In this paper, we calculate cohomology of a classical Lie algebra of type $A_2$ over an algebraically field $k$ of characteristic $p=3$ with coefficients in simple modules. To describe their structure, we will consider them as modules over an algebraic group $SL_3(k).$ In the case of characteristic $p=3,$ there are only two peculiar simple modules: a simple module isomorphic to the quotient module of the adjoint module by the center, and a one-dimensional trivial module. The results on the cohomology of simple nontrivial module are used for calculate the coomology of the adjoint module. We also calculate cohomology of the simple quotient algebra Lie of $A_2$ by the center.
We calculate the cohomology of $\mathfrak{sl}_3(k)$ over an algebraically closed field $k$ of characteristic $p>3$ with coefficients in simple modules and Weyl modules. We also give descriptions of the corresponding cohomology of $\mathfrak{gl}_3(k)$.
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.
Let $ 0\rightarrow \mathfrak{a} \rightarrow \mathfrak{e} \rightarrow \mathfrak{g} \rightarrow 0$ be an abelian extension of the Lie superalgebra $\mathfrak{g}$. In this article we consider the problems of extending endomorphisms of $\mathfrak{a}$ and lifting endomorphisms of $\mathfrak{g}$ to certain endomorphisms of $\mathfrak{e}$. We connect these problems to the cohomology of $\mathfrak{g}$ with coefficients in $\mathfrak{a}$ through construction of two exact sequences, which is our main result, involving various endomorphism groups and the second cohomology. The first exact sequence is obt…
Let ${\mathfrak g}$ be a classical simple Lie superalgebra. In this paper, the author studies the cohomology groups for the subalgebra $\mathfrak{n}^{+}$ relative to the BBW parabolic subalgebras constructed by D. Grantcharov, N. Grantcharov, Nakano and Wu. These classical Lie superalgebras have a triangular decomposition ${\mathfrak g}={\mathfrak n}^{-}\oplus {\mathfrak f} \oplus {\mathfrak n}^{+}$ where $\mathfrak f$ is a detecting subalgebra as introduced by Boe, Kujawa and Nakano. It is shown that there exists a Hochschild-Serre spectral sequence that collapses for all infinite families of…
The paper is devoted to the study of pro-solvable Lie algebras whose maximal pro-nilpotent ideal is either $\mathfrak{m}_0$ or $\mathfrak{m}_2$. Namely, we describe such Lie algebras and establish their completeness. Triviality of the second cohomology group for one of the obtained algebra is established.
Categorification of Verma modules and indecomposable projective modules in the category $\mathcal I_{\mathfrak{g}}(\mathfrak{sl}_2)$ for $\mathfrak{sl}_2$
利用T. T的一个张量积分解定理，对$\mathfrak{sl}_2$的Verma和不可分解投射模进行范畴$\mathcal I_{\mathfrak{g}}(\mathfrak{sl}_2)$的分类. J.Enright和J.Chuang和R. Rouquier，A. Licata和A. Savage（Hecke代数、有限一般线性群和Heisenberg分类）和M. Khovanov（Heisenberg Algebra and a Graphical Calculus）.
Mathematics Institute, Georg-August-University of Göttingen（哥廷根大学数学研究所）；Department of Mathematics, Aristotle University of Thessaloniki（萨洛尼卡亚里士多德大学数学系）；Institut für Mathematik, Julius-Maximilians-Universität Würzburg（维尔茨堡大学数学研究所）
Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $ξ$ of $\mathbb{E}$-triangles. In this paper, we study the balance of complete cohomology in $(\mathcal{C},\mathbb{E},\mathfrak{s})$, which is motivated by a result of Nucinkis that complete cohomology of modules is not balanced in the way the absolute cohomology Ext is balanced. As an application, we give some criteria for identifying a triangulated catgory to be Gorenstein and an artin algebra to be $F$-Gorenstein.
In this paper we explore the possibility of endowing simple infinite-dimensional ${\mathfrak{sl}_2(\mathbb{C})}$-modules by the structure of the graded module. The gradings on finite-dimensional simple module over simple Lie algebras has been studied in [arXiv:1308.6089] and [arXiv:1601.03008].
For a commutative algebra $A$ over $\mathbb{C}$,denote $\mathfrak{g}=\text{Der}(A)$. A module over the smash product $A\# U(\mathfrak{g})$ is called a jet $\mathfrak{g}$-module, where $U(\mathfrak{g})$ is the universal enveloping algebra of $\mathfrak{g}$.In the present paper, we study jet modules in the case of $A=\mathbb{C}[t_1^{\pm 1},t_2]$.We show that $A\#U(\mathfrak{g})\cong\mathcal{D}\otimes U(L)$, where $\mathcal{D}$ is the Weyl algebra $\mathbb{C}[t_1^{\pm 1},t_2, \frac{\partial}{\partial t_1},\frac{\partial}{\partial t_2}]$, and $L$ is a Lie subalgebra of $A\# U(\mathfrak{g})$ called…
The rank $n$ symplectic oscillator Lie algebra $\mathfrak{g}_n$ is the semidirect product of the symplectic Lie algebra $\mathfrak{sp}_{2n}$ and the Heisenberg Lie algebra $H_n$. In this paper, we study weight modules with finite dimensional weight spaces over $\mathfrak{g}_n$. When $\dot z\neq 0$, it is shown that there is an equivalence between the full subcategory $\mathcal{O}_{\mathfrak{g}_n}[\dot z]$ of the BGG category $\mathcal{O}_{\mathfrak{g}_n}$ for $\mathfrak{g}_n$ and the BGG category $\mathcal{O}_{\mathfrak{sp}_{2n}}$ for $\mathfrak{sp}_{2n}$. Then using the technique of localizat…
