Yucai Su, Jianhua Zhou
We determine the isomorphism classes of the first family of infinite dimensional simple Lie algebras recently introduced by Xu. The structure space of these algebras is given explicitly. The derivations of these algebras are also determined.
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Irreducibilities of Verma modules over a class of Block type Lie algebras are completely determined. The approach developed in the present paper can be used to deal with non-weight modules.
We investigate some properties of Rota-Baxter operators on BiHom-Lie algebras. Along the way, we introduce BiHom analogues of pre-Lie and Leibniz algebras.
We construct an associative product on the symmetric module S(L) of any pre-Lie algebra L. Then we proove that in the case of rooted trees our construction is dual to that of Connes and Kreimer. We also show that symmetric brace algebras and pre-Lie algebras are the same. Finally, using brace algebras instead of pre-Lie algebras, we give a similar interpretation of Foissy's Hopf algebra of planar rooted trees.
We associate elliptic affine Lie algebras with what are called vertex $\C((z))$-algebras and their modules in a certain category. In the course, we construct two families of Lie algebras closely related to elliptic affine Lie algebras.
We introduce a notion of Hopf-Lie-Rinehart algebra and show that the universal algebra of a Hopf-Lie-Rinehart algebra acquires an ordinary Hopf algebra structure.
In this paper, we study a family of infinite-dimensional Lie algebras $\widehat{X}_{S}$, where $X$ stands for the type: $A,B,C,D$, and $S$ is an abelian group, which generalize the $A,B,C,D$ series of trigonometric Lie algebras. Among the main results, we identify $\widehat{X}_{S}$ with what are called the covariant algebras of the affine Lie algebra $\widehat{\mathcal{L}_{S}}$ with respect to some automorphism groups, where $\mathcal{L}_{S}$ is an explicitly defined associative algebra viewed as a Lie algebra. We then show that restricted $\widehat{X}_{S}$-modules of level $\ell$ naturally co…
In this paper, we introduce two new families of infinite-dimensional simple Lie algebras and a new family of infinite-dimensional simple Lie superalgebras. These algebras can be viewed as generalizations of the Block algebras.
In this paper, we study nullity-2 toroidal extended affine Lie algebras in the context of vertex algebras and their $ϕ$-coordinated modules. Among the main results, we introduce a variant of toroidal extended affine Lie algebras, associate vertex algebras to the variant Lie algebras, and establish a canonical connection between modules for toroidal extended affine Lie algebras and $ϕ$-coordinated modules for these vertex algebras. Furthermore, by employing some results of Billig, we obtain an explicit realization of irreducible modules for the variant Lie algebras.
Universal enveloping algebras of braided m-Lie algebras and PBW theorem are obtained by means of combinatorics on words.
The aim of this paper is to study a Lie conformal algebra of Block type. In this paper, conformal derivation, conformal module of rank 1 and low-dimensional comohology of the Lie conformal algebra of Block type are studied. Also, the vertex Poisson algebra structure associated with the Lie conformal algebra of Block type is constructed.
Let L be the Lie algebra of Block type over a field F of characteristic zero, defined with basis {L_{i,j} | i,j\in Z} and relations [L_{i,j},L_{k,l}]=((j+1)k-(l+1)i)L_{i+k,j+l}. Then L is Z^2-graded. In this paper, Z^2-graded L-modules of the intermediate series are classified.
Let ${\mathcal N}$ be the Lie algebra of all $n\times n$ strictly block upper triangular matrices over a field ${\mathbb F}$ relative to a given partition. In this paper, we give an explicit description of all derivations of ${\mathcal N}$.
