The cohomology of the Heisenberg Lie algebras over fields of finite characteristic

We give explicit formulas for the cohomology of the Heisenberg Lie algebras over fields of finite characteristic. We use this to show that in characteristic two, unlike all other cases, the Betti numbers are unimodal.

     

Metric Lie algebras with maximal isotropic centre

A metric Lie algebra is a Lie algebra equipped with an invariant non-degenerate symmetric bilinear form. It is called indecomposable if it is not the direct sum of two metric Lie algebras. We are interested in describing the isomorphism classes of indecomposable metric Lie algebras. In the present paper we restrict ourselves to a certain class of solvable metric Lie algebras which includes all indecomposable metric Lie algebras with maximal isotropic centre. We will see that each metric Lie algebra belonging to this class is a twofold extension associated with an orthogonal representation of an abelian Lie algebra. We will describe equivalence classes of such extensions by a certain cohomology set. In particular we obtain a classification scheme for indecomposable metric Lie algebras with maximal isotropic centre and the classification of metric Lie algebras of index 2.     

Cohomology of Oriented Tree Diagram Lie Algebras

Xu introduced a family of root-tree-diagram nilpotent Lie algebras of differential operators, in connection with evolution partial differential equations. We generalized his notion to more general oriented tree diagrams. These algebras are natural analogues of the maximal nilpotent Lie subalgebras of finite-dimensional simple Lie algebras. In this article, we use Hodge Laplacian to study the cohomology of these Lie algebras. The “total rank conjecture” and “b ₂-conjecture” for the algebras are proved. Moreover, we find the generating functions of the Betti numbers by means of Young tableaux for the Lie algebras associated with certain tree diagrams of single branch point. By these functions and Euler–Poincaré principle, we obtain analogues of the denominator identity for finite-dimensional simple Lie algebras. The result is a natural generalization of the Bott's classical result in the case of special linear Lie algebras.     

A Program for Computing the Cohomology of Lie Superalgebras of Vector Fields

The cohomology of Lie (super)algebras has many important applications in mathematics and physics. At present, since the required algebraic computations are very tedious, the cohomology is explicitly computed only in a few cases for different classes of Lie (super)algebras. That is why application of computer algebra methods is important for this problem. We describe an algorithm (and its C implementation) for computing the cohomology of Lie algebras and superalgebras. In elaborating the algorithm, we focused mainly on the cohomology with coefficients in trivial, adjoint, and coadjoint modules for Lie (super)algebras of the formal vector fields. These algebras have many applications to modern supersymmetric models of theoretical and mathematical physics. As an example, we consider the cohomology of the Poisson algebra Po(2) with coefficients in the trivial module and present 3- and 5-cocycles found by a computer. Bibliography: 6 titles.     

Isomorphism classes of finite dimensional connected Hopf algebras in positive characteristic

We classify all finite-dimensional connected Hopf algebras with large abelian primitive spaces. We show that they are Hopf algebra extensions of restricted enveloping algebras of certain restricted Lie algebras. For any abelian matched pair associated with these extensions, we construct a cohomology group, which classifies all the extensions up to equivalence. Moreover, we present a 1–1 correspondence between the isomorphism classes and a group quotient of the cohomology group deleting some exceptional points, where the group respects the automorphisms of the abelian matched pair and the exceptional points represent those restricted Lie algebra extensions.     

Cohomology of graded lie algebras of maximal class with coefficients in the adjoint representation

We compute explicitly the adjoint cohomology of two ℕ-graded Lie algebras of maximal class (infinite-dimensional filiform Lie algebras) m₀ and m₂. It is known that up to an isomorphism there are only three ℕ-graded Lie algebras of maximal class. The third algebra from this list is the “positive” part L ₁ of the Witt (or Virasoro) algebra, and its adjoint cohomology was computed earlier by Feigin and Fuchs. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 263, pp. 106–119.     

On non-Abelian Leibniz cohomology

In this note, by using a generalized notion of the Leibniz algebra of derivations, we present the constructions of the zero, first, and second non-Abelian Leibniz cohomologies with coefficients in crossed modules, which generalize the classical zero, first, and second Leibniz cohomology. For Lie algebras we compare the non-Abelian Leibniz and Lie cohomologies. We describe the second non-Abelian Leibniz cohomology via extensions of Leibniz algebras by crossed modules.     

On the Semisimplicity of the Outer Derivations of Monomial Algebras

We show that the Hochschild cohomology of a monomial algebra over a field of characteristic zero vanishes from degree two if the first Hochschild cohomology is semisimple as a Lie algebra. We also prove that first Hochschild cohomology of a radical square zero algebra is reductive as a Lie algebra. In the case of the multiple loops quiver, we obtain the Lie algebra of square matrices of size equal to the number of loops.     

Low Dimensional Cohomology of Hom-Lie Algebras and q-deformed W（2, 2） Algebra

This paper aims to study low dimensional cohomology of Hom-Lie algebras and the qdeformed W(2, 2) algebra. We show that the q-deformed W(2, 2) algebra is a Hom-Lie algebra. Also,we establish a one-to-one correspondence between the equivalence classes of one-dimensional central extensions of a Hom-Lie algebra and its second cohomology group, leading us to determine the second cohomology group of the q-deformed W(2, 2) algebra. In addition, we generalize some results of derivations of finitely generated Lie algebras with values in graded modules to Hom-Lie algebras.As application, we compute all αk-derivations and in particular the first cohomology group of the q-deformed W(2, 2) algebra.     

