Dong Liu, Naihong Hu
There are several researches on Lie algebras and Lie superalgebras graded by finite root systems. In this paper, we study Leibniz algebras graded by finite root systems and obtain some results in simply-laced cases.
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The structure of Lie algebras, Lie superalgebras and Leibniz algebras graded by finite root systems has been studied by several researchers since 1992. In this paper, we study the structure of Leibniz superalgebras graded by finite root systems, which gives an approach to study various classes of Leibniz superalgebras.
Dialgebras are generalizations of associative algebras which give rise to Leibniz algebras instead of Lie algebras. In this paper we study super dialgebras and Leibniz superalgebras, which are $\z_2$-graded dialgebras and Leibniz algebras. We also study universal central extesions of Leibniz superalgebras and obtain some results as in the case of Lie superalgebras. We determine universal central extensions of basic classical Lie superalgebras in the category of Leibniz superalgebras. They play a key role in studying Leibniz superalgebras graded by finite root systems.
The main goals for this paper is i) to study of an algebraic structure of $\mathbb{N}$-graded vertex algebras $V_B$ associated to vertex $A$-algebroids $B$ when $B$ are cyclic non-Lie left Leibniz algebras, and ii) to explore relations between the vertex algebras $V_B$ and the rank one Heisenberg vertex operator algebra. To achieve these goals, we first classify vertex $A$-algebroids $B$ associated to given cyclic non-Lie left Leibniz algebras $B$. Next, we use the constructed vertex $A$-algebroids $B$ to create a family of indecomposable non-simple vertex algebras $V_B$. Finally, we use the a…
Lie bialgebras constructed from Zinbiel bialgebras and Leibniz bialgebras
本文基于Leibniz双代数与Zinbiel双代数构造Lie双代数，证明其张量积存在相应结构，并探讨经典Yang-Baxter方程解与quasi-triangular性质的关系，最终提出构造quasi-Frobenius Lie代数的方法。
In this paper, we study an impact of Leibniz algebras on the algebraic structure of $\mathbb{N}$-graded vertex algebras. We provide easy ways to characterize indecomposable non-simple $\mathbb{N}$-graded vertex algebras $\oplus_{n=0}^{\infty}V_{(n)}$ such that $\dim V_{(0)}\geq 2$. Also, we examine the algebraic structure of $\mathbb{N}$-graded vertex algebras $V=\oplus_{n=0}^{\infty}V_{(n)}$ such that $\dim~V_{(0)}\geq 2$ and $V_{(1)}$ is a (semi)simple Leibniz algebra that has $sl_2$ as its Levi factor. We show that under suitable conditions this type of vertex algebra is indecomposable but …
On compatible Leibniz algebras
该文研究相容Leibniz代数，通过微分分次李代数的Maurer-Cartan元刻画其结构，定义对应上同调理论以控制单参数形式形变，并进一步探讨其阿贝尔扩张问题。
The notion of pre-Leibniz algebras was recently introduced in the study of Rota-Baxter operators on Leibniz algebras. In this paper, we first construct a graded Lie algebra whose Maurer-Cartan elements are pre-Leibniz algebras. Using this characterization, we define the cohomology of a pre-Leibniz algebra with coefficients in a representation. This cohomology is shown to split the Loday-Pirashvili cohomology of Leibniz algebras. As applications of our cohomology, we study formal and finite order deformations of a pre-Leibniz algebra. Finally, we define homotopy pre-Leibniz algebras and classif…
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.
