Alex Cameron, Vincent E. Coll,, Nicholas Mayers, Nicholas Russoniello
The breadth of a Lie algebra $L$ is defined to be the maximal dimension of the image of $ad_x=[x,-]:L\to L$, for $x\in L$. Here, we initiate an investigation into the breadth of three families of Lie algebras defined by posets and provide combinatorial breadth formulas for members of each family.
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The notion of breadth of a nilpotent Lie algebra was introduced by B. Khuhirun, K.C. Misra and E. Stitzinger and used to approach problems of classification up to isomorphism. In the present paper, we study this invariant in terms of characteristic sequence, another invariant introduced by M. Goze and J.M. Ancochea-Bermudez. This permits to complete the determination of Lie algebras of breadth 2 and to begin the work for Lie algebras with breadth greater than 2.
A Lie algebra $L$ is said to be of breadth $k$ if the maximal dimension of the images of left multiplication by elements of the algebra is $k$. In this paper we give characterization of finite dimensional nilpotent Lie algebras of breadth less than or equal to two. Furthermore, using these characterizations we determined the isomorphism classes of these algebras.
On solvable Lie algebras of small breadth
本文研究了有限维可解李代数中广度小于等于2的结构，并对复数域上纯且非 nilpotent 的李代数进行了完整分类。
We provide a combinatorial recipe for constructing all posets of height at most two for which the corresponding type-A Lie poset algebra is contact. In the case that such posets are connected, a discrete Morse theory argument establishes that the posets' simplicial realizations are contractible. It follows from a cohomological result of Coll and Gerstenhaber on Lie semi-direct products that the corresponding contact Lie algebras are absolutely rigid.
Nilpotent Lie Algebras of breadth type $(0,3)$
本文分类了奇数特征有限域上宽度类型为 (0,3) 的有限维幂零李代数，并给出偶数特征域及实复数域上的部分分类，同时讨论了 2-步幂零 Camina 李代数。
We show that the isomorphism class of a two-step solvable Lie poset subalgebra of a semisimple Lie algebra is determined by its dimension. We further establish that all such algebras are absolutely rigid.
We establish combinatorial formulas for the index of a class of matrix Lie algebras whose matrix forms are encoded by strict partial orderings.
When $\mathfrak h$ is a toral subalgebra of a Lie algebra $\mathfrak g$ over a field $\mathbf k$, and $M$ a $\mathfrak g$-module on which $\mathfrak h$ also acts torally, the Hochschild-Serre filtration of the Chevalley-Eilenberg cochain complex admits a stronger form than for an arbitrary subalgebra. For a semidirect product $\mathfrak g = \mathfrak h \ltimes \mathfrak k$ with $\mathfrak h$ toral one has $H^*(\mathfrak g, M) \cong \bigwedge\mathfrak h^{\vee} \bigotimes H^*(\mathfrak k,M)^{\mathfrak h} = H^*(\mathfrak h, \mathbf k)\bigotimes H^*(\mathfrak k,M)^{\mathfrak h}$, and for a Lie pos…
We introduce a family of posets which generate Lie poset subalgebras of $A_{n-1}=\mathfrak{sl}(n)$ whose index can be realized topologically. In particular, if $\mathcal{P}$ is such a \textit{toral poset}, then it has a simplicial realization which is homotopic to a wedge sum of $d$ one-spheres, where $d$ is the index of the corresponding type-A Lie poset algebra $\mathfrak{g}_A(\mathcal{P})$. Moreover, when $\mathfrak{g}_A(\mathcal{P})$ is Frobenius, its spectrum is \textit{binary}; that is, consists of an equal number of 0's and 1's. We also find that all Frobenius, type-A Lie poset algebras…
Contact Lie poset algebras of types B, C, and D
本文推广 B、C、D 型 Lie 偏序集代数的组合指标公式，并据此用关联图刻画高度为一偏序集对应的接触 Lie 偏序集代数。
On toral posets and contact Lie algebras
本文针对切触李代数的构造问题，提出组合程序生成关联任意基数链偏序集的切触A型李偏序代数，并猜想该构造可实现切触李偏序代数的完整刻画，拓展了相关领域研究。
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.
This paper concerns the algebraic structure of finite-dimensional complex Leibniz algebras. In particular, we introduce left central and symmetric Leibniz algebras, and study the poset of Lie subalgebras using an associative bilinear pairing taking values in the Leibniz kernel.
We investigate the (co)homological properties of two classes of Lie algebras that are constructed from any finite poset: the solvable class $\frak{gl}^\preceq$ and the nilpotent class $\frak{gl}^\prec$. We confirm the conjecture of Jollenbeck that says: every prime power $p^r\!\leq\!n\!-\!2$ appears as torsion in $H_\ast(\frak{nil}_n;\mathbb{Z})$, and every prime power $p^r\!\leq\!n\!-\!1$ appears as torsion in $H_\ast(\frak{sol}_n;\mathbb{Z})$. If $\preceq$ is a bounded poset, then the (co)homology of $\frak{gl}^\preceq$ is \emph{torsion-convex}, i.e. if it contains $p$-torsion, then it also …
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.
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 $I(X,K)$.
Posets of finite GK-dimensional graded pre-Nichols algebras of diagonal type
本文分类了具有有限 Gelfand-Kirillov 维数的对角型分次 pre-Nichols 代数，通过正根子集的偏序集同构进行刻画，推广了小量子群的中心扩张，并分类了相关幂么代数群函数代数的分次商。
Let $X$ be a finite connected poset, $F$ a field and $I(X,F)$ the incidence algebra of $X$ over $F$. We describe the bijective linear idempotent preservers $φ:I(X,F)\to I(X,F)$. Namely, we prove that, whenever $\mathrm{char}(F)\ne 2$, $φ$ is either an automorphism or an anti-automorphism of $I(X,F)$. If $\mathrm{char}(F)=2$ and $|F|>2$, then $φ$ is a (in general, non-proper) Lie automorphism of $I(X,F)$. Finally, if $F=\mathbb{Z}_2$, then $φ$ is the composition of a bijective shift map and a Lie automorphism of $I(X,F)$. Under certain restrictions on the characteristic of $F$ we also obtain de…
Transposed Poisson structures on Lie incidence algebras
本文研究了特征零域上有限连通偏序集关联代数李代数结构，证明了$\ rac{1}{2}$-导数的分解定理，并据此得出任意转置Poisson结构可分解为Poisson型、变异型和特定映射确定的结构之和。
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