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Zhankui Xiao

Publications and source records attributed to Zhankui Xiao.

17 recordsLinked to original sources
arXiv2026

Functional identities of degree 2 at two-sided zero products on incidence algebras

Let $R$ be a commutative ring with unity such that $\frac{1}{2}\in R$. Let $X$ be a connected finite poset with $|X|>2$ and $I(X,R)$ be the incidence algebra of $X$ over $R$. In this paper, we characterize the forms of linear maps $F_1,F_2,F_3,F_4:I(X,R)\to I(X,R)$ satisfying \[ F_1(f)g+fF_2(g)+F_3(g)f+gF_4(f)=0, \] whenever $fg=gf=0$. We prove that the $F_i$'s are of the so-called standard form if and only if any two edges in the comparability graph of $X$ are contained in one cycle. The ingredients of the proof contain a characterization of $2$-connectedness in comparability graph and the two-sided zero product determined property of incidence algebras.

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arXiv2026

A categorification of the Brenti--Welker identity

The paper aims to provide a categorification of the Brenti--Welker identity involving Eulerian numbers in (Adv. Appl. Math. 42 (2009): 545--556) by lifting it from an enumerative equality to an isomorphism of symmetric group representations. To do so, we study the decomposition of the tensor product of $(\mathbb{C}^r)^{\otimes n}$ and modules affording Foulkes characters as modules of the symmetric group. The main ingredient of the proof is a combinatorial identity which may be of independent interest.

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arXiv2026

Finite dimensional zero Jordan product determined algebras are generated by idempotents

Bre\v{s}ar showed that a finite dimensional unital associative algebra is zero product determined if and only if it is generated by idempotents. For the analogue of zero Jordan product determined algebras, only one direction was known: over a field of characteristic not 2, every algebra generated by idempotents is zero Jordan product determined. Whether the converse holds has remained an open problem. In this paper, we answer this question affirmatively in the finite dimensional case. Some related open problems are stated at the end.

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arXiv2026

Commuting maps of inflated algebras

Commuting maps on a class of algebras called inflated algebras are investigated. In particular, we can prove that every commuting map $\theta$ on such an algebra is of the form $\theta(x)=c x+\mu(x)$, where $c$ belongs to the base field $K$ of characteristic not 2, and $\mu$ is a central-valued linear map.

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arXiv2026

Dual partially harmonic tensors and quantized Schur--Weyl duality

Let $V$ be a $2m$-dimensional symplectic space over an infinite field $K$. Let $\mathfrak{B}^{(f)}_{n,K}$ be the two-sided ideal of the Birman--Murakami--Wenzl algebra $\mathfrak{B}_{n,K}$ generated by $E_1E_3\cdots E_{2f-1}$ with $1\leq f\leq\left\lfloor \frac n2 \right\rfloor$. In this paper, using the diagram category of framed tangles and canonical basis, we prove that the natural homomorphism from $\mathfrak{B}_{n,K}/\mathfrak{B}^{(f)}_{n,K}$ to $ \mathrm{End}_{U_q(\mathfrak{sp}_{2m})}\left(V^{\otimes n}/\left(V^{\otimes n}\cdot \mathfrak{B}^{(f)}_{n,K}\right)\right)$ is always surjective.

math.QA↗
arXiv2021

Tilting modules, dominant dimensions and Brauer-Schur-Weyl duality

Let $A$ be a standardly stratified algebra over a field $K$ and $T$ a tilting module over $A$. Let $Λ^+$ be an indexing set of all simple modules in $A\lmod$. We show that if there is an integer $r\in\N$ such that for any $λ\inΛ^+$, there is an embedding $Δ(λ)\hookrightarrow T^{\oplus r}$ as well as an epimorphism $T^{\oplus r}\twoheadrightarrow\overline{\nabla}(λ)$ as $A$-modules, then $T$ is a faithful $A$-module and $A$ has the double centraliser property with respect to $T$. As applications, we prove that if $A$ is quasi-hereditary with a simple preserving duality and $T$ a given faithful tilting $A$-module, then $A$ has the double centralizer property with respect to $T$. This provides a simple and useful criterion which can be applied in many situations in algebraic Lie theory. We affirmatively answer a question of Mazorchuk and Stroppel by proving the existence of a unique minimal basic tilting module $T$ over $A$ for which $A=\End_{\End_A(T)}(T)$. We also establish a Schur-Weyl duality between the symplectic Schur algebra $S^{sy}(m,n)$ and $\bb_{n}/\mathfrak{B}_{n}^{(f)}$ on $V^{\otimes n}/V^{\otimes n}\mathfrak{B}_{n}^{(f)}$ when $\cha K>\min\{n-f+m,n\}$, where $V$ is a $2m$-dimensional symplectic space over $K$, $\mathfrak{B}_{n}^{(f)}$ is the two-sided ideal of the Brauer algebra $\bb_{n}(-2m)$ generated by $e_1e_3\cdots e_{2f-1}$ with $1\leq f\leq [\frac{n}{2}]$.

