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Jiawen Shan

Publications and source records attributed to Jiawen Shan.

8 recordsLinked to original sources
arXiv2026

Homology of non-matching complexes under edge additions and applications to their Stanley-Reisner ideals

For a bipartite graph $G$ and an integer $t\geq2$, let $\NM_t(G)$ be its $t$-non-matching complex. We prove that adding an edge while preserving bipartiteness induces an injection on reduced homology in degree $2t-3$. Combined with the cyclic-polytope model for non-matching complexes of cycles, this shows that $\NM_t(G)$ has Leray number $2t-2$ whenever $G$ contains a cycle of length at least $2t$. Under the same hypothesis, Hochster's formula yields regularity $2t-1$ for the Stanley-Reisner ideal $I_{\NM_t(G)}$, together with explicit lower bounds for the Betti numbers on its top regularity strand and for its projective dimension. If $G$ contains a $2t$-cycle, we also determine the maximal shifts of $I_{\NM_t(G)}$ through homological degree $|E(G)|-2t+1$. For $G=K_{r,s}$ with $2\leq t\leq r\leq s$, we determine the depth, projective dimension, all maximal shifts, and the unique extremal Betti number of $I_{\NM_t(K_{r,s})}$, thereby settling a conjecture on facet ideals of chessboard complexes.

math.AC↗
arXiv2026

Further results on monomial ideals of projective dimension one

We prove that a monomial ideal has projective dimension one if and only if its minimal monomial generators can be ordered so that each successive colon ideal is principal, and show that this characterization is equivalent to the monomial version of the Hilbert-Burch Lemma. Furthermore, we prove that any squarefree monomial ideal of projective dimension one with a linear resolution has the property that all its powers \(I^s\) admit linear quotients, and we provide a partial classification of such ideals.

math.AC↗
arXiv2026

Rota-Baxter operators on $ω$-Lie algebras

This article explores Rota-Baxter operators on finite-dimensional $ω$-Lie algebras over a field of characteristic not 2. We provide several methods for constructing left-symmetric algebras, $ω$-Lie algebras, and Hom-Lie algebras via compatible Rota-Baxter operators on a given $ω$-Lie algebra. We also study the geometric structures of compatible Rota-Baxter operators of weight $0$ and isometric Rota-Baxter operators of weight $1$ over the field of complex numbers. In particular, we prove that the affine variety of all isometric Rota-Baxter operators of weight $1$ on any finite-dimensional non-Lie complex simple $ω$-Lie algebra is $1$-dimensional. Furthermore, we show that for every $4$-dimensional non-Lie complex $ω$-Lie algebra, there always exists a nilpotent compatible Rota-Baxter operator of weight $0$ such that the induced Hom-Lie algebra is nonabelian but solvable.

math.RA↗
arXiv2025

Generalized derivations of $ω$-Lie algebras

This article explores the structure theory of compatible generalized derivations of finite-dimensional $ω$-Lie algebras over a field $\mathbb{K}$. We prove that any compatible quasiderivation of an $ω$-Lie algebra can be embedded as a compatible derivation into a larger $ω$-Lie algebra, refining the general result established by Leger and Luks in 2000 for finite-dimensional nonassociative algebras. We also provide an approach to explicitly compute (compatible) generalized derivations and quasiderivations for all $3$-dimensional non-Lie complex $ω$-Lie algebras.

math.RA↗
arXiv2025

Poisson Yang-Baxter equations and $\mathcal{O}$-operators of Poisson superalgebras

We investigate connections between $\mathcal {O}$-operators of Poisson superalgebras and skew-symmetric solutions of the Poisson Yang-Baxter equation (PYBE). We prove that a skew-symmetric solution of the PYBE on a Poisson superalgebra can be interpreted as an $\mathcal {O}$-operator associated to the co-regular representation. We show that this connection can be enhanced with symplectic forms when considering non-degenerate skew-symmetric solutions. We also show that $\mathcal {O}$-operators associated to a general representation could give skew-symmetric solutions of the PYBE in certain semi-direct product of Poisson superalgebras.

math.RA↗
arXiv2024

Regularity of powers of path ideals of line graphs

Let $L_n$ be a line graph with $n$ vertices and let $I$ be its $t$-path ideal. It is shown that $I^s$ has a linear resolution for some $s\geq 1$ (or equivalently for all $s\geq 1$) if and only if $I^s$ has linear quotients for some $s\geq 1$ (or equivalently for all $s\geq 1$) if and only if $t\leq n\leq 2t$. In addition, we present an explicit formula for the regularity of $I^s$ for all $s\geq 1$. It turns out it is linear in $s$ from the very beginning.

math.AC↗
arXiv2024

Multiplicity of powers of path ideals of a line graph

Let $S=K[x_1,\ldots,x_n]$ and let $I$ be the $t$-path ideal of the line graph $L_n$ with $n$-vertices. It is shown that the set of associated prime ideals of $I^s$ is equal to the set of minimal prime ideals of $I$ for all $s\geq 1$, and we provide an explicit description of these prime ideals. Additionally, as the main contribution of this paper, we derive an explicit formula for the multiplicity of $S/I^s$ for all $s\geq 1$, revealing that it is a polynomial in $s$ from the beginning.

math.AC↗