Sergei O. Ivanov, Roman Mikhailov, Anatoly Zaikovski
In this paper, an explicit construction of a countable parafree Lie algebra over $\mathbb Z/2$ with nonzero second homology is given. It is also shown that the cohomological dimension of the pronilpotent completion of a free noncyclic finitely generated Lie algebra over $\mathbb Z$ is greater than two. Moreover, it is proven that there exists a countable parafree group with nontrivial $H_2$.
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We develop a version of the Bass-Serre theory for Lie algebras (over a field $k$) via a homological approach. We define the notion of fundamental Lie algebra of a graph of Lie algebras and show that this construction yields Mayer-Vietoris sequences. We extend some well known results in group theory to $\mathbb{N}$-graded Lie algebras: for example, we show that one relator $\mathbb{N}$-graded Lie algebras are iterated HNN extensions with free bases which can be used for cohomology computations and apply the Mayer-Vietoris sequence to give some results about coherence of Lie algebras.
Generators of the homological Goldman Lie algebra
确定了由曲面的第一个同调群的元素组成的曲面的同调Goldman李代数的生成元的最小个数。
We consider homological finiteness properties $FP_n$ of certain $\mathbb{N}$-graded Lie algebras. After proving some general results, see Theorem A, Corollary B and Corollary C, we concentrate on a family that can be considered as the Lie algebra version of the generalized Bestvina-Brady groups associated to a graph $Γ$. We prove that the homological finiteness properties of these Lie algebras can be determined in terms of the graph in the same way as in the group case. In the last version we have corrected some missprints, in particular the statement of Theorem D (from $n-1$-acyclicity to $n-…
We generalise notions of Gorenstein homological algebra for rings to the context of arbitrary abelian categories. The results are strongest for module categories of rngs with enough idempotents. We also reformulate the notion of Frobenius extensions of noetherian rings into a setting which allows for direct generalisation to arbitrary abelian categories. The abstract theory is then applied to the BGG category O for Lie superalgebras, which can now be seen as a "Frobenius extension" of the corresponding category for the underlying Lie algebra and is therefore "Gorenstein". In particular we obta…
We show Lie algebra versions of some results on homological finiteness properties of subdirect products of groups, including a version of the 1-2-3 Theorem.
The Arens-Michael envelope of a solvable Lie algebra is a homological epimorphism
本文证明了有限维复李代数的universal enveloping algebra的Arens-Michael包当且仅当该李代数为可解时是同调上模，填补了Pirkovskii之前证明的必要条件。
Homological properties of color Lie superalgebras
设$\mathcal{L}=\mathcal{L}_{+}\oplus \mathcal{L}_{-}$是特征为0的域上的有限维色李超代数，其泛包络代数为$U(\mathcal{L})$ .我们证明了$\limfunc{gldim}(U(\mathcal{L}_{+}))= \limfunc{lFPD}(U(\mathcal{L}))= \limfunc{rFPD}(U(\mathcal{L}))= \limfunc{injdim}_{U(\mathcal{L})}(U(\mathcal{L}))= \dim (\mathcal{L}_{+})$ .我们还证明了$U(\mathcal{L})$是Auslander-Gorenstein和Cohen-Macaulay，从而它有一个QF经典商环.