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arXiv2019

Commuting maps on certain incidence algebras

Let $\mathcal{R}$ be a $2$-torsion free commutative ring with unity, $X$ a locally finite pre-ordered set and $I(X,\mathcal{R})$ the incidence algebra of $X$ over $\mathcal{R}$. If $X$ consists of a finite number of connected components, in this paper we give a sufficient and necessary condition for each commuting map on $I(X,\mathcal{R})$ being proper.

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arXiv2019

Lie Triple Derivations of Incidence Algebras

Let $\mathcal{R}$ be a $2$-torsion free commutative ring with unity, $X$ a locally finite pre-ordered set and $I(X,\mathcal{R})$ the incidence algebra of $X$ over $\mathcal{R}$. If $X$ consists of a finite number of connected components, we prove in this paper that every Lie triple derivation of $I(X,\mathcal{R})$ is proper.

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arXiv2017

Zero action determined modules for associative algebras

Let $A$ be a unital associative algebra over a field $F$ and $V$ be a unital left $A$-module. The module $V$ is called zero action determined if every bilinear map $f: A\times V\rightarrow F$ with the property that $f(a,m)=0$ whenever $am=0$ is of the form $f(x,v)=Φ(xv)$ for some linear map $Φ: V\rightarrow F$. In this paper, we classify the finite dimensional irreducible and principal projective zero action determined modules of $A$. As an application, two classes of zero product determined algebras are shown: some semiperfect algebras (infinite dimensional in general); quasi-hereditary cellular algebras.

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arXiv2014

On Tensor Spaces for Rook Monoid Algebras

Let $m,n\in \mathbb{N}$, and $V$ be a $m$-dimensional vector space over a field $F$ of characteristic $0$. Let $U=F\oplus V$ and $R_n$ be the rook monoid. In this paper, we construct a certain quasi-idempotent in the annihilator of $U^{\otimes n}$ in $FR_n$, which comes from some one-dimensional two-sided ideal of rook monoid algebra. We show that the two-sided ideal generated by this element is indeed the whole annihilator of $U^{\otimes n}$ in $FR_n$.

math.RA↗
arXiv2014

Centralizing traces and Lie triple isomorphisms on generalized matrix algebras

Let $\mathcal{G}$ be a generalized matrix algebra over a commutative ring $\mathcal{R}$ and $\mathcal{Z(G)}$ be the center of $\mathcal{G}$. Suppose that ${\mathfrak q}\colon \mathcal{G}\times \mathcal{G}\longrightarrow \mathcal{G}$ is an $\mathcal{R}$-bilinear mapping and ${\mathfrak T}_{\mathfrak q}\colon \mathcal{G}\longrightarrow \mathcal{G}$ is the trace of $\mathfrak{q}$. We describe the form of ${\mathfrak T}_{\mathfrak q}$ satisfying the condition $[{\mathfrak T}_{\mathfrak q}(G), G]\in \mathcal{Z(G)}$ for all $G\in \mathcal{G}$. The question of when ${\mathfrak T}_{\mathfrak q}$ has the proper form is considered. Using the aforementioned trace function, we establish sufficient conditions for each Lie triple isomorphism of $\mathcal{G}$ to be almost standard. As applications we characterize Lie triple isomorphisms of full matrix algebras, of triangular algebras and of certain unital algebras with nontrivial idempotents. Some topics for future research closely related to our current work are proposed at the end of this article.

