A class of non‐graded left‐symmetric algebraic structures on the Witt algebra

We classify the compatible left‐symmetric algebraic structures on the Witt algebra satisfying certain non‐graded conditions. It is unexpected that they are Novikov algebras. Furthermore, as applications, we study the induced non‐graded modules of the Witt algebra and the induced Lie algebras by Novikov‐Poisson algebras’ approach and Balinskii‐Novikov's construction.     

Left-symmetric algebra structures on the twisted Heisenberg-Virasoro algebra

The compatible left-symmetric algebra structures on the twisted Heisenberg-Virasoro algebra with some natural grading conditions are completely determined. The results of the earlier work on left-symmetric algebra structures on the Virasoro algebra play an essential role in determining these compatible structures. As a corollary, any such left-symmetric algebra contains an infinite-dimensional nontrivial subalgebra that is also a submodule of the regular module.     

Length two extensions of modules for the Witt algebra

In this paper, we establish an explicit classification of length two extensions of tensor modules for the Witt algebra using the cohomology of the Witt algebra with coe?cients in the module of the space of homomorphisms between the two modules of interest. To do this we extended our module to a module that has a compatible action of the commutative algebra of Laurent polynomials in one variable. In this setting, we are be able to directly compute all possible 1-cocycles.     

Commuting variety of Witt algebra

Let \(\mathfrak{g} = W_1 \) be the Witt algebra over an algebraically closed field k of characteristic p > 3; and let Open image in new window be the commuting variety of g. In contrast with the case of classical Lie algebras of P. Levy [J. Algebra, 2002, 250: 473–484], we show that the variety Open image in new window is reducible, and not equidimensional. Irreducible components of Open image in new window and their dimensions are precisely given. As a consequence, the variety Open image in new window is not normal.     

Witt型与Virasoro型Lie双代数的对偶

本文给出单边的Witt 型、Witt 型和Virasoro 型Lie 双代数的对偶Lie 双代数结构. 由此, 本文得到一系列无限维Lie 代数.     

关于Witt代数基本子代数的一点注记

设g=W_1是特征p3的代数闭域k上的Witt代数.本文确定了g的极大基本子代数.进一步具体给出了最大维数的基本子代数的G共轭类,这里G是g的自同构群.从而证明了最大维数为(p-1)/2的基本子代数射影簇E((p-1)/2,g)是不可约的且是一维的.更进一步,证明了E(1,g)是不可约的,具有维数p-2,而E(2,g)是等维的,共有(p-3)/2个不可约分支,且每个不可约分支的维数是p-4.而当3≤r≤(p-3)/2时,E(r,g)是可约的.给出了E(r,g)(3≤r≤(p-3)/2)维数的一个下界.     

A note on the fundamental theorem of algebra for the octonions

This paper is a revision of a portion of the author's doctoral dissertation submitted to the University of Oregon. Using elementary concepts of K$K$-theory, the Brouwer degree of the power map in the octonions is computed. Later, a proof of a weaker version of the fundamental theorem of algebra for polynomials with coefficients in the octonions is given. As a partial complement, a lower bound to the number of solutions of a homogeneous monomial equation over the octonions is obtained.     

On the Restricted Lie Algebra Structure of the Witt Lie Algebra in Finite Characteristic

The article contains an explicit formula for the restricted Lie algebra structure in the Witt Lie algebra over a field of finite characteristic. Some combinatorial lemmas can be of independent interest.     

Witt代数和Virasoro代数上的Poisson代数结构

Poisson代数是指同时具有结合代数结构和李代数结构的一类代数,其结合代数结构和李代数结构满足Leibniz法则.确定了特征为0和特征为p>0的基域上的Witt代数和Virasoro代数上的Poisson代数结构.     

K-theory for the maximal Roe algebra of certain expanders

We study in this paper the maximal version of the coarse Baum-Connes assembly map for families of expanding graphs arising from residually finite groups. Unlike for the usual Roe algebra, we show that this assembly map is closely related to the (maximal) Baum-Connes assembly map for the group and is an isomorphism for a class of expanders. We also introduce a quantitative Baum-Connes assembly map and discuss its connections to K-theory of (maximal) Roe 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.     

Lie bialgebras of generalized Witt type

In this paper, all Lie bialgebra structures on the Lie algebras of generalized Witt type are considered. It is proved that, for any Lie algebra W of generalized Witt type, all Lie bialgebras on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group H1(W, W(?)W) is trivial.     

Nilpotent Orbits in the Witt Algebra W 1

In this article, the nilpotent orbits of the Witt algebra W ₁ are determined under the automorphism group over an algebraically closed field F of characteristic p > 3. In contrast with a finite number of nilpotent orbits in a classical simple Lie algebra (cf. [5 Jantzen , J. C. ( 2003 ). Nilpotent orbits in representation theory . Progr. Math. 1 – 211 . [Google Scholar]]), there is an infinite number of nilpotent orbits in W ₁. A set of representatives of nilpotent orbits, as well as their dimensions, are precisely presented.     

Projective push-forwards in the Witt theory of algebraic varieties

We define push-forwards along projective morphisms in the Witt theory of smooth quasi-projective varieties over a field. We prove that they have standard properties such as functoriality, compatibility with pull-backs and projection formulas.