Non-finitely Presented Lie Algebras of Type $FP_2$
该研究基于前人工作，证明合适Bestvina–Brady群复群代数的Steinberg李代数为$FP_2$型且非有限表示，同时解答了李代数的同调0-1-2问题。
Homological properties of quantized coordinate rings of semisimple groups
证明了对每个连通半单李群$G$，广义量子坐标环$\mathcal{O}_q(G)$是Auslander正则的，Cohen-Macaulay的，悬链线的.这回答了布朗、勒纳根和第一作者提出的问题。证明了在正规元存在的条件下，Noether霍普夫代数是Auslander-Gorenstein和Cohen-Macaulay.这为布朗和第一作者的问题提供了一组新的积极案例。
Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures
我们在非交换环上的余代数上的代数上的模的无界导出范畴（适当定义）中发展了同调代数的基本结构（我们称之为核上的半代数）。定义了半张量积和半同态函子的双边导出函子SemiTor和SemiExt，并构造了半模的奇异导出范畴与半模的奇异范型之间的等价关系.某些（余）平坦性和/或（余）投射性条件必须施加在核和半代数上，以使模范畴阿贝尔（和余张量积结合）。此外，由于一些技术上的原因，我们大多不得不假设基本环具有有限的同调维数（关于核和半代数没有这样的假设）。在最后一节中，我们构造了半（反）模复形范畴上的模型范畴结构，并发展了滤子半代数和拟微分核环的相对非齐次Koszul对偶理论。我们的例子来自半无限上同调理论。与Tate李代数和分次结合代数的半无限（上）同调在附录中建立了比较，并定义了局部紧拓扑群相对于开profinite子群的半无限同调.应用于无穷维李代数在互补中心电荷能级（Virasoro的$c$和$26-c$）上表示的复形之间的对应关系。
We introduce the notion of virtual endomorphisms of Lie algebras and use it as an approach for constructing self-similarity of Lie algebras. This is done in particular for a class of metabelian Lie algebras having homological type F Pn, which are variants of lamp-lighter groups. We establish several criteria when the existence of virtual endomorphism implies a self-similar Lie structure. Furthermore, we prove that the classical Lie algebra sln(k), where char(k) does not divide n affords non-trivial faithful self-similarity.
Cohomological dimension of a Lie algebra homomorphism
该研究引入李代数同态的上同调维数和同调维数概念，建立其基本性质，获上同调维数链同伦刻画，并计算自由和阿贝尔李代数间同态的这些不变量。
We study the analogue of Sidki's weak commutativity construction, defined originally for groups, in the category of Lie algebras. This is the quotient $χ(\mathfrak{g})$ of the Lie algebra freely generated by two isomorphic copies $\mathfrak{g}$ and $\mathfrak{g}^ψ$ of a fixed Lie algebra by the ideal generated by the brackets $[x,x^ψ]$, for all $x$. We exhibit an abelian ideal of $χ(\mathfrak{g})$ whose associated quotient is a subdirect sum in $\mathfrak{g} \oplus \mathfrak{g} \oplus \mathfrak{g}$ and we give conditions for this ideal to be finite dimensional. We show that $χ(\mathfrak{g})$ h…
On universal central extensions of Hom-Lie algebras
本文发展了Hom-李代数的泛中心扩张理论。李代数的泛中心扩张的经典结果不能完全推广到Hom-李代数的情形，因为两个中心扩张的合成不是中心的。这一事实导致引入通用$α$ -中心扩展的概念。经典的关于完备Hom-李代数存在泛中心扩张的结果仍然成立，而关于泛中心扩张的中间项的中心扩张分裂的结果只对$α$ -中心扩张成立.给出泛（$α$）-中心扩张的一个同调刻画.
How Euler would compute the Euler-Poincaré characteristic of a Lie superalgebra
有限维李代数的欧拉-庞加莱特征消失。如果我们想把这个结果推广到李超代数，我们应该处理无穷和。我们观察到，一个合适的求和方法，可以追溯到欧拉，在一定程度上允许这样做。它背后的数学很简单，我们只是把初等同调代数、一年级微积分和行人组合学的碎片粘在一起，并以一种（希望）连贯的方式呈现出来。
Surface topology and involutive bimodules
本文讨论了对合李双代数及其对合双模同调方面的一些基本事实，并提出了与这些事实有关的曲面拓扑问题。
(Co)Homology of Poset Lie Algebras
研究了由任意有限偏序集构造的两类李代数：可解类$\frak{gl}^\preceq$和幂零类$\frak{gl}^\prec$的（余）同调性质.我们证实了Jollenbeck的猜想：每一个素幂$p^r\!\leq\!n\!-\!2$在$H_\ast(\frak{nil}_n;\mathbb{Z})$中表现为挠，每一个素幂$p^r\!\leq\!n\!-\!1$在$H_\ast(\frak{sol}_n;\mathbb{Z})$中表现为挠.如果$\preceq$是有界偏序集，则$\frak{gl}^\preceq$的（上）同调是\emph{torsion-convex}，即如果它包含$p$ -挠，则对于每个素数$p'\!<\!p$，它也包含$p'$ -挠。本文给出了任意域上某些族的（上）同调的新的显式公式。其中有可解的非幂零类似物的海森堡李代数从凯恩斯和Jambor的文章，2步李代数从阿姆斯特朗和凯恩斯和Jessup的文章，严格块三角李代数等所产生的生成函数和组合数学如何获得它们本身是有趣的。\par所有这一切都是通过使用AMT（代数莫尔斯理论）。这篇文章作为一个来源的例子，如何构建有用的非循环匹配，其中每一个反过来又诱导引人注目的组合问题和解决方案。它也使图论可以用于同调代数。
We study the degeneration relations on the varieties of associative and Lie algebra structures on a fixed finite dimensional vector space and give a description of them in terms of Gerstenhaber formal deformations. We use this result to show how the orbit closure of the $3$-dimensional Lie algebras can be determined using homological algebra. For the case of finite dimensional associative algebras, we prove that the $N$-Koszul property is preserved under the degeneration relation for all $N\geq2$.
