Vladimir Dergachev
In this paper we prove several theorems about the behavior of index of Lie algebras derived from associative algebras under tensor products of underlying associative algebras.
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The index of nilpotent Lie algebras
本文研究多类幂零李代数的指数，并针对自由幂零、图及丝状幂零李代数给出显式计算公式。
The filiform and the quasi-filiform Lie algebras form a special class of nilpotent Lie algebras. The aim of this paper is to compute the index and provide regular vectors of this two class of nilpotent Lie algebras. we consider the graded filiform Lie algebras $L_{n}$, $Q_{n}$, the $n$-dimensional filiform Lie algebras for $n$ $<8$, also the graded quasi-filiform Lie algebras and finally Lie algebras whose nilradical is $Q_{2n}$.
We present a new method of analysis of associative algebras. This method bears a certain resemblance to the famous analysis of commutative $C^*$-algebras in which an important role is played by multiplicative functionals over the algebra. These functionals can be considered as "rank 1" in the framework of our method. In our approach a special position is taken by generic (maximal rank) functionals, which reveal a wealth of information on the structure of non-commutative algebras. We apply this method to derive interesting properties of index of Lie algebras and to analyze finite-dimensional as…
The ideals of the Lie algebras of unitriangular polynomial derivations are classified. An isomorphism criterion is given for the Lie factor algebras of the Lie algebras of unitriangular polynomial derivations.
The aim of this paper is to compare the structure and the cohomology spaces of Lie algebras and induced $3$-Lie algebras.
Lie nilpotency index of skew symmetric elements in group algebras
针对换位子群阶为$p$的有限奇素数$p$-群$G$的群代数，证明其李幂零指数由斜对称元构成的李子代数决定，并描述了该子代数的下李中心序列结构。
We introduce the method of calculation of index of Lie algebras that are factors of the unitriangular Lie algebra with respect to ideals spanned by subsets of root vectors.
The purpose of this paper is to introduce twisted $\mathcal{O}$-operators on $3$-Lie algebras. We define a cohomology of a twisted $\mathcal{O}$-operator $T$ as the Chevalley-Eilenberg cohomology of a certain $3$-Lie algebra induced by $T$ with coefficients in a suitable representation. Then we consider infinitesimal and formal deformations of twisted $\mathcal{O}$-operators from cohomological points of view. Furthermore, we introduce and study $3$-NS-Lie-algebras as the underlying structure of twisted $\mathcal{O}$-operators on $3$-Lie algebras. Finally, we investigate twisted $\mathcal{O}$-o…
The main purpose of this paper is to introduce the notion of $n$-L-dendriform algebra which can be seen as a dendrification of $n$-pre-Lie algebras by means of $\mathcal{O}$-operators. We investigate the representation theory of $n$-pre-Lie algebras and provide some related constructions. Furthermore, we introduce the notion of phase space of a $n$-Lie algebra and show that a $n$-Lie algebra has a phase space if and only if it is sub-adjacent to a $n$-pre-Lie algebra. Moreover, we present a procedure to construct $(n + 1)$-pre-Lie algebras from $n$-pre-Lie algebras equipped with a generalized …
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.
The aim of this paper is to introduce and study quadratic Hom-Lie algebras, which are Hom-Lie algebras with symmetric invariant nondegenerate bilinear forms. We provide several constructions leading to examples and extend the double extension theory to Hom-Lie algebras. We reduce the case where the twist map is invertible to the study of involutive quadratic Lie algebras. We establish a correspondence between the class of involutive quadratic Hom-Lie algebras and quadratic simple Lie algebras with symmetric involution. Centerless involutive quadratic Hom-Lie algebras are characterized. Also el…
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.
Lie conformal algebras $\mathcal{W}(a,b)$ are the semi-direct sums of Virasoro Lie conformal algebra and its nontrivial conformal modules of rank one. In this paper, we give a complete classification of extensions of finite irreducible conformal modules of $\mathcal{W}(a,b)$. With a similar method, we characterize all extensions of finite irreducible conformal modules of Schrödinger-Virasoro type Lie conformal algebras $TSV(a,b)$ and $TSV(c)$.
