N. S. Khripchenko, B. V. Novikov
We consider the functions in two variables on an arbitrary poset, for which the convolution operation is defined. We obtain the generalization of incidence algebra and describe its properties: invertibility, the Jackobson radical, idempotents, regular elements. As a consequence a positive solution of the isomorphism problem for such algebras is obtained.
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We give a full description of the Poisson structures on the finitary incidence algebra $FI(P,R)$ of an arbitrary poset $P$ over a commutative unital ring $R$.
Let $P$ be an arbitrary partially ordered set, $R$ a commutative ring with identity and $FI(P,R)$ the finitary incidence algebra of $P$ over $R$. Under some natural assumption on $R$, we prove that each Lie-type derivation of $FI(P,R)$ is proper.
Let $\mathcal{C}$ be a pocategory, $FI(\mathcal{C})$ the finitary incidence ring of $\mathcal{C}$ and $φ$ a Jordan isomorphism of $FI(\mathcal{C})$ onto an associative ring $A$. We study the problem of decomposition of $φ$ into the (near-)sum of a homomorphism and an anti-homomorphism.
Let $X$ be a partially ordered set, $R$ a commutative $2$-torsionfree unital ring and $FI(X,R)$ the finitary incidence algebra of $X$ over $R$. In this note we prove that each $R$-linear Jordan isomorphism of $FI(X,R)$ onto an $R$-algebra $A$ is the near-sum of a homomorphism and an anti-homomorphism.
Let $FI(X,K)$ be the finitary incidence algebra of a non-connected partially ordered set $X$ over a field $K$ of characteristic different from $2$. For the case where every multiplicative automorphism of $FI(X,K)$ is inner, we present necessary and sufficient conditions for two involutions on $FI(X,K)$ to be equivalent.
Let $P$ be a partially ordered set, $R$ a commutative ring with identity and $FI(P,R)$ the finitary incidence algebra of $P$ over $R$. In this note we prove that each $R$-linear local derivation of $FI(P,R)$ is a derivation, which partially generalizes a result by Nowicki and Nowosad.
This is a continuation of our earlier works \cite{KhrypchenkoWei, Yang20211, Yang20212} with respect to (non-)linear Lie-type derivations of finitary incidence algebras. Let $X$ be a pre-ordered set, $\mathcal{R}$ be a $2$-torsionfree and $(n-1)$-torsionfree commutative ring with identity, where $n\geq 2$ is an integer. Let $FI(X,\mathcal{R})$ be the finitary incidence algebra of $X$ over $\mathcal{R}$. In this paper, a complete clarification is obtained for the structure of nonlinear Lie-type derivations of $FI(X,\mathcal{R})$. We introduce a new class of derivations on $FI(X,\mathcal{R})$ na…
Let $P$ be a partially ordered set, $R$ a commutative unital ring and $FI(P,R)$ the finitary incidence algebra of $P$ over $R$. We prove that each $R$-linear higher derivation of $FI(P,R)$ decomposes into the product of an inner higher derivation of $FI(P,R)$ and the higher derivation of $FI(P,R)$ induced by a higher transitive map on the set of segments of $P$.
Let $P$ be a preordered set, $R$ a ring and $FI(P,R)$ the finitary incidence ring of $P$ over $R$. We find a criterion for all Jordan derivations of $FI(P,R)$ to be derivations and generalize Theorem 3.3 from arXiv:1411.6123. In particular, we prove that each Jordan derivation of the ring $RFM_I(R)$ of row-finite $I\times I$-matrices over $R$ is a derivation, if $|I|>1$.
Invertibility preservers of finitary incidence algebras
该研究针对任意偏序集$X$与任意域$K$，通过幂集代数映射与线性映射刻画了有限关联代数$FI(X,K)$的线性单位可逆性保持映射，丰富了关联代数的映射理论。
Let $R$ be a commutative, indecomposable ring with identity and $(P,\le)$ a partially ordered set. Let $FI(P)$ denote the finitary incidence algebra of $(P,\le)$ over $R$. We will show that, in most cases, local automorphisms of $FI(P)$ are actually $R$-algebra automorphisms. In fact, the existence of local automorphisms which fail to be $R$-algebra automorphisms will depend on the chosen model of set theory and will require the existence of measurable cardinals. We will discuss local automorphisms of cartesian products as a special case in preparation of the general result.