math.RA↗
arXiv2014

Jordan Derivations of Incidence Algebras

Let $\mathcal{R}$ be a commutative ring with identity, $I(X,\mathcal{R})$ be the incidence algebra of a locally finite pre-ordered set $X$. In this note, we characterise the derivations of $I(X,\mathcal{R})$ and prove that every Jordan derivation of $I(X,\mathcal{R})$ is a derivation provided that $\mathcal{R}$ is $2$-torsion free.

math.RA↗
arXiv2013

Centralizing Traces and Lie Triple Isomorphisms on Triangular Algebras

Let $\mathcal{T}$ be a triangular algebra over a commutative ring $\mathcal{R}$ and $\mathcal{Z(T)}$ be the center of $\mathcal{T}$. Suppose that ${\mathfrak q}\colon \mathcal{T}\times \mathcal{T}\longrightarrow \mathcal{T}$ is an $\mathcal{R}$-bilinear mapping and that ${\mathfrak T}_{\mathfrak q}\colon: \mathcal{T}\longrightarrow \mathcal{T}$ is a trace of $\mathfrak{q}$. We describe the form of ${\mathfrak T}_{\mathfrak q}$ satisfying the condition $[{\mathfrak T}_{\mathfrak q}(T), T]\in \mathcal{Z(T)}$ for all $T\in \mathcal{T}$. The question of when ${\mathfrak T}_{\mathfrak q}$ has the proper form will be addressed. Using the aforementioned trace function, we establish sufficient conditions for each Lie triple isomorphism on $\mathcal{T}$ to be almost standard. As applications we characterize Lie triple isomorphisms of triangular matrix algebras and nest algebras. Some further research topics related to current work are proposed at the end of this article.

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arXiv2012

Commuting Traces and Lie Isomorphisms on Generalized Matrix Algebras

Let $\mathcal{G}$ be a generalized matrix algebra over a commutative ring $\mathcal{R}$, ${\mathfrak q}\colon \mathcal{G}\times\mathcal{G}\longrightarrow \mathcal{G}$ be an $\mathcal{R}$-bilinear mapping and ${\mathfrak T}_{\mathfrak q}\colon:\mathcal{G}\longrightarrow \mathcal{G}$ be a trace of $\mathfrak{q}$. We describe the form of ${\mathfrak T}_{\mathfrak q}$ satisfying the condition ${\mathfrak T}_{\mathfrak q}(G)G=G{\mathfrak T}_{\mathfrak q}(G)$ for all $G\in \mathcal{G}$. The question of when ${\mathfrak T}_{\mathfrak q}$ has the proper form is considered. Using the aforementioned trace function, we establish sufficient conditions for each Lie isomorphism of $\mathcal{G}$ to be almost standard. As applications we characterize Lie isomorphisms of full matrix algebras, of triangular algebras and of certain unital algebras with nontrivial idempotents. Some further research topics related to current work are proposed at the end of this article.

math.RA↗
arXiv2012

Nonlinear Lie type Derivations of Von Neumann Algebras

Let $A$ be a von Neumann algebra with no central summands of type $I_1$. We will show that every nonlinear Lie $n$-derivation on $A$ is of the standard form, i.e. it can be expressed as a sum of an additive derivation and a central-valued mapping which annihilates each $(n-1)$-th commutator of $A$.

math.RA↗
arXiv2011

On a theorem of Lehrer and Zhang

Let $K$ be an arbitrary field of characteristic not equal to 2. Let $m, n\in\N$ and $V$ an $m$ dimensional orthogonal space over $K$. There is a right action of the Brauer algebra $\bb_n(m)$ on the $n$-tensor space $V^{\otimes n}$ which centralizes the left action of the orthogonal group $O(V)$. Recently G.I. Lehrer and R.B. Zhang defined certain quasi-idempotents $E_i$ in $\bb_n(m)$ (see (\ref{keydfn})) and proved that the annihilator of $V^{\otimes n}$ in $\bb_n(m)$ is always equal to the two-sided ideal generated by $E_{[(m+1)/2]}$ if $\ch K=0$ or $\ch K>2(m+1)$. In this paper we extend this theorem to arbitrary field $K$ with $\ch K\neq 2$ as conjectured by Lehrer and Zhang. As a byproduct, we discover a combinatorial identity which relates to the dimensions of Specht modules over symmetric groups of different sizes and a new integral basis for the annihilator of $V^{\otimes m+1}$ in $\bb_{m+1}(m)$.

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