Lie's Third Theorem for Lie $\infty$-Algebras
本文通过局部最小模型理论，首次证明了有限型Lie ∞-代数的Lie第三定理，并给出了弦Lie 2群的新模型。
A Lie algebra that can be written as a sum of two nilpotent subalgebras, is solvable
这是一张旧纸，放在这里是为了考古。证明了特征p>5的域上的有限维李代数是可解的，它可以表示为两个幂零子代数的向量空间（不一定是直的）和。同样的结果（但也涵盖了案件的低特征）是独立成立的V. Panyukov（俄罗斯数学。45（1990），N4，181-182），并且证明中使用的同调方法后来在arXiv：math/0204004中发展。英文翻译中的许多不准确之处都得到了纠正，否则文本与出版的版本相同。
We provide a short alternative homological argument showing that any invariant symmetric bilinear form on simple modular generalized Jacobson-Witt algebras vanishes, and outline another, deformation-theoretic one, allowing to describe such forms on simple modular Lie algebras of contact type.
Tilting modules for the current algebra of a simple Lie algebra
目前和仿射李代数的水平零表示类股票的许多其他知名的类别，出现在李理论和代数群的特性p和在本文中，我们探讨进一步的相似之处。标准模和余标准模的作用分别由有限维局部Weyl模和无限维全局Weyl模的对偶起。定义了当前代数的分次模的正则滤子。在$\mathfrak g$是$\mathfrak{sl}_{n+1}$型的情况下，我们表明，著名的必要和充分的同调条件的一个典型的过滤是一个很好的（或$\nabla$ -过滤）在我们的情况下也成立。最后，我们构造了范畴中的不可分解倾斜模，并证明了任何倾斜模同构于不可分解倾斜模的直和。
For the exceptional finite-dimensional modular Lie superalgebras $\mathfrak{g}(A)$ with indecomposable Cartan matrix $A$, and their simple subquotients, we computed non-isomorphic Lie superalgebras constituting the homologies of the odd elements with zero square. These homologies are~key ingredients in the Duflo--Serganova approach to the representation theory. There were two definitions of defect of Lie superalgebras in the literature with different ranges of application. We suggest a third definition and an easy-to-use way to find its value. In positive characteristic, we found out one more …
Let $X$ be a simply connected closed oriented manifold of rationally elliptic homotopy type. We prove that the string topology bracket on the $S^1$-equivariant homology $\overline{H}_{\ast}^{S^1}(\mathcal{L}X,\mathbb{Q}) $ of the free loop space of $X$ preserves the Hodge decomposition of $\overline{H}_{\ast}^{S^1}(\mathcal{L}X,\mathbb{Q}) $ , making it a bigraded Lie algebra. We deduce this result from a general theorem on derived Poisson structures on the universal enveloping algebras of homologically nilpotent finite-dimensional DG Lie algebras. Our theorem settles a conjecture proposed in …
The Auslander-Gorenstein property for Z-algebras
我们提供了一个框架的同调理论的一部分Z-代数及其推广，针对类似物的Auslander-Gorenstein条件和相关的双Ext谱序列，是有用的包络代数的李代数和相关的环。作为应用，我们证明了Z-代数的一个不可约表示的特征簇的等维性，以及量子辛分解上的相关表示的等维性。在特殊情况下的Cherednik代数的类型A，这回答了一个问题提出的作者。
Integral coefficient rings and homological dimensions of algebras
该研究定义了模的积分轮廓与有限维复代数的积分系数环，证明积分轮廓是模的完全不变量，还得出积分系数环的Morita不变性及代数整体维数、有限维数有限的等价条件等结果。
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