophthalmologists, the performance of the proposed sequential-level deep model achieved 80.00% diagnostic accuracy, far better than the 49.27% diagnostic accuracy achieved by ophthalmologists over 120 test images.

eess.IV↗
arXiv2018

Non-weight modules over algebras related to the Virasoro algebra

In this paper, we study a class of non-weight modules over two kinds of algebras related to the Virasoro algebra, i.e., the loop-Virasoro algebras $\mathfrak{L}$ and a class of Block type Lie algebras $\mathfrak{B(q)}$, where $q$ is a nonzero complex number. We determine those modules whose restriction to the Cartan subalgebra (modulo center) are free of rank one. We also provide a sufficient and necessary condition for such modules to be simple, and determine their isomorphism classes. Moreover, we obtain the simplicity of modules over loop-Virasoro algebras by taking tensor products of some irreducible modules mentioned above with irreducible highest weight modules or Whittaker modules.

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arXiv2014

A criterion for irreducibility of parabolic baby Verma modules of reductive Lie algebras

Let $G$ be a connected, reductive algebraic group over an algebraically closed field $k$ of prime characteristic $p$ and $\mathfrak{g}=Lie(G)$. In this paper, we study representations of $\mathfrak{g}$ with a $p$-character $χ$ of standard Levi form. When $\mathfrak{g}$ is of type $A_n, B_n, C_n$ or $D_n$, a sufficient condition for the irreducibility of standard parabolic baby Verma $\mathfrak{g}$-modules is obtained. This partially answers a question raised by Friedlander and Parshall in [Friedlander E. M. and Parshall B. J., Deformations of Lie algebra representations, Amer. J. Math. 112 (1990), 375-395]. Moreover, as an application, in the special case that $\mathfrak{g}$ is of type $A_n$ or $B_n$, and $χ$ lies in the sub-regular nilpotent orbit, we recover a result of Jantzen in [Jantzen J. C., Subregular nilpotent representations of $sl_n$ and $so_{2n+1}$, Math. Proc. Cambridge Philos. Soc. 126 (1999), 223-257].

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arXiv2013

Nilpotent commuting varieties of the Witt algebra

Let $\mathfrak{g}$ be the $p$-dimensional Witt algebra over an algebraically closed field $k$ of characteristic $p>3$. Let $\mathscr{N}={x\in\ggg\mid x^{[p]}=0}$ be the nilpotent variety of $\mathfrak{g}$, and $\mathscr{C}(\mathscr{N}):=\{(x,y)\in \mathscr{N}\times\mathscr{N}\mid [x,y]=0\}$ the nilpotent commuting variety of $\mathfrak{g}$. As an analogue of Premet's result in the case of classical Lie algebras [A. Premet, Nilpotent commuting varieties of reductive Lie algebras. Invent. Math., 154, 653-683, 2003.], we show that the variety $\mathscr{C}(\mathscr{N})$ is reducible and equidimensional. Irreducible components of $\mathscr{C}(\mathscr{N})$ and their dimension are precisely given. Furthermore, the nilpotent commuting varieties of Borel subalgebras are also determined.

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arXiv2012

Borel subalgebras of the Witt algebra $W_1$

Let $\mathbb{F}$ be an algebraically closed field of characteristic $p>3$, and $\ggg$ the $p$-dimensional Witt algebra over $\mathbb{F}$. Let $\N$ be the nilpotent cone of $\ggg$. Explicit description of $\N$ is given, so that the conjugacy classes of Borel subalgebras of $\ggg$ under the automorphism group are determined. In contrast with only one conjugacy class of Borel subalgebras in a classical simple Lie algebra, there are two conjugacy classes of Borel subalgebras in $\ggg$. The representatives of conjugacy classes of Borel subalgebras, i.e., the so-called standard Borel subalgebras, are precisely given.

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