Askar Dzhumadil'daev, Pasha Zusmanovich
On Lie algebras, we study commutative 2-cocycles, i.e., symmetric bilinear forms satisfying the usual cocycle equation. We note their relationship with antiderivations and compute them for some classes of Lie algebras, including finite-dimensional semisimple, current and Kac-Moody algebras.
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Hom-Poisson superalgebras with nondegenerate bilinear forms and compatible Hom-anti-pre-Poisson superalgebras
本文引入Hom-anti-Zinbiel超代数等概念，通过非退化超交换上循环建立Hom-Poisson超代数与相容Hom-anti-pre-Poisson超代数的联系，为两类超代数提供统一研究框架。
Semigroups in semi-simple Lie groups: Flag type and estimation of cocycles
本文研究非紧半单李群半群的旗型，证明可通过极大旗流形上上循环的下界存在性恢复该代数不变量。
Unifying various generalizations of the important notions of Reynolds operators, the relative cocycle weighted Reynolds operators are studied. Here cocycle weighted means the weight of the operators is given by a 2-cocycle rather than by a scaler as in the classical case. We show that the operators and 2-cocycles uniquely determine each other. We further give a characterization of relative cocycle weighted Reynolds operators in the context of pre-Lie algebras. Using a method of Liu, we construct an explicit graded Lie algebra whose Maurer-Cartan elements are given by a relative cocycle weighte…
We study relations between finite-dimensional representations of color Lie algebras and their cocycle twists. Main tools are the universal enveloping algebras and their FCR-properties (finite-dimensional representations are completely reducible.) Cocycle twist preserves the FCR-property. As an application, we compute all finite dimensional representations (up to isomorphism) of the color Lie algebra $sl_2^c$.
We introduce the conception of matched pairs of $(H, β)$-Lie algebras, construct an $(H, β)$-Lie algebra through them. We prove that the cocycle twist of a matched pair of $(H, β)$-Lie algebras can also be matched.
In this work, we recall that every filiform Lie superalgebra is a deformation of the superalgebra $L_{n,m}$. We study the even cocycles which give this nilpotent deformations. A family of independent bilinear maps will help us to describe this cocycles. At the end an evaluation of the dimension of the space $Z_0^2(L_{n,m},L_{n,m})$ is established.
Classical contact Lie algebras are the fundamental algebraic structures on the manifolds of contact elements of configuration spaces in classical mechanics. Xu introduced a large category of contact simple Lie algebras which are related to locally finite derivations and are in general not finitely graded. The isomorphism classes of these Lie algebras were determined in a previous paper by Xu and Su. In this paper, the derivation algebras and the 2-cohomology groups of these Lie algebras are determined, and it is obtained that the 2-cohomology groups of these Lie algebras are all trivial.
We construct large families of characteristically nilpotent Lie algebras by considering deformations of the Lie algebra g_{m,m-1}^{4} of type Q_{n},and which arises as a central extension fo the filiform Lie algebra L_{n}. By studying the graded cohomology spaces we obtain that the sill algebras are isomorphic to the nilradicals of solvable, complete Lie algebra laws. For extremal cocycles these laws are also rigid. Considering supplementary cocycles we construcy, for dimensions n>8, nonfiliform characteristically nilpotent Lie algebras and show that for certain deformations these are compatib…
We show that any CPA-structure (commutative post-Lie algebra structure) on a perfect Lie algebra is trivial. Furthermore we give a general decomposition of inner CPA-structures, and classify all CPA-structures on parabolic subalgebras of simple Lie algebras.
We determine commutative post-Lie algebra structures on some infinite-dimensional Lie algebras. We show that all commutative post-Lie algebra structures on loop algebras are trivial. This extends the results for finite-dimensional perfect Lie algebras. Furthermore we show that all commutative post-Lie algebra structures on affine Kac--Moody Lie algebras are "almost trivial".
We give an explicit description of commutative post-Lie algebra structures on some classes of nilpotent Lie algebras. For non-metabelian filiform nilpotent Lie algebras and Lie algebras of strictly upper-triangular matrices we show that all CPA-structures are associative and induce an associated Poisson-admissible algebra.
In this paper, we characterize the biderivations of Schrödinger-Virasoro Lie algebra. We get a classes of non-inner and non-skewsymmetric biderivations. As application, we characterize the commutative post-Lie algebra structures on Schrödinger-Virasoro Lie algebra.
We develop Lie's correspondence and an explicit Baker-Campbell-Hausdorff formula for commutative automorphic formal loops.