Let $K$ be a field and $X$ a partially ordered set (poset). Let $FI(X,K)$ and $I(X,K)$ be the finitary incidence algebra and the incidence space of $X$ over $K$, respectively, and let $D(X,K)=FI(X,K)(+)I(X,K)$ be the idealization of the $FI(X,K)$-bimodule $I(X,K)$. In the first part of this paper, we show that $D(X,K)$ has an anti-automorphism (involution) if and only if $X$ has an anti-automorphism (involution). We also present a characterization of the anti-automorphisms and involutions on $D(X,K)$. In the second part, we obtain the classification of involutions on $D(X,K)$ to the case when …
Identities and derived lengths of finitary incidence algebras and their group of units
本文研究了域$K$上偏序集$X$的有限关联代数$FI(X,K)$，确定了其满足多项式恒等式及单位群满足群恒等式的条件，并求出了其李导出长度与单位群的导出长度。
Involutions of the second kind on finitary incidence algebras
本文研究有限入射代数上的第二类对合，给出了其存在的充要条件、特征描述，以及在特定条件下两个此类对合等价的判定条件。
We describe gradings by finite abelian groups on the associative algebras of infinite matrices with finitely many nonzero entries, over an algebraically closed field of characteristic zero.
We classify, up to isomorphism, all gradings by an arbitrary abelian group on simple finitary Lie algebras of linear transformations (special linear, orthogonal and symplectic) on infinite-dimensional vector spaces over an algebraically closed field of characteristic different from 2.
An extremal element $x$ in a Lie algebra $\mathfrak{g}$ is an element for which the space $[x, [x, \mathfrak{g}]]$ is contained in the linear span of $x$. Long root elements in classical Lie algebras are examples of extremal elements. Lie algebras generated by extremal elements lead to geometries with as points the $1$-spaces generate by extremal elements an as lines the $2$-spaces whose non-zero elements are pairwise commuting extremal elements. In this paper we show that the finitary special linear Lie algebras can be characterized by their extremal geometry. Moreover, we also show that the …
Images of Multilinear Polynomials in the Algebra of Finitary Matrices Contain Trace Zero Matrices
本研究针对无限域上非零多重线性多项式在有限矩阵代数中的像问题，证明对任意正整数d存在s使其像含所有d阶迹零矩阵，进而得出其在全体有限矩阵代数中的像包含所有迹零有限矩阵的结论。
Maximal ideals in the finitary symmetric group algebra in characteristic two
本文解决了特征2域上有限对称群代数极大理想分类的悬而未决问题，通过关联其素理想与高阶Verlinde范畴，借助研究一般线性群倾斜模的厚张量理想完成了该工作。
With any locally finite partially ordered set $K$ its incidence algebra $Ω(K)$ is associated. We shall consider algebras over fields with characteristic zero. In this case there is a correspondence $K \leftrightarrow Ω(K)$ such that the poset $K$ can be reconstructed from its incidence algebra up to an isomorphism -- due to Stanley theorem. In the meantime, a monotone mapping between two posets in general induces no homomorphism of their incidence algebras. In this paper I show that if the class of posets is confined to simplicial complexes then their incidence algebras acquire the structure o…
The graph algebra is a commutative, cocommutative, graded, connected incidence Hopf algebra, whose basis elements correspond to finite simple graphs and whose Hopf product and coproduct admit simple combinatorial descriptions. We give a new formula for the antipode in the graph algebra in terms of acyclic orientations; our formula contains many fewer terms than Takeuchi's and Schmitt's more general formulas for the antipode in an incidence Hopf algebra. Applications include several formulas (some old and some new) for evaluations of the Tutte polynomial.