ON THE STRUCTURE OF COHOMOLOGY RINGS OF p-NILPOTENT LIE ALGEBRAS

In this paper the authors investigate the structure of the restricted Lie algebra cohomology of p-nilpotent Lie algebras with trivial p-power operation. Our study is facilitated by a spectral sequence whose E ₂-term is the tensor product of the symmetric algebra on the dual of the Lie algebra with the ordinary Lie algebra cohomology and converges to the restricted cohomology ring. In many cases this spectral sequence collapses, and thus, the restricted Lie algebra cohomology is Cohen–Macaulay. A stronger result involves the collapsing of the spectral sequence and the cohomology ring identifying as a ring with the E ₂-term. We present criteria for the collapsing of this spectral sequence and provide some examples where the ring isomorphism fails. Furthermore, we show that there are instances when the spectral sequence does not collapse and yields cohomology rings which are not Cohen-Macaulay.     

On Exact Courant Algebras

In this note, we will show that exact Courant algebras over a Lie algebra 𝔤 can be characterized via Leibniz 2-cocycles, and the automorphism group of a given exact Courant algebra is in a one-to-one correspondence with first Leibniz cohomology space of 𝔤.     

Metric Lie algebras and quadratic extensions

The present paper contains a systematic study of the structure of metric Lie algebras, i.e., finite-dimensional real Lie algebras equipped with a nondegenerate invariant symmetric bilinear form. We show that any metric Lie algebra g without simple ideals has the structure of a so called balanced quadratic extension of an auxiliary Lie algebra l by an orthogonal l-module a in a canonical way. Identifying equivalence classes of quadratic extensions of l by a with a certain cohomology set H²_Q(l,a), we obtain a classification scheme for general metric Lie algebras and a complete classification of metric Lie algebras of index 3.     

着色李超代数与左着色对称结构

本文研究了着色李超代数上的左着色对称结构问题.利用着色李超代数的两种仿射表示和1-上同调群,得出左着色对称结构存在的几个充分或必要条件,推广了文[2]的结论.     

A note on Sen��s theory in the imperfect residue field case

In Sen??s theory in the imperfect residue field case, Brinon defined a functor from the category of ${{\mathbb C}_p}$ -representations to the category of linear representations of a certain Lie algebra. We give a comparison theorem between the continuous Galois cohomology of ${{\mathbb C}_p}$ -representations and the Lie algebra cohomology of the associated representations. The key ingredients of the proof are Hyodo??s calculation of Galois cohomology and the effaceability of Lie algebra cohomology for solvable Lie algebras.     

DEGENERATIONS OF 7-DIMENSIONAL NILPOTENT LIE ALGEBRAS

ABSTRACT

We study the varieties of Lie algebra laws and their subvarieties of nilpotent Lie algebra laws. We classify all degenerations of (almost all) five-step and six-step nilpotent seven-dimensional complex Lie algebras. One of the main tools is the use of trivial and adjoint cohomology of these algebras. In addition, we give some new results on the varieties of complex Lie algebra laws in low dimension.     

Foundations of the Theory of Dual Lie Algebras

In this paper we introduce and study dual Lie algebras, i.e., Lie algebras over the algebra of dual numbers. We establish some fundamental properties of such Lie algebras and compare them with the corresponding properties of real and complex Lie algebras. We discuss the question of classification of dual Lie algebras of small dimension and consider the connection of dual Lie algebras with approximate Lie algebras.     

Novikov algebras and Novikov structures on Lie algebras

We study ideals of Novikov algebras and Novikov structures on finite-dimensional Lie algebras. We present the first example of a three-step nilpotent Lie algebra which does not admit a Novikov structure. On the other hand we show that any free three-step nilpotent Lie algebra admits a Novikov structure. We study the existence question also for Lie algebras of triangular matrices. Finally we show that there are families of Lie algebras of arbitrary high solvability class which admit Novikov structures.     

Solvable quadratic Lie algebras

A Lie algebra endowed with a nondegenerate, symmetric, invariant bilinear form is called a quadratic Lie algebra. In this paper, the author investigates the structure of solvable quadratic Lie algebras, in particular, the solvable quadratic Lie algebras whose Cartan subalgebras consist of semi-simple elements, the author presents a procedure to construct a class of quadratic Lie algebras from the point of view of cohomology and shows that all solvable quadratic Lie algebras can be obtained in this way.     

Hochschild Cohomology of <Emphasis Type="Italic">q</Emphasis>-Schur Algebras

We compute the Hochschild cohomology of any block of q-Schur algebras. We focus on the even part of this Hochschild cohomology ring. To compute the Hochschild cohomology of q-Schur algebras, we prove the following two results: first, we construct two graded algebra surjections between the Hochschild cohomologies of quasi-hereditary algebras because all q-Schur algebras over a field are quasi-hereditary. Second, we give the graded algebra isomorphism of Hochschild cohomologies by using a certain derive equivalence.