Motivated by recent developments of $\infty$-categorical theories related to differential graded (dg for short) Lie algebras, we develop a general framework for locally finite $\infty$-$\mathfrak{g}$-modules over a dg Lie algebra $\mathfrak{g}$. We show that the category of such locally finite $\infty$-$\mathfrak{g}$-modules is almost a model category in the sense of Vallette. As a homotopy theoretical generalization of Loday and Pirashvili's Lie algebra objects in the tensor category of linear maps, we further study weak Loday-Pirashvili modules consisting of $\infty$-morphisms from locally f…
In this paper, we first construct the controlling algebras of embedding tensors and Lie-Leibniz triples, which turn out to be a graded Lie algebra and an $L_\infty$-algebra respectively. Then we introduce representations and cohomologies of embedding tensors and Lie-Leibniz triples, and show that there is a long exact sequence connecting various cohomologies. As applications, we classify infinitesimal deformations and central extensions using the second cohomology groups. Finally, we introduce the notion of a homotopy embedding tensor which will induce a Leibniz$_\infty$-algebra. We realize Ko…
We develop general results on centroids of Lie algebras and apply them to determine the centroid of extended affine Lie algebras, loop-like and Kac-Moody Lie algebras, and Lie algebras graded by finite root systems.
As a first application of the double affine Hecke algebra with unequal parameters on Weyl orbits to representation theory of semisimple Lie algebras, we find the graded multiplicities of the trivial module and of the little adjoint module in the exterior algebra of the little adjoint module of a simple Lie algebra $\,\mathfrak{g}\,$ with a non-simply laced Dynkin diagram. We prove that in type $\,B, C\,$ or $\,F\,$ these multiplicities can be expressed in terms of special exponents of positive long roots in the dual root system of $\,\mathfrak{g}.\,$
We give a closed formula for the graded decomposition numbers of the blob algebra over a field of characteristic zero at a root of unity.
Graded Lie superalgebras from embedding tensors
本文通过嵌入张量构造分层李超代数，并揭示其与李代数模之间的关系。
Compatible root graded anti-pre-Lie algebraic structures on finite-dimensional complex simple Lie algebras
本研究借助${\rm sl_2(\C)}$表示理论探究有限维复单李代数的相容根阶化反预李代数结构，证明仅${\rm sl_2(\C)}$上存在且仅存在一个此类结构，其余有限维复单李代数均不存在该结构。
In this paper we introduce (weakly) root graded Banach--Lie algebras and corresponding Lie groups as natural generalizations of group like $\GL_n(A)$ for a Banach algebra $A$ or groups like $C(X,K)$ of continuous maps of a compact space $X$ into a complex semisimple Lie group $K$. We study holomorphic induction from holomorphic Banach representations of so-called parabolic subgroups $P$ to representations of $G$ on holomorphic sections of homogeneous vector bundles over $G/P$. One of our main results is an algebraic characterization of the space of sections which is used to show that this spac…
The essential feature of a root-graded Lie algebra L is the existence of a split semisimple subalgebra g with respect to which L is an integrable module with weights in a possibly non-reduced root system S of the same rank as the root system R of g. Examples include map algebras (maps from an affine scheme to g, S = R), matrix algebras like sl_n(A) for a unital associative algebra A (S = R = A_{n-1}), finite-dimensional isotropic central-simple Lie algebras (S properly contains R in general), and some equivariant map algebras. In this paper we study the category of representations of a root-gr…
In this review article, first we give the concrete formulas of representations and cohomologies of associative algebras, Lie algebras, pre-Lie algebras, Leibniz algebras and 3-Lie algebras and some of their strong homotopy analogues. Then we recall the graded Lie algebras and graded associative algebras that characterize these algebraic structures as Maurer-Cartan elements. The corresponding Maurer-Cartan element equips the graded Lie or associative algebra with a differential. Then the deformations of the given algebraic structures are characterized as the Maurer-Cartan elements of the result…