Totally compatible structures on the radical of an incidence algebra
研究有限偏序集上发生代数极小理想中的完全兼容结构，证明此类结构通常非正规。
Bell polynomials appear in several combinatorial constructions throughout mathematics. Perhaps most naturally in the combinatorics of set partitions, but also when studying compositions of diffeomorphisms on vector spaces and manifolds, and in the study of cumulants and moments in probability theory. We construct commutative and noncommutative Bell polynomials and explain how they give rise to Faà di Bruno Hopf algebras. We use the language of incidence Hopf algebras, and along the way provide a new description of antipodes in noncommutative incidence Hopf algebras, involving quasideterminants…
Let $X$ be a finite connected poset and $K$ a field. We study the question, when all Lie automorphisms of the incidence algebra $I(X,K)$ are proper. Without any restriction on the length of $X$ we find only a sufficient condition involving certain equivalence relation on the set of maximal chains of $X$. For some classes of posets of length one, such as finite connected crownless posets (i.e., without weak crown subposets), crowns and ordinal sums of two antichains we give a complete answer.
We study Galois extensions Coinv(M)<M for M an H-comodule algebra and H a Frobenius Hopf algebroid. We obtain generalizations of various theorems in Hopf-Galois theory by Kreimer-Takeuchi, Doi-Takeuchi and Cohen-Fischman-Montgomery. An algebra extension is Galois precisely if it is balanced, depth 2, and Frobenius. Then we show that Yetter-Drinfeld categories over H are always braided and their braided commutative algebras play the role of noncommutative scalar extensions by the Brzezinski-Militaru Theorem. Contravariant "fiber functors" are used to prove an analogue of Ulbrich's Theorem and t…
We introduce the notion of an ACP process algebra and the notion of a meadow enriched ACP process algebra. The former notion originates from the models of the axiom system ACP. The latter notion is a simple generalization of the former notion to processes in which data are involved, the mathematical structure of data being a meadow. Moreover, for all associative operators from the signature of meadow enriched ACP process algebras that are not of an auxiliary nature, we introduce variable-binding operators as generalizations. These variable-binding operators, which give rise to comprehended ter…
The clone of term operations of an algebraic structure consists of all operations that can be expressed by a term in the language of the structure. We consider bounds for the length and the height of the terms expressing these functions, and we show that these bounds are often robust against the change of the basic operations of the structure.
The algebraic monoid structure of an incidence algebra is investigated. We show that the multiplicative structure alone determines the algebra automorphisms of the incidence algebra. We present a formula that expresses the complexity of the incidence monoid with respect to the two sided action of its maximal torus in terms of the zeta polynomial of the poset. In addition, we characterize the finite (connected) posets whose incidence monoids have complexity $\leq 1$. Finally, we determine the covering relations of the adherence order on the incidence monoid of a star poset.
Group gradings on finite-dimensional incidence algebras. II
本文完成了有限维关联代数上群分次的描述，并对能实现为带有群分次的关联代数的有限维分次代数进行了分类。
Polynomial identities for quivers via incidence algebras
研究有向图的路径代数与矩阵代数的多项式恒等式，证明其PI等价性。
We provide a unified approach, via deformations of incidence algebras, to several important types of representations with finiteness conditions, as well as the combinatorial algebras which produce them. We show that over finite dimensional algebras, representations with finitely many orbits, or finitely many invariant subspaces, or distributive coincide, and further coincide with thin modules in the acyclic case. Incidence algebras produce examples of such modules, and we show that algebras which are locally hereditary, and whose projective are distributive, or equivalently, which have finitel…
Let $R$ be a ring with identity and $I(X,R)$ be the incidence algebra of a locally finite partially ordered set $X$ over $R.$ In this paper, we compute the socle and the singular ideal of the incidence ring for some $X$ in terms of the socle of $R$ and the singular ideal of $R$, respectively.
Let $X$ be a finite connected poset and $K$ a field. We give a full description of the Lie automorphisms of the incidence algebra $I(X,K)$. In particular, we show that they are in general not proper.
