Joerg Feldvoss
In this paper we establish a connection between the cohomology of a modular Lie algebra and its p-envelopes. We also compute the cohomology of Zassenhaus algebras and their minimal p-envelopes with coefficients in generalized baby Verma modules and in simple modules over fields of characteristic p>2.
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We classify one-dimensional restricted central extensions of the modular Witt Lie algebra in characteristic $p>3$.
Modular intersection cohomology of Drinfeld's compactifications
本文计算了与连通约化群相关的Zastava概形和Drinfeld紧化上交叉上同调茎的维数，并证明其不依赖于系数域特征（在好特征条件下）。
We use the twisted topological trace formula developed in an earlier paper to understand liftings from symplectic to general linear groups. We analyse the lift from $\SP_{2g}$ to $\PGL_{2g+1}$ over the ground field $\Q$ in further detail, and we get a description of the image of this lift for the $L^2$ cohomology of $\SP_{2g}$ (which is related to the intersection cohomology of the Shimura variety attached to $\GSp_{2g}$) in terms of the Eisenstein cohomology of the general linear group, whose building constituents are cuspidal representations of Levi groups. This description may be used to un…
We give new applications of graded Lie algebras to: identities of standard polynomials, deformation theory of quadratic Lie algebras, cyclic cohomology of quadratic Lie algebras, $2k$-Lie algebras, generalized Poisson brackets and so on.
Vogan raised the idea of Dirac cohomology to study representations of semisimple Lie groups and Lie algebras. He conjectured that the infinitesimal character of Harish-Chandra modules are determined by their Dirac cohomology. Huang and Pandžić proved this conjecture and initiated the research on Dirac cohomology for Lie superalgebras based on Kostant's results. The aim of the present paper is to study Dirac cohomology of unitary representations for the general linear superalgebra and its relation to nilpotent Lie superalgebra cohomology.
Dirac cohomology is a new tool to study unitary and admissible representations of semisimple Lie groups. It was introduced by Vogan and further studied by Kostant and ourselves \cite{V2}, \cite{HP1}, \cite{Kdircoh}. The aim of this paper is to study the Dirac cohomology for the Kostant cubic Dirac operator and its relation to Lie algebra cohomology. We show that the Dirac cohomology coincides with the corresponding nilpotent Lie algebra cohomology in many cases, while in general it has better algebraic behavior and it is more accessible for calculation.
The author establishes finite-generation of the cohomology ring of a classical Lie superalgebra relative to an even subsuperalgebra. A spectral sequence is constructed to provide conditions for when this relative cohomology ring is Cohen-Macaulay. With finite generation established, support varieties for modules are defined via the relative cohomology, which generalize those of [BKN-1]
In this work we consider all metric Lie algebras, having a nondegenerate symmetric invariant bilinear form, over \C and \R up to dimension 5 and all metric Lie algebras over \C in dimension 6. We introduce cyclic and reduced cyclic cohomology to identify their metric deformations.
Irreducible unitary representations with non-zero relative Lie algebra cohomology of a Lie group of type f4(4)
本文确定了f4(4)型李群具有非零相对李代数上同调的不可约酉表示及其庞加莱多项式，给出了离散系列表示的充要条件，并识别了离散系列及Borel-de Siebenthal离散系列表示。
In this paper we introduce generalized Spencer cohomology for finite depth Z-graded Lie (super)algebras. We develop a method of finding filtered deformations of such Z-graded Lie (super)algebras based on this cohomology. As an application we determine all simple filtered deformations of certain Z-graded Lie superalgebras classified in [K3], thus completing the last step in the classification of simple infinite-dimensional linearly compact Lie superalgebras.
Support varieties for Lie superalgebras over the complex numbers were introduced in \cite{BKN1} using the relative cohomology. In this paper we discuss finite generation of the relative cohomology rings for Lie superalgebras, we formulate a definition for subalgebras which detect the cohomology, also discuss realizability of support varieties. In the last section as an application we compute the relative cohomology ring of the Lie superalgebra $\overline{S}(n)$ relative to the graded zero component $\overline{S}(n)_0$ and show that this ring is finitely generated. We also compute support varie…
Restricted Cohomology of Heisenberg Lie Algebras
本文在正特征代数闭域上，利用普通上同调计算了海森堡李代数的受限1-和2-上同调空间。
It has been shown in previous work that the modular group acts projectively on the center of a factorizable ribbon Hopf algebra. The center is the zeroth Hochschild cohomology group. In this article, we extend this projective action of the modular group to an arbitrary Hochschild cohomology group of a factorizable ribbon Hopf algebra, in fact up to homotopy even to a projective action on the entire Hochschild cochain complex.