Combinatorics of graded module categories over skew polynomial algebras at roots of unity
本文引入了$\mathbb{Z}/\ell\mathbb{Z}$上斜对称矩阵的switching运算与模Euler矩阵，证明其与单位根处斜多项式代数的分次模范畴密切相关，并以此研究了三次单位根处斜多项式代数的点单纯复形。
We introduce and study the class of groups graded by root systems. We prove that if Φ is an irreducible classical root system of rank at least 2 and G is a group graded by Φ, then under certain natural conditions on the grading, the union of the root subgroups is a Kazhdan subset of G. As the main application of this theorem we prove that for any reduced irreducible classical root system Φ of rank at least 2 and a finitely generated commutative ring R with 1, the Steinberg group St_Φ(R) and the elementary Chevalley group E_Φ(R) have property (T). We also show that there exists a group with pro…
Let $\mathbb{F}$ be a field of characteristic zero and let $\mathfrak{g}$ be a non-zero finite-dimensional split semisimple Lie algebra with root system $Δ$. Let $Γ$ be a finite set of integral weights of $\mathfrak{g}$ containing $Δ$ and $\{0\}$. Following [2,10], we say that a Lie algebra $L$ over $\mathbb{F}$ is \emph{generalized root graded}, or more exactly $(Γ,\mathfrak{g})$-\emph{graded}, if $L$ contains a semisimple subalgebra isomorphic to $\mathfrak{g}$, the $\mathfrak{g}$-module $L$ is the direct sum of its weight subspaces $L_α$ ($α\inΓ$) and $L$ is generated by all $L_α$ with $α\n…
This thesis is concerned with superconformal Chern-Simons theories with matter in 3 dimensions. The interest in these theories is two-fold. On the one hand, it is a new family of theories in which to test the AdS/CFT correspondence and on the other, they are important to study one of the main objects of M-theory (M2-branes). All these theories have something in common: they can be written in terms of 3-Leibniz algebras. Here we study the structure theory of such algebras, paying special attention to a subclass of them that gives rise to maximal supersymmetry and that was the first to appear in…
We determine the Lie superalgebras over fields of characteristic zero that are graded by the root system A(n,n) of the special linear Lie superalgebra psl(n+1,n+1).
We look to gradations of Kac-Moody Lie algebras by Kac-Moody root systems with finite dimensional weight spaces. We extend, to general Kac-Moody Lie algebras, the notion of C-admissible pair as introduced by H. Rubenthaler and J. Nervi for semi-simple and affine Lie algebras. If g is a Kac-Moody Lie algebra (with Dynkin diagram indexed by I) and (I,J) is such a C-admissible pair, we construct a C-admissible subalgebra g^J, which is a Kac-Moody Lie algebra of the same type as g, and whose root system Σgrades finitely the Lie algebra g. For an admissible quotient ρ: I \rightarrow I we build also…
On the derived Hall algebra of a graded gentle one-cycle algebra I: the triangle structure
本文在温和条件下将分级温和单圈代数的两类导出范畴描述为无限箭图的扭曲根范畴，并证明其三角结构由底层加性范畴唯一决定。
Lie algebras formed via semi-direct sums of the Witt algebra $\text{Der}(\mathbb{C}[t,t^{-1}])$ and its modules have become increasingly prominent in both physics and mathematics in recent years. In this paper, we complete the study of (Leibniz) central extensions, derivations and automorphisms of the Lie algebras formed from the semi-direct sum of the Witt algebra and its indecomposable intermediate series modules (that is, graded modules with one-dimensional graded components). Our techniques exploit the internal grading of the Witt algebra, which can be applied to a wider class of graded Li…
We show that the Temperley-Lieb algebra of type $A$ and the blob algebra (also known as the Temperley-Lieb algebra of type $ B$) at roots of unity are $ \mathbb Z$-graded algebras.We moreover show that they are graded cellular algebras, thus making their cell modules, or standard modules, graded modules for the algebras.
We provide a graded and quantum version of the category of rooted cluster algebras introduced by Assem, Dupont and Schiffler and show that every graded quantum cluster algebra of infinite rank can be written as a colimit of graded quantum cluster algebras of finite rank. As an application, for each k we construct a graded quantum infinite Grassmannian admitting a cluster algebra structure, extending an earlier construction of the authors for k=2.