Let $X$ be a locally finite preordered set, $\mathcal R$ a commutative ring with identity and $I(X,\mathcal R)$ the incidence algebra of $X$ over $\mathcal R$. In this note we prove that each Lie derivation of $I(X,\mathcal R)$ is proper, provided that $\mathcal R$ is $2$-torsion free.
Acyclic categories were introduced by Kozlov and can be viewed as generalised posets. Similar to posets, one can define their incidence algebras and a related topological complex. We consider the incidence algebra of either a poset or acyclic category as the quotient of a path algebra by the parallel ideal. We show that this ideal has a quadratic Groeobner basis with a lexicographic monomial order if and only if the poset or acyclic category is lex-shellable.
Let P a locally finite partially ordered set, F a field, G a group, and I(P,F) the incidence algebra of P over F. We describe all the inequivalent elementary G-gradings on this algebra. If P is bounded, F is a infinite field of characteristic zero, and A, B are both elementary G-graded incidence algebras satisfying the same G-graded polynomial identities, and the automorphisms group of P acts transitively on the maximal chains of P , we show that A and B are graded isomorphic.
We fully characterize regular Hom-Lie structures on the incidence algebra $I(X,K)$ of a finite connected poset $X$ over a field $K$. We prove that such a structure is the sum of a central-valued linear map annihilating the Jacobson radical of $I(X,K)$ with the composition of certain inner and multiplicative automorphisms of $I(X,K)$.
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.
We give various formulas to compute the number of all involutions, i.e. elements of order 2, in an incidence algebra $I(X,\mathbb{K})$, where $X$ is a finite poset (star, Y and Rhombuses) and $\mathbb{K}$ is a finite field of characteristic different from 2. Using the techniques describing here we show an algorithm to calculate the number of involutions on any finite poset.
Given an algebra A we associate an incidence algebra A(Σ) and compare their Hochschild cohomology groups.
Let $P$ and $Q$ be finite posets and $R$ a commutative unital ring. In the case where $R$ is indecomposable, we prove that the $R$-linear isomorphisms between partial flag incidence algebras $I^3(P,R)$ and $I^3(Q,R)$ are exactly those induced by poset isomorphisms between $P$ and $Q$. We also show that the $R$-linear derivations of $I^3(P,R)$ are trivial.
We give a full description of all the canonical algebras over an algebraically closed field that are derived equivalent to incidence algebras of finite posets. These are the canonical algebras whose number of weights is either 2 or 3.
We compute the Hochschild cohomology of the reduced incidence algebras such as the algebra of formal power series, the algebra of exponential power series, the algebra of Eulerian power series, and the algebra of formal Dirichlet series. We achieve the result by carrying out the computation on the coalgebra ${\rm Cotor}$-groups of their pre-dual coalgebras. Using the same coalgebraic machinery, we further identify the Hochschild cohomology groups of an incidence algebra associated to a quiver with the ${\rm Ext}$-groups of the incidence algebra associated to a suspension of the quiver.
Group gradings on finite dimensional incidence algebras
本文在特定域条件下分类了有限维关联代数的群分次，研究了阿贝尔群下的分次双模结构，得到更深刻的分次结构结果并分类了分次同构类。
This inquiry is based on both the construction of generalized incidence rings due to Gene Abrams and the construction of good group gradings of incidence algebras due to Molli Jones. We provide conditions for a generalized incidence ring to be graded isomorphic to a subring of an incidence ring over a preorder. We also extend Jones's construction to good group gradings for incidence algebras over preorders with crosscuts of length one or two.
Multiplicative Automorphisms of Incidence Algebras
该文研究有限连通偏序集上关联代数的乘性自同构结构，证明内乘性自同构子群是其直因子，并由此得到二者匹配的若干判定准则，丰富了关联代数的自同构理论。
Jordan $*$-derivations of incidence algebras
本文研究局部有限偏序集上关联代数的 Jordan *-导子，证明其均为内 *-导子且为转置 Jordan *-导子，并给出存在非 *-导子的 Jordan *-导子的结论。
Let $\mathbb{K}$ be a field such that $char(\mathbb{K})\nmid k$ and $char(\mathbb{K})\nmid k+1$. We describe all $(k+1)$-potent matrices over the group of upper triangular matrix. In the case that $\mathbb{K}$ is a finite field we show how to compute the number of these elements in triangular matrix groups and use this formula to compute the number of $(k+1)$-potent elements in the Incidence Algebra $\mathcal{I}(X,\mathbb{K})$ where $X$ is a finite poset.