In this paper, by using Composition-Diamond lemma for Lie algebras, we give a Gröbner-Shirshov basis for free partially commutative Lie algebra over a commutative ring with unit. As an application, we obtain a normal form for such a Lie algebra.
In this paper, the complete description of centralizers of elements in partially commutative Lie algebras is obtained. The description is given explicitly in the terms of generators.
Splitting partially commutative Lie algebras into direct sums
本文证明了部分交换李代数及其亚交换、幂零情形可分裂为两个子代数直和的充要条件是定义图的补图不连通。
In this paper, we consider partially commutative metabelian Lie algebras whose defining graphs are cycles. We show that such algebras are universally equivalent iff the corresponding cycles have the same length. Moreover, we give an example showing that the class of partially commutative metabelian Lie algebras such that their defining graphs are trees is not separable by universal theory in the class of all partially commutative metabelian Lie algebras.
This paper begins by introducing the concept of a quasi-hom-Lie algebra which is a natural generalization of hom-Lie algebras introduced in a previous paper by the authors. Quasi-hom-Lie algebras include also as special cases (color) Lie algebras and superalgebras, and can be seen as deformations of these by homomorphisms, twisting the Jacobi identity and skew-symmetry. The natural realm for these quasi-hom-Lie algebras is as a generalization-deformation of the Witt algebra $\Witt$ of derivations on the Laurent polynomials $\C[t,t^{-1}]$. We also develop a theory of central extensions for qhl-…
Let $R$ be a finite commutative ring with identity. In this paper, we give a necessary condition for the existence of an orthogonal decomposition of the special linear Lie algebra over $R$. Additionally, we study orthogonal decompositions of the symplectic Lie algebra and the special orthogonal Lie algebra over $R$.
For $a,b\in \mathbb{C}$, the Lie algebra $\mathcal{W}(a,b)$ is the semidirect product of the Witt algebra and a module of the intermediate series. In this paper, all biderivations of $\mathcal{W}(a,b)$ are determined. Surprisingly, these Lie algebras have symmetric (and skewsymmetric) non-inner biderivations. As an applications, commutative post-Lie algebra structures on $\mathcal{W}(a,b)$ are obtained.
In this paper, we begin a systematic study of modified Rota-Baxter algebras, as an associative analogue of the modified classical Yang-Baxter equation. We construct free commutative modified Rota-Baxter algebras by a variation of the shuffle product and describe the structure both recursively and explicitly. We then provide these algebras with a Hopf algebra structure by applying a Hochschild cocycle.
From the four normed division algebras--the real numbers, complex numbers, quaternions and octonions, of dimension k=1, 2, 4 and 8, respectively--a systematic procedure gives a 3-cocycle on the Poincare superalgebra in dimensions k+2=3, 4, 6 and 10, and a 4-cocycle on the Poincare superalgebra in dimensions k+3=4, 5, 7 and 11. The existence of these cocycles follow from spinor identities that hold only in these dimensions, and which are closely related to the existence of the superstring in dimensions k+2, and the super-2-brane in dimensions k+3. In general, an (n+1)-cocycle on a Lie superalge…
In this paper we establish a Gröbner-Shirshov bases theory for Lie algebras over commutative rings. As applications we give some new examples of special Lie algebras (those embeddable in associative algebras over the same ring) and non-special Lie algebras (following a suggestion of P.M. Cohn (1963) \cite{Conh}). In particular, Cohn's Lie algebras over the characteristic $p$ are non-special when $p=2,\ 3,\ 5$. We present an algorithm that one can check for any $p$, whether Cohn's Lie algebras is non-special. Also we prove that any finitely or countably generated Lie algebra is embeddable in a …
One of the natural ways to prove that the Hall words (Philip Hall, 1933) consist of a basis of a free Lie algebra is a direct construction: to start with a linear space spanned by Hall words, to define the Lie product of Hall words, and then to check that the product yields the Lie identities (Marshall Hall, 1950). Here we suggest another way using the Composition-Diamond lemma for free anti-commutative (non-associative) algebras (A.I. Shirshov, 1962).