In the computation of the intersection cohomology of Shimura varieties, or of the $L^2$ cohomology of equal rank locally symmetric spaces, combinatorial identities involving averaged discrete series characters of real reductive groups play a large technical role. These identities can become very complicated and are not always well-understood (see for example the appendix of [8]). We propose a geometric approach to these identities in the case of Siegel modular varieties using the combinatorial properties of the Coxeter complex of the symmetric group. Apart from some introductory remarks about …
The aim of this paper is to compare the structure and the cohomology spaces of Lie algebras and induced $3$-Lie algebras.
A morphism Lie algebra is a triple $(\mathfrak{g}, \mathfrak{h}, ϕ)$ consisting of two Lie algebras $\mathfrak{g}, \mathfrak{h}$ and a Lie algebra homomorphism $ϕ: \mathfrak{g} \rightarrow \mathfrak{h}$. We define representations and cohomology of morphism Lie algebras. As applications of our cohomology, we study some aspects of deformations, abelian extensions of morphism Lie algebras and classify skeletal morphism sh Lie algebras. Finally, we consider the cohomology of morphism Lie groups and find a relation with the cohomology of morphism Lie algebras.
We investigate a new cohomology of Lie superalgebras, which may be compared to a de Rham cohomology of Lie supergroups involving both differential and integral forms. It is defined by a BRST complex of Lie superalgebra modules, which is formulated in terms of a Weyl superalgebra and incorporates inequivalent representations of the bosonic Weyl subalgebra. The new cohomology includes the standard Lie superalgebra cohomology as a special case. Examples of new cohomology groups are computed.
We study the locally analytic vectors in the completed cohomology of modular curves and determine the eigenvectors of a rational Borel subalgebra of $\mathfrak{gl}_2(\mathbb{Q}_p)$. As applications, we prove a classicality result for overconvergent eigenforms of weight one and give a new proof of the Fontaine-Mazur conjecture in the irregular case under some mild hypotheses. For an overconvergent eigenform of weight $k$, we show its corresponding Galois representation has Hodge-Tate-Sen weights $0,k-1$ and prove a converse result.
This note presents a general theorem about the cohomology of finite dimensional Lie algebras of arbitrary characteristic. As an application we compute the cohomology of the Borel subalgebra of sl(N).
In this paper, we define representations and cohomology of weighted Rota-Baxter Lie algebras. As applications of cohomology, we study abelian extensions and formal $1$-parameter deformations weighted Rota-Baxter Lie algebras. Finally, we consider weighted Rota-Baxter paired operators that induces a weighted Rota-Baxter Lie algebra and a representation of it. We define suitable cohomology for such paired operators that govern deformation.
We compute the cohomology with trivial coefficients of two graded infinite-dimensional Lie algebras of maximal class, give explicit formulas for their representative cocycles. Also we discuss the relations with combinatorics and representation theory.
The aim of this paper is to study the cohomology of Hom-Leibniz superalgebras. We construct the $q$-deformed Heisenberg-Virasoro superalgebra of Hom-type and provide as application the computations of the derivations and second cohomology group. Moreover, we extend to graded case the Takhtajan's construction of a cohomology of $n$-ary Hom-Nambu-Lie algebras starting from cohomology of Hom-Leibniz algebras.
We investigate the second cohomology space of the Lie superalgebra $\mathcal{K}(2)$ with coefficients in the superspace of weighted densities on the (1,2)-dimentional real superspace. We explicitly give cocycles spanning this cohomology space.
We describe an essential improvement of our recent algorithm for computing cohomology of Lie (super)algebra based on partition of the whole cochain complex into minimal subcomplexes. We replace the arithmetic of rational numbers or integers by a much cheaper arithmetic of a modular field and use the inequality between the dimensions of cohomology H over any modular field F_p = Z/pZ and over Q: dim H(F_p) >= dim H(Q). With this inequality we can, by computing over arbitrary F_p, quickly find the (usually, rare) subcomplexes for which dim H(F_p) > 0 and then carry out the full computation over Q…
In this paper the authors investigate the structure the restricted Lie algebra cohomology of p-nilpotent Lie algebras with trivial p-power operation. Our study is facilitated by a spectral sequence whose $E_{2}$-term is the tensor product of the symmetric algebra on the dual of the Lie algebra with the ordinary Lie algebra cohomology and converges to the restricted cohomology ring. In many cases this spectral sequence collapses, and thus, the restricted Lie algebra cohomology is Cohen-Macaulay. A stronger result involves the collapsing of the spectral sequence and the cohomology ring identifyi…
In this paper, we establish the cohomology of relative Rota-Baxter operators on Lie-Yamaguti algebras via the Yamaguti cohomology. Then we use this type of cohomology to characterize deformations of relative Rota-Baxter operators on Lie-Yamaguti algebras. We show that if two linear or formal deformations of a relative Rota-Baxter operator are equivalent, then their infinitesimals are in the same cohomology class in the first cohomology group. Moreover, an order $n$ deformation of a relative Rota-Baxter operator can be extended to an order $n+1$ deformation if and only if the obstruction class …
Let $(S,L)$ be a Lie-Rinehart pair such that $L$ is $S$-projective and let $U$ be its universal enveloping algebra. The purpose of this paper is to present a spectral sequence which converges to the Hochschild cohomology of $U$ and whose second page involves the Lie-Rinehart cohomology of the pair and the Hochschild cohomology of $S$ with values on $U$.