In this paper, we investigate the subset obtained by evaluations of a fixed multilinear polynomial on a given algebra. We provide an example of a multilinear polynomial, whose image is not a vector subspace; namely, the product of two commutators need not to be a subspace whenever evaluated on certain subalgebras of upper triangular matrices (the so-called incidence algebras). In the last part of the paper, given that the field is infinite, we reduce the problem of the description of the image of a polynomial evaluated on an incidence algebra to the study of evaluations of a certain family of …
Let $I(X,K)$ be the incidence algebra of a finite connected poset $X$ over a field $K$ and $D(X,K)$ its subalgebra consisting of diagonal elements. We describe the bijective linear maps $φ:I(X,K)\to I(X,K)$ that strongly preserve the commutativity and satisfy $φ(D(X,K))=D(X,K)$. We prove that such a map $φ$ is a composition of a commutativity preserver of shift type and a commutativity preserver associated to a quadruple $(θ,σ,c,κ)$ of simpler maps $θ$, $σ$, $c$ and a sequence $κ$ of elements of $K$.
Incidence Rings: Automorphisms and Derivations
本文研究预序集$X$与交换环$T$上代数$R$构成的关联代数$I(X,R)$的自同构、同构及导子性质。
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.
Additive Biderivations of Incidence Algebras
本文研究了局部有限偏序集上关联代数的加性双导子，证明其均可表示为若干内双导子与极值双导子之和，并在极大链无限时简化为仅由内双导子构成。
Triangle equivalences between Gorenstein tiled orders and incidence algebras of posets
本文研究了Gorenstein铺砌阶环与有限偏序集的incidence算代数之间的三角等价关系，并探讨了相关偏序集在分级Morita等价和覆盖下的行为，最后对满足特定条件的Gorenstein铺砌阶环进行了分类。
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.
Skew derivations of incidence algebras
本文研究局部有限偏序集上关联代数的斜导子，证明其可分解，且斜导子空间模内斜导子子空间与第一上同调群同构。
Functional identities of degree 2 at two-sided zero products on incidence algebras
该论文刻画关联代数上满足双侧零积处二次泛函恒等式的线性映射形式，证明其为标准形式的充要条件，涉及可比图2-连通性刻画与关联代数的双侧零积确定性质。
Let $A$, $B$ be algebras and $a\in A$, $b\in B$ a fixed pair of elements. We say that a map $φ:A\to B$ preserves products equal to $a$ and $b$ if for all $a_1,a_2\in A$ the equality $a_1a_2=a$ implies $φ(a_1)φ(a_2)=b$. In this paper we study bijective linear maps $φ:I(X,F)\to I(X,F)$ preserving products equal to primitive idempotents of $I(X,F)$, where $I(X,F)$ is the incidence algebra of a finite connected poset $X$ over a field $F$. We fully characterize the situation, when such a map $φ$ exists, and whenever it does, $φ$ is either an automorphism of $I(X,F)$ or the negative of an automorphi…
Let $\A$ be a finitary hereditary abelian category. We give a Hall algebra presentation of Kashaev's theorem on the relation between Drinfeld double and Heisenberg double. As applications, we obtain realizations of the Drinfeld double Hall algebra of $\A$ via its derived Hall algebra and Bridgeland Hall algebra of $m$-cyclic complexes.
Projective resolutions of simple modules and Hochschild cohomology for incidence algebras
本文提出计算有限维 incidence 代数Λ简单模投射化解析的算法方法，并应用于计算Ext群、霍奇斯同调群和有限T0拓扑空间的奇异同调群。
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