We show that for a given nilpotent Lie algebra $\mathfrak{g}$ with $Z(\mathfrak{g})\subseteq [\mathfrak{g},\mathfrak{g}]$ all commutative post-Lie algebra structures, or CPA-structures, on $\mathfrak{g}$ are complete. This means that all left and all right multiplication operators in the algebra are nilpotent. Then we study CPA-structures on free-nilpotent Lie algebras $F_{g,c}$ and discover a strong relationship to solving systems of linear equations of type $[x,u]+[y,v]=0$ for generator pairs $x,y\in F_{g,c}$. We use results of Remeslennikov and Stöhr concerning these equations to prove that…
A bialgebra theory for transposed Poisson algebras via anti-pre-Lie bialgebras and anti-pre-Lie-Poisson bialgebras
本文通过引入反预李双代数和反预李-泊松双代数，利用李代数上交换2-上循环的Manin三元组，为转置泊松代数建立了双代数理论，并研究了共边情形及相关结构。
In the present paper we prove that every 2-local inner derivation on the Lie ring of skew-adjoint matrices over a commutative $*$-ring is an inner derivation. We also apply our technique to various Lie algebras of infinite-dimensional skew-adjoint matrix-valued maps on a set and prove that every 2-local spatial derivation on such algebras is a spatial derivation. We also show that every local spatial derivation on the above Lie algebras is a derivation.
We consider the three-dimensional associative algebra H consisting of the 3x3 strictly upper triangular matrices whose the commutator is the Heisenberg Lie algebra. We determine the solutions of the Yang-Baxter associative equation in H. For the antisymmetric solutions, the corresponding bialgebraic structures, double constructions of Frobenius algebras and properties are given explicitly.Besides, we determine some related compatible dendriform algebras and solutions of their D-equations. Using symmetric solutions of these equations, we build the double constructions of related Connes cocycles…
The algebraic and geometric classification of commutative post-Lie algebras
本文从代数和几何角度研究交换后李代数的分类，发现其代数结构特性并给出3、4维nilpotent CPA的分类结果。
In this work, we extend the definition of the graded prime ideals from those in commutative graded rings to the ideals over graded Lie algebras. We prove some facts about graded prime Lie ideals in arbitrary Lie algebras that are similar to those about graded prime ideals over a commutative or non-commutative ring.In addition, the ideas of graded semiprime Lie ideals and graded total prime Lie ideals will be introduced.
We discuss a version of the Chevalley--Eilenberg cohomology in characteristic $2$, where the alternating cochains are replaced by symmetric ones.
This is a survey of results on partially commutative groups and partially commutative algebras.
1-cocycles of the Witt algebra with coefficients in tensor product of modules
本文分类系数在两个任意张量密度模张量积中的 Witt 代数 1-上循环，恢复 Ng–Taft 定理，并据此确定若干含 Witt 代数的无限维李代数的李双代数结构。
Twisting $\mathcal{O}$-operators by $(2,3)$-Cocycle of Hom-Lie-Yamaguti Algebras with Representations
本文在 Hom-Lie-Yamaguti 代数上引入扭变 $\mathcal{O}$-算子，证明其诱导 Hom-Lie-Yamaguti 结构，定义其上同调，并研究其底层 Hom-NS-Lie-Yamaguti 代数及由 Hom-Lie 代数诱导的情形。
In this paper, we find a criterium for universal equivalence of partially commutative Lie algebras whose defining graphs are trees. Besides, we obtain bases for partially commutative metabelian Lie algebras.
We consider some special type extensions of an arbitrary Lie algebra, which we call universal extensions. We show that these extensions are in one-to-one correspondence with finite dimensional associative commutative algebras. We also construct a special kind of these extensions, that correspond to a finite commutative monoids.
In this paper, linear bases for the partially commutative Lie algebras are found. The method of the Gröbner--Shirshov bases is used. It easily follows from the structure that the equality problem is algorithmically solvable for the partially commutative Lie algebras.
Starting from the four normed division algebras - the real numbers, complex numbers, quaternions and octonions - a systematic procedure gives a 3-cocycle on the Poincare Lie superalgebra in dimensions 3, 4, 6 and 10. A related procedure gives a 4-cocycle on the Poincare Lie superalgebra in dimensions 4, 5, 7 and 11. In general, an (n+1)-cocycle on a Lie superalgebra yields a "Lie n-superalgebra": that is, roughly speaking, an n-term chain complex equipped with a bracket satisfying the axioms of a Lie superalgebra up to chain homotopy. We thus obtain Lie 2-superalgebras extending the Poincare s…
We obtain the functions that bound the dimensions of finite dimensional nilpotent associative or Lie algebras of class 2 over an algebraically closed field in terms of the dimensions of their commutative subalgebras. As a result, we also compute a similar function for complex nilpotent Lie groups of class 2.
We consider the Casimir Invariants related to some a special kind of Lie-algebra extensions, called universal extensions. We show that these invariants can be studied using the equivalence between the universal extensions and the commutative algebras and consider in detail the so called coextension structures, arising in the calculation of the Casimir functions.