The aim of this paper is to provide cohomologies of $n$-ary Hom-Nambu-Lie algebras governing central extensions and one parameter formal deformations. We generalize to $n$-ary algebras the notions of derivations and representation introduced by Sheng for Hom-Lie algebras. Also we show that a cohomology of $n$-ary Hom-Nambu-Lie algebras could be derived from the cohomology of Hom-Leibniz algebras.
Hochschild Cohomology, Modular Tensor Categories, and Mapping Class Groups
本文针对有限模张量范畴，将带边界紧曲面关联至上链复形，证明曲面映射类群在其上同调群上射影作用，零度时与手征共形块空间已知作用一致，并在环面与可因子化带状Hopf代数情形下恢复先前结果。
Let g be a finite-dimensional complex semi simple Lie algebra. We present a new calculation of the continuous cohomology of the Lie algebra z g[[z]]. In particular, we shall give an explicit formula for the Laplacian on the Lie algebra cochains, from which we can deduce that the cohomology in each dimension is a finite-dimensional representation of g which contains any irreducible representation of g at most once.
A LieYRep pair consists of a Lie-Yamaguti algebra and its representation. In this paper, we establish the cohomology theory of LieYRep pairs and characterize their linear deformations by the second cohomology group. Then we introduce the notion of relative Rota-Baxter-Nijenhuis structures on LieYRep pairs, investigate their properties, and prove that a relative Rota-Baxter-Nijenhuis structure gives rise to a pair of compatible relative Rota-Baxter operators under a certain condition. Finally, we show the equivalence between $r$-matrix-Nijenhuis structures and Rota-Baxter-Nijenhuis structures o…
In this paper, we use geometrical methods adapted from the Borel-Weil-Bott theory to compute the character of every finite dimensional simple module over a basic classical Lie superalgebra.
We study cohomology for classical Lie superalgebras $\mathfrak{g}$ (e.g. gl(m|n)) over the complex numbers. Using results from invariant theory, we show that there exist subsuperalgebras which detect the cohomology of $\mathfrak{g}.$ Furthermore, drawing inspiration from finite groups and restricted Lie algebras, we introduce the notion of support varieties in this context. One can prove these varieties have many of the desirable properties of such a theory. They also appear to provide new interpretations for certain combinatorics associated with Lie superalgebras.
We compute explicitly the adjoint cohomology of two N-graded Lie algebras of maximal class (infinite dimensional filiform Lie algebras) m_0 and m_2. It is known that up to an isomorphism there are only three N-graded Lie algebras of the maximal class. The third algebra from this list is the "positive" part L_1 of the Witt (or Virasoro) algebra and its adjoint cohomology was computed earlier by Feigin and Fukhs. We show that the total space H*(m_j,m_j) is "almost" isomorphic to the completed tensor product of the algebra m_j by scalar cohomology space H^*(m_j), j=0,2.
The main purpose of this paper is to define representations and a cohomology of color Hom-Lie algebras and to study some key constructions and properties. We describe Hartwig-Larsson-Silvestrov Theorem in the case of $Γ$-graded algebras, study one-parameter formal deformations, discuss $α^{k}$-generalized derivation and provide examples.
We study the Belkale-Kumar family of cup products on the cohomology of a generalized flag variety. We give an alternative construction of the family using relative Lie algebra cohomology, and in particular, identify the Belkale-Kumar cup product with a relative Lie algebra cohomology ring for every value of the parameter. As a consequence, we extend a fundamental disjointness result of Kostant to a family of Lie algebras. In an appendix, written jointly with Edward Richmond, we extend a Levi movability result of Belkale and Kumar to arbitrary parameters.
We introduce a generalization of Brauer character to allow arbitrary finite length modules over discrete valuation rings. We show that the generalized super Brauer character of Tate cohomology is a linear combination of trace functions. Using this result, we find a counterexample to a conjecture of Borcherds about vanishing of Tate cohomology for Fricke elements of the Monster.