Let M be a maximal subalgebra of the Lie algebra L. A subalgebra C of L is said to be a completion for M if C is not contained in M but every proper subalgebra of C that is an ideal of L is contained in M. The set I(M) of all completions of M is called the index complex of M in L. We use this concept to investigate the influence of the maximal subalgebras on the structure of a Lie algebra, in particular finding new characterisations of solvable and supersolvable Lie algebras.
same as arXiv:0904.1778
We give an upper bound for the index of certain Lie algebras, called of seaweed type, introduced by V. Dergachev, A. Kirillov and D. Panyushev. We deduce from this a conjecture of D. Panyushev stated in "Inductive formulas for the index of seaweed Lie algebra, Moscow Math. Journal (2001) 221-241". We conjecture that our upper bound gives in fact a formula for the index of such Lie algebras, and we prove it in certain cases. Finally, we obtain a result concerning the index of parabolic subalgebras of a semisimple Lie algebra.
We characterize, in a purely algebraic manner, certain linear forms, called stable, on a Lie algebra. As an application, we determine the index of a Borel subalgebra of a semi-simple Lie algebra. Finally, we give an example of a parabolic subalgebra of a semi-simple Lie algebra which does not admit any stable linear form.
This is a continuation of arXiv:0903.0398 [math.RT]. Let g be a simple Lie algebra. In this note, we provide simple formulae for the index of sl(2)-subalgebras in the classical Lie algebras and a new formula for the index of the principal sl(2). We also compute the difference, D, of the indices of principal and subregular sl(2)-subalgebras. Our formula for D involves some data related to the McKay correspondence for g. Using the index of sl(2)-subalgebras of classical Lie algebras, we also obtain three series of interesting combinatorial identities parameterised by partitions.
For a finite dimensional complex Lie algebra, its index is the minimal dimension of stabilizers for the coadjoint action. A famous conjecture due to Elashvili says that the index of the centralizer of an element of a reductive Lie algebra is equal to the rank. That conjecture caught attention of several Lie theorists for years. In this paper we give an almost general proof of that conjecture.
This is a survey on extended affine Lie algebras and related types of Lie algebras, which generalize affine Lie algebras.
Three kinds of universal central extension are considered for a perfect Lie algebra. More precisely, one can consider such a Lie algebra as a Lie triple system, or a Leibniz algebra and construct appropriate central extensions. We show that Lie triple system universal central extension is isomorphic either to Lie or Leibniz algebra universal central extension considered as a Lie triple system in appropriate way.
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.
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。
The first aim of this paper is to show that any finite-dimensional reductive Lie algebra and its finite-dimensional completely reducible representation can be embedded into some PC Lie algebra. The second aim is to find the structure of a PC Lie algebra.
It is shown that any generalized Kac-Moody Lie algebra g that has no mutually orthogonal imaginary simple roots can be written as the vector space direct sum of a Kac-Moody subalgebra and subalgebras isomorphic to free Lie algebras over certain modules for the Kac-Moody subalgebra. Also included is a detailed discussion of Borcherds' construction of the Monster Lie algebra from a vertex algebra and an elementary proof of Borcherds' theorem relating Lie algebras with `an almost positive definite bilinear form' to generalized Kac-Moody algebras. (Preprint version 1996)
In this paper, first we introduce the notion of a twilled 3-Lie algebra, and construct an $L_\infty$-algebra, whose Maurer-Cartan elements give rise to new twilled 3-Lie algebras by twisting. In particular, we recover the Lie $3$-algebra whose Maurer-Cartan elements are O-operators (also called relative Rota-Baxter operators) on 3-Lie algebras. Then we introduce the notion of generalized matched pairs of 3-Lie algebras using generalized representations of 3-Lie algebras, which will give rise to twilled 3-Lie algebras. The usual matched pairs of 3-Lie algebras correspond to a special class of t…
We will construct standard pentads which are analogues of Cartan subalgebras, and moreover, we will study graded Lie algebras corresponding to these standard pentads. We call such pentads pentads of Cartan type and describe them by two positive integers and three matrices. The structures of their corresponding Lie algebras are related with contragredient Lie algebras.
In this paper, we study the index for several natural classes of non-reductive subalgebras of semisimple Lie algebras. Namely, we look at parabolic subalgebras, centralisers of nilpotent elements, and the normalisers of the centralisers. We discuss a conjecture of Elashvili to the effect that the index of any centraliser is equal to the rank of the semisimple algebra in question. It is shown that Elashvili's conjecture is true for `small' and `large' orbits. Some properties of the index for the normaliser of the centraliser are proved. In particular, we prove that, for a regular nilpotent elem…
The theory of standard pentads is the theory aims to construct a graded Lie algebra whose local part consists of a given Lie algebra and its representation. In other words, using standard pentads, we can embed given Lie algebra and its representation into a larger graded Lie algebra. As special cases of Lie algebras associated with standard pentads, we have the notion of PC Lie algebras. Our aim of this paper is to show that the class of PC Lie algebras contains the class of Kac-Moody algebras, that is, to show that the notion of PC Lie algebras is an extension of Kac-Moody algebras.