In this paper, we study the structure of 3-Lie algebras with involutive derivations. We prove that if $A$ is an $m$-dimensional 3-Lie algebra with an involutive derivation $D$, then there exists a compatible 3-pre-Lie algebra $(A, \{ , , , \}_D)$ such that $A$ is the sub-adjacent 3-Lie algebra, and there is a local cocycle $3$-Lie bialgebraic structure on the $2m$-dimensional semi-direct product 3-Lie algebra $A\ltimes_{ad^*} A^*$, which is associated to the adjoint representation $(A, ad)$. By means of involutive derivations, the skew-symmetric solution of the 3-Lie classical Yang-Baxter equa…
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.
Recent work applying higher gauge theory to the superstring has indicated the presence of `higher symmetry'. Infinitesimally, this is realized by a `Lie 2-superalgebra' extending the Poincare superalgebra in precisely the dimensions where the classical supersymmetric string makes sense: 3, 4, 6 and 10. In the previous paper in this series, we constructed this Lie 2-superalgebra using the normed division algebras. In this paper, we use an elegant geometric technique to integrate this Lie 2-superalgebra to a `Lie 2-supergroup' extending the Poincare supergroup in the same dimensions. Briefly, a …
Non-commutative torsors (equivalently, two-cocycles) for a Hopf algebra can be used to twist comodule algebras. After surveying and extending the literature on the subject, we prove a theorem that affords a presentation by generators and relations for the algebras obtained by such twisting. We give a number of examples, including new constructions of the quantum affine spaces and the quantum tori.
Motivated by the Hopf algebra structures established on free commutative Rota-Baxter algebras, we explore Hopf algebra related structures on free commutative Nijenhuis algebras. Applying a cocycle condition, we first prove that a free commutative Nijenhuis algebra on a left counital bialgebra (in the sense that the right-sided counicity needs not hold) can be enriched to a left counital bialgebra. We then establish a general result that a connected graded left counital bialgebra is a left counital right antipode Hopf algebra in the sense that the antipode is also only right-sided. We finally a…
We establish results about the second cohomology with coefficients in the trivial module, symmetric invariant bilinear forms and derivations of a Lie algebra extended over a commutative associative algebra without unit. These results provide a simple unified approach to a number of questions treated earlier in completely separated ways: periodization of semisimple Lie algebras (Anna Larsson), derivation algebras, with prescribed semisimple part, of nilpotent Lie algebras (Benoist), and presentations of affine Kac-Moody algebras.
Let $\mathbb{K}$ be an infinite field. We prove that if a variety of anti-commutative $\mathbb{K}$-algebras - not necessarily associative, where $xx=0$ is an identity - is locally algebraically cartesian closed, then it must be a variety of Lie algebras over $\mathbb{K}$. In particular, $\mathsf{Lie}_{\mathbb{K}}$ is the largest such. Thus, for a given variety of anti-commutative $\mathbb{K}$-algebras, the Jacobi identity becomes equivalent to a categorical condition: it is an identity in~$\mathcal{V}$ if and only if $\mathcal{V}$ is a subvariety of a locally algebraically cartesian closed var…
Anti-Leibniz algebras: A non-commutative version of mock-Lie algebra
本文提出反莱布尼茨代数作为模拟李代数的非交换推广，完成低维分类，并利用平均算子与嵌入张量构造出反结合二代数、反结合三代数及反莱布尼茨三代数等新结构。
Let $R$ be a commutative integral unital domain and $L$ a free non-commutative Lie algebra over $R$. In this paper we show that the ring $R$ and its action on $L$ are 0-interpretable in $L$, viewed as a ring with the standard ring language $+, \cdot,0$. Furthermore, if $R$ has characteristic zero then we prove that the elementary theory $Th(L)$ of $L$ in the standard ring language is undecidable. To do so we show that the arithmetic ${\bf N} = \langle{\bf N}, +,\cdot,0 \rangle$ is 0-interpretable in $L$. This implies that the theory of $Th(L)$ has the independence property. These results answe…
Manifolds of Lie-Group-Valued Cocycles and Discrete Cohomology
本文证明紧群作用下的李群值相对连续上循环空间构成解析子流形，并分解为纤维丛，由此得出上同调空间离散，并应用于态射空间与射影表示空间的流形结构。
Multi-point algebras of Krichever Novikov type for higher genus Riemann surfaces are generalisations of the Virasoro algebra and its related algebras. Complete existence and uniqueness results for local 2-cocycles defining almost-graded central extensions of the functions algebra, the vector field algebra, and the differential operator algebra (of degree \le 1) are shown. This is applied to the higher genus, multi-point affine algebras to obtain uniqueness for almost-graded central extensions of the current algebra of a simple finite-dimensional Lie algebra. An earlier conjecture of the author…
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