This paper is devoted to studying deformation, cohomology theory of Rota-Baxter pre-Lie algebras of arbitrary weights. First we give the notion of a new representation of a Rota-Baxter pre-Lie algebra of arbitrary weight and define the cohomology theory of a Rota-Baxter pre-Lie algebra of arbitrary weight. Then we study formal deformations by lower degree cohomology groups. Finally, we classify abelian extensions of Rota-Baxter pre-Lie algebras of arbitrary weight using the second cohomology group, and classify skeletal Rota-Baxter pre-Lie 2-algebra of arbitrary weight using the third cohomolo…
In this paper, we introduce the notion of BiHom-Lie conformal superalgebras. We develop its representation theory and define the cohomology group with coefficients in a module. Finally, we introduce conformal derivations of BiHom-Lie conformal superalgebras and study some of their properties.
We prove that the cohomology algebra of a conilpotent Lie coalgebra is generated in degree 1 as an A-infinity algebra. By dualizing, the same is true about cohomology algebras of finite dimensional nilpotent Lie algebras. In the process, we provide a proof of the conilpotent version of the dual Poincare-Birkhoff-Witt theorem.
A cohomology theory of weighted Rota-Baxter $3$-Lie algebras is introduced. Formal deformations, abelian extensions, skeletal weighted Rota-Baxter $3$-Lie 2-algebras and crossed modules of weighted Rota-Baxter 3-Lie algebras are interpreted by using lower degree cohomology groups.
An FI-module $V$ over a commutative ring $\bf{k}$ encodes a sequence $(V_n)_{n \geq 0}$ of representations of the symmetric groups $(\mathfrak{S}_n)_{n \geq 0}$ over $\bf{k}$. In this paper, we show that for a "finitely generated" FI-module $V$ over a field of characteristic $p$, the cohomology groups $H^t(\mathfrak{S}_n, V_n)$ are eventually periodic in $n$. We describe a recursive way to calculate the period and the periodicity range and show that the period is always a power of $p$. As an application, we show that if $\mathcal{M}$ is a compact, connected, oriented manifold of dimension $\ge…
In the present paper, we define the new class of representation on $n$-Lie algebra that is called as generalized representation. We study the cohomology theory corresponding to generalized representations of $n$-Lie algebras and show its relation with the cohomology corresponding to the usual representations. Furthermore, we provide the computation for the low dimensional cocycles.
Representations and cohomology of a matched pair of Lie algebras, and $L_\infty$-algebras
本文引入李代数匹配对的表示与上同调，证明其与李双代数上同调的态射关系，可用于无穷小形变与阿贝尔扩张研究，还定义了$L_\infty$-代数匹配对并构造双交叉积，揭示骨架$L_\infty$-代数匹配对与该上同调的关联。
The purpose of this paper is to define the representation and the cohomology of Hom-Lie superalgebras. Moreover we study Central extensions and provide as application the computations of the derivations and second cohomology group of q-deformed Witt superalgebra.
In this note we show that the theory of non abelian extensions of a Lie algebra $\mathfrak{g}$ by a Lie algebra $\mathfrak{h}$ can be understood in terms of a differential graded Lie algebra $L$. More precisely we show that the non-abelian cohomology $H^2_{nab}(\mathfrak{g},\mathfrak{h})$ is $\mathcal{MC}(L)$, the $π_0$ of the Deligne groupoid of $L$.
Let $k$ be an algebraically closed field of characteristic 0, let $R$ be a commutative $k$-algebra, and let $M$ be a torsion free $R$-module of rank one with a connection $\nabla$. We consider the Lie-Rinehart cohomology with values in $End_{R}(M)$ with its induced connection, and give an interpretation of this cohomology in terms of the integrable connections on $M$. When $R$ is an isolated singularity of dimension $d\geq2$, we relate the Lie-Rinehart cohomology to the topological cohomology of the link of the singularity, and when $R$ is a quasi-homogenous hypersurface of dimension two, we g…
The second de Rham cohomology groups of nilpotent orbits in non-compact real forms of classical complex simple Lie algebras are explicitly computed. Furthermore, the first de Rham cohomology groups of nilpotent orbits in non-compact classical simple Lie algebras are computed; they are proven to be zero for nilpotent orbits in all the complex simple Lie algebras. A key component in these computations is a description of the second and first cohomology groups of homogeneous spaces of general connected Lie groups which is obtained here. This description, which generalizes a previous theorem of th…
Hochschild Cohomology, Modular Tensor Categories, and Mapping Class Groups II
该研究在正特征域上有限群的Drinfel'd双情形下，计算映射类群在导出块空间上的作用，证明其与普通块空间上的映射类群表示通常不同，拓展了映射类群作用的相关研究。
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