Classical invariants for representations of one Lie group can often be related to invariants of some other Lie group. Physics suggests that the right objects to consider for these questions are certain refinements of classical invariants known as conformal blocks. Conformal blocks appear in algebraic geometry as spaces of global sections of line bundles on the moduli stack of parabolic bundles on a smooth curve. Rank-level duality connects a conformal block associated to one Lie algebra to a conformal block for a different Lie algebra. In this paper we prove a rank-level duality for type so(2r…
The notions of Lie color 2-algebras and 2-term color L-infty-algebras over a group-graded vector space are introduced and studied. It is proved that the category of Lie color 2-algebras and the category of 2-term color L1-algebras are equivalent. We construct Lie color 2-algebras from omni-Lie color algebras and Leibniz color algebras. Some example of $Z^2_2$ -graded Lie color 2-algebras are given.
Representations of color Hom-Lie algebras are reviewed, and it is shown that there exist a series of coboundary operators. We also introduce the notion of a color omni-Hom-Lie algebra associated to a vector space and an even invertible linear map. We show how regular color Hom-Lie algebra structures on a vector space can be characterized. Moreover, it is shown that the underlying algebraic structure of the color omni-Hom-Lie algebra is a color Hom-Leibniz algebra.
We prove that free pre-Lie algebras, when considered as Lie algebras, are free. Working in the category of S-modules, we define a natural filtration on the space of generators. We also relate the symmetric group action on generators with the structure of the anticyclic PreLie operad.
In this paper, we explore natural connections among trigonometric Lie algebras, (general) affine Lie algebras, and vertex algebras. Among the main results, we obtain a realization of trigonometric Lie algebras as what were called the covariant algebras of the affine Lie algebra $\widehat{\mathcal{A}}$ of Lie algebra $\mathcal{A}=\frak{gl}_{\infty}\oplus\frak{gl}_{\infty}$ with respect to certain automorphism groups. We then prove that restricted modules of level $\ell$ for trigonometric Lie algebras naturally correspond to equivariant quasi modules for the affine vertex algebras $V_{\widehat{\…
Let B be the Lie algebra with basis {L_{i,j},C|i,j\in Z} and relations [L_{i,j},L_{k,l}]=((j+1)k-i(l+1))L_{i+k,j+l}+iδ_{i,-k}δ_{j+l,-2}C, [C,L_{i,j}]=0. It is proved that an irreducible highest weight B-module is quasifinite if and only if it is a proper quotient of a Verma module. For an additive subgroup G of the base field F, there corresponds to a Lie algebra B(G) of Block type. Given a totalorder \succ on G and a weight Λ, a Verma B(G)-module M(Λ,\succ) is defined. The irreducibility of M(Λ,\succ) is completely determined.
In this paper,we will show how to kill the obstructions to Lie algebra deformations via a method which essentially embeds a Lie algebra into Strong homotopy Lie algebra or $L_{\infty}$ algebra. All such obstructions have been transfered to the revelvant $L_{\infty}$ algebras which contain only three terms
Lie-Rinehart algebras, also known as Lie algebroids, give rise to Hopf algebroids by a universal enveloping algebra construction, much as the universal enveloping algebra of an ordinary Lie algebra gives a Hopf algebra, of infinite dimension. In finite characteristic, the universal enveloping algebra of a restricted Lie algebra admits a quotient Hopf algebra which is finite-dimensional if the Lie algebra is. Rumynin has shown that suitably defined restricted Lie algebroids allow to define restricted universal enveloping algebras that are finitely generated projective if the Lie algebroid is. T…
The purpose of this paper is to generalize some results on $n$-Lie algebras and $n$-Hom-Lie algebras to $n$-Hom-Lie color algebras. Then we introduce and give some constructions of $n$-Hom-Lie color algebras.