We show that the p-operator in the Witt algebra (the restricted Lie algebra of derivations of the quotient of the polynomial algebra over a field of characteristic p by the ideal generated by the p-th power of the indeterminant) is given by multiplication by a scalar.
In a previous paper by the authors, we obtain the first example of a finitely freely generated simple $\mathbb Z$-graded Lie conformal algebra of linear growth that cannot be embedded into any general Lie conformal algebra. In this paper, we obtain, as a byproduct, another class of such Lie conformal algebras by classifying $\mathbb Z$-graded simple Lie conformal algebras ${\cal G}=\oplus_{i=-1}^\infty{\cal G}_i$ satisfying the following, (1) ${\cal G}_0\cong{\rm Vir}$, the Virasoro conformal algebra; (2) Each ${\cal G}_i$ for $i\ge-1$ is a ${\rm Vir}$-module of rank one. These algebras includ…
We define and investigate nilpotent Lie algebras associated with quadratic forms. We also present their connections with Lie algebras and Ringel-Hall algebras associated with representation directed algebras.
Using $n$ finite order automorphisms on a simple complex Lie algebra we construct twisted $n$-toroidal Lie algebras. Thus obtaining Lie algebras wich have a rootspace decomposition. For the case $n=2$ we list certain simple Lie algebras and their automorphisms, which produce twisted 2-toroidal algebras. In this way we obtain Lie algebras that are related to all Extended Affine Root Systems of K. Saito.
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。
Let $Γ$ be a generic subgroup of the multiplicative group $\mathbb{C}^*$ of nonzero complex numbers. We define a class of Lie algebras associated to $Γ$, called twisted $Γ$-Lie algebras, which is a natural generalization of the twisted affine Lie algebras. Starting from an arbitrary even sublattice $Q$ of $\mathbb Z^N$ and an arbitrary finite order isometry of $\mathbb Z^N$ preserving $Q$, we construct a family of twisted $Γ$-vertex operators acting on generalized Fock spaces which afford irreducible representations for certain twisted $Γ$-Lie algebras. As application, this recovers a number o…
Using a K-theory point of view, Bott related the Atiyah-Singer index theorem for elliptic operators on compact homogeneous spaces to the Weyl character formula. This article explains how to prove the local index theorem for compact homogenous spaces using Lie algebra methods. The method follows in outline the proof of the local index theorem due to Berline and Vergne. But the use of Kostant's cubic Dirac operator in place of the Riemannian Dirac operator leads to substantial simplifications. An important role is also played by the quantum Weil algebra of Alekseev and Meinrenken.
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{\…
We consider the natural Lie algebra structure on the (associative) group algebra of a finite group $G$, and show that the Lie subalgebras associated to natural involutive antiautomorphisms of this group algebra are reductive ones. We give a decomposition in simple factors of these Lie algebras, in terms of the ordinary representations of $G$.
In this paper, we introduce the notions of a $3$-$Lie_\infty$-algebra and a 3-Lie 2-algebra. The former is a model for a 3-Lie algebra that satisfy the fundamental identity up to all higher homotopies, and the latter is the categorification of a 3-Lie algebra. We prove that the 2-category of 2-term $3$-$Lie_\infty$-algebras is equivalent to the 2-category of 3-Lie 2-algebras. Skeletal and strict 3-Lie 2-algebras are studied in detail. A construction of a 3-Lie 2-algebra from a symplectic 3-Lie algebra is given.
We show that a certain symmetry exists in the stable irreducible decomposition of the Lie algebra consisting of symplectic derivations of the free Lie algebra generated by the first homology group of compact oriented surfaces.
F-Lie algebras are natural generalisations of Lie algebras (F=1) and Lie superalgebras (F=2). We give finite dimensional examples of F-Lie algebras obtained by an inductive process from Lie algebras and Lie superalgebras. Matrix realizations of the $F-$Lie algebras constructed in this way from osp(2|m) are given. We obtain a non-trivial extension of the Poincaré algebra by an Inönü-Wigner contraction of a certain $F-$Lie algebras with $F>2$.
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。
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