Embedding tensors on Hom-Lie algebras
本文在 Hom-Lie 代数上引入了嵌入张量，构造了刻画它的分次 Lie 代数与控制 $L_\infty$-代数，定义了相关上同调与 Hom-Lie-Leibniz 三元组，并探讨了同伦嵌入张量与 $HLeib_\infty$-代数的关系。
There is a $p$-differential on the triply-graded Khovanov--Rozansky homology of knots and links over a field of positive characteristic $p$ that gives rise to an invariant in the homotopy category finite-dimensional $p$-complexes. A differential on triply-graded homology discovered by Cautis is compatible with the $p$-differential structure. As a consequence we get a categorification of the colored Jones polynomial evaluated at a $2p$th root of unity.
The Z-grading determined by a long simple root of an affine or finite type Lie algebra arises from an adjoint or cominuscule representation of a lower rank semi-simple complex Lie algebra. Analysis of the relationship between the grading and the representation leads to an extension of Kac's construction of nontwisted affine Lie algebras.
We equip the odd nilHecke algebra and its associated thick calculus category with digrammatically local differentials. The resulting differential graded Grothendieck groups are isomorphic to two different forms of the positive part of quantum sl(2) at a fourth root of unity.
Character Sheaves for Graded Lie Algebras: Stable Gradings
本研究针对稳定分次李代数，通过分析[GVX2]中邻近周期层的傅里叶变换，构造出推测为尖点特征层的满支撑特征层，涉及单位根处复反射群相关Hecke代数的不可约表示。
On $\mathbb{N}$-graded vertex algebras associated with Gorenstein algebras
本文研究与Gorenstein代数相关的不可分解$\n$分次顶点代数的结构，证明局部性、不可分解性与半共形顶点代数结构条件的等价性，分析$V_0$、$V_1$的结构特征及双线性型的影响，给出秩一Heisenberg顶点算子代数的嵌入条件。
The paper concerns a certain subcategory of the category of representations for a semisimple algebraic group $G$ in characteristic $p$, which arise from the semisimple modules for the corresponding quantum group at a $p$-th root of unity. The subcategory, thus, records the cohomological difference between quantum groups and algebraic groups. We define translation functors in this category and use them to obtain information on the irreducible characters for $G$ when the Lusztig character formula does not hold.
We state a conjecture which gives a combinatorial parametrization of the irreducible tempered representations with real central character of a graded Hecke algebra with unequal labels, associated to a root sytem of type B or C. This conjecture is based on a combinatorial generalization of the Springer correspondence in the classical (equal label) case. In particular, the described modules turn out to have a natural grading for the action of W_0, and are completely determined by their central character together with the W_0-representation in the top degree. This latter is an irreducible W_0-cha…
We describe a general setting for the definition of semi-infinite cohomology of finite dimensional algebras, and provide its categorical interpretation. We apply this interpretation to compute semi-infinite cohomology of some modules over the small group at a root of unity, generalizing an earlier result of S. Arkhipov (conjectured by B. Feigin).
Structure and Representation Theory of basic simple $\mathbb{Z}_2\times \mathbb{Z}_2$-graded color Lie algebras
本文将复半单李代数理论方法应用于发展一类简单 $\mathbb{Z}_2\times \mathbb{Z}_2$-graded 色李代数的根理论，通过最高权定理和完全可约性定理对有限维表示进行分类。
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。
In this paper, first we show that $(\g,[\cdot,\cdot],α)$ is a hom-Lie algebra if and only if $(Λ\g^*,α^*,d)$ is an $(α^*,α^*)$-differential graded commutative algebra. Then, we revisit representations of hom-Lie algebras, and show that there are a series of coboundary operators. We also introduce the notion of an omni-hom-Lie algebra associated to a vector space and an invertible linear map. We show that regular hom-Lie algebra structures on a vector space can be characterized by Dirac structures in the corresponding omni-hom-Lie algebra. The underlying algebraic structure of the omni-hom-Lie …
Finite-dimensional $\mathbb{Z}$-graded Lie algebras
本文研究了有限维 $\mathbb{Z}$-分次 Lie 代数的结构与表示理论，扩展了有限维半单 Lie 代数理论，并揭示了 Heisenberg 与共形 Galilei 代数等物理实例的分次结构应用潜力。
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