Braided m-Lie algebras induced by multiplication are introduced, which generalize Lie algebras, Lie color algebras and quantum Lie algebras. The necessary and sufficient conditions for the braided m-Lie algebras to be strict Jacobi braided Lie algebras are given. Two classes of braided m-Lie algebras are given, which are generalized matrix braided m-Lie algebras and braided m-Lie subalgebras of $End_F M$, where $M$ is a Yetter-Drinfeld module over $B$ with dim $B< \infty $ . In particular, generalized classical braided m-Lie algebras $sl_{q, f}(GM_G(A), F)$ and $osp_{q, t} (GM_G(A), M, F)$ of …
In this paper the universal enveloping algebra of color hom-Lie algebras is studied. A construction of the free involutive hom-associative color algebra on a hom-module is described and applied to obtain the universal enveloping algebra of an involutive hom-Lie color algebra. Finally, the construction is applied to obtain the well-known Poincar{é}-Birkhoff-Witt theorem for Lie algebras to the enveloping algebra of an involutive color hom-Lie algebra.
A pre-Lie-Rinehart algebra is an algebraic generalization of the notion of a left-symmetric algebroid. We construct pre-Lie-Rinehart algebras from r-matrices through Lie algebra actions. We study cohomologies of pre-Lie-Rinehart algebras and show that abelian extensions of pre-Lie-Rinehart algebras are classified by the second cohomology groups. We introduce the notion of crossed modules for pre-Lie-Rinehart algebras and show that they are classified by the third cohomology groups of pre-Lie-Rinehart algebras. At last, we use (pre-)Lie-Rinehart 2-algebras to characterize the crossed modules fo…
All bialgebra structures on twodimensional Galilei algebra are classified. The corresponding Lie-Poisson structures on Galilei group are found.
This is a survey on extended affine Lie algebras and related types of Lie algebras, which generalize affine Lie algebras.
Homological blocks with simple Lie algebras and Witten--Reshetikhin--Turaev invariants
本文针对任意Seifert纤维化同调3-球，引入具有简单李代数的同调块，通过发展渐近公式与渐近系数消没结果，证明其径向极限等同于Witten--Reshetikhin--Turaev不变量。
The notions of vertex Lie algebra and vertex Poisson algebra are presented and connections among vertex Lie algebras, vertex Poisson algebras and vertex algebras are discussed.
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。
We study (quasi-)twilled pre-Lie algebras and the associated $L_\infty$-algebras and differential graded Lie algebras. Then we show that certain twisting transformations on (quasi-)twilled pre-Lie algbras can be characterized by the solutions of Maurer-Cartan equations of the associated differential graded Lie algebras ($L_\infty$-algebras). Furthermore, we show that $\mathcal{O}$-operators and twisted $\mathcal{O}$-operators are solutions of the Maurer-Cartan equations. As applications, we study (quasi-)pre-Lie bialgebras using the associated differential graded Lie algebras ($L_\infty$-algeb…
Conformal Lie 2-algebras and conformal omni-Lie algebras
本文引入并研究了共形李2-代数与共形全李代数的概念，证明了共形李2-代数范畴与2项共形$L_{\infty}$-代数范畴等价，并从共形全李代数和莱布尼茨共形代数构造了共形李2-代数。
Anti-pre-Lie algebras, Novikov algebras and commutative 2-cocycles on Lie algebras
本文引入反预李代数作为李代数上非退化交换2-上闭链的底层结构，揭示其与预李代数的类比，并建立容许Novikov代数与Novikov代数及相应泊松型结构之间的对应关系。
Lie groups and Lie algebras
本文是作者在麻省理工学院讲授的关于Lie群与Lie代数的两个学期课程的扩展笔记。
In this paper, first we introduce the notion of quadratic Rota-Baxter Lie algebras of arbitrary weight, and show that there is a one-to-one correspondence between factorizable Lie bialgebras and quadratic Rota-Baxter Lie algebras of nonzero weight. Then we introduce the notions of matched pairs, bialgebras and Manin triples of Rota-Baxter Lie algebras of arbitrary weight, and show that Rota-Baxter Lie bialgebras, Manin triples of Rota-Baxter Lie algebras and certain matched pairs of Rota-Baxter Lie algebras are equivalent. The coadjoint representations and quadratic Rota-Baxter Lie algebras pl…
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