Nilpotent Bicone and Characteristic Submodule of a Reductive Lie Algebra

Let be a finite-dimensional complex reductive Lie algebra and S() its symmetric algebra. The nilpotent bicone of is the subset of elements (x, y) of whose subspace generated by x and y is contained in the nilpotent cone. The nilpotent bicone is naturally endowed with a scheme structure, as nullvariety of the augmentation ideal of the subalgebra of generated by the 2-order polarizations of invariants of . The main result of this paper is that the nilpotent bicone is a complete intersection of dimension , where and are the dimensions of Borel subalgebras and the rank of , respectively. This affirmatively answers a conjecture of Kraft and Wallach concerning the nullcone [KrW2]. In addition, we introduce and study in this paper the characteristic submodule of . The properties of the nilpotent bicone and the characteristic submodule are known to be very important for the understanding of the commuting variety and its ideal of definition. The main difficulty encountered for this work is that the nilpotent bicone is not reduced. To deal with this problem, we introduce an auxiliary reduced variety, the principal bicone. The nilpotent bicone, as well as the principal bicone, are linked to jet schemes. We study their dimensions using arguments from motivic integration. Namely, we follow methods developed by Mustaţǎ in [Mu]. Finally, we give applications of our results to invariant theory.     

On Symmetric Functions Related to Witt Vectors and the Free Lie Algebra

With the notations of Macdonald we define symmetric functions q_n by Eq. (2.1). We conjecture that for n≥2, −q_n is a sum of Schur functions and thus is the characteristic function of some representation of S_n. A first result is the "orthogonality relation" of Theorem 3.1, where l_n is the symmetric function corresponding to the nth free Lie algebra representation. The conjecture is deduced when n is a power of 2 (Corollary 3.5). When n is odd, a Hall basis construction shows that −q_n has positive coefficients (Corollary 4.7); when n is a power of an odd prime, the construction of a functor embedded in the free Lie algebra implies the conjecture in this case (Corollary 5.2).     

Automorphisms Group of an Invariant Subalgebra of a Reductive Lie Algebra

We prove that any automorphism of an invariant subalgebra of a reductive Lie algebra over a field of zero characteristic is a standard automorphism. Mathematics Subject Classifications (2000) 17B20, 17B40.     

On the Generalized Enveloping Algebra of a Color Lie Algebra

Let G be an abelian group, ε an anti-bicharacter of G and L a G-graded ε Lie algebra (color Lie algebra) over a field of characteristic zero. We prove that for all G-graded, positively filtered A such that the associated graded algebra is isomorphic to the G-graded ε-symmetric algebra S(L), there is a G- graded ε-Lie algebra L and a G-graded scalar two cocycle , such that A is isomorphic to U _ω (L) the generalized enveloping algebra of L associated with ω. We also prove there is an isomorphism of graded spaces between the Hochschild cohomology of the generalized universal enveloping algebra U(L) and the generalized cohomology of the color Lie algebra L. Supported by the EC project Liegrits MCRTN 505078.     

On the Simplicity of the Lie Algebra of a Leavitt Path Algebra

For a field F and a row-finite directed graph Γ, let L(Γ) be the associated Leavitt path algebra. We find necessary and sufficient conditions for the Lie algebra [L(Γ), L(Γ)] to be simple.     

New Properties of the Lie Algebra Variety N2A

We continue to consider the properties of the almost polynomial growth variety of Lie algebras over a field of characteristic zero defined by the identity (x ₁ x ₂)(x ₃ x ₄)(x ₅ x ₆)?≡?0. Here we have constructed the bases of its multilinear parts and proved the formulas for the colength and codimension sequences of this variety.     

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.     

On Lie Algebra Crossed Modules

Abstract

The goal of this article is to construct a crossed module representing the cocycle 〈[,],〉 generating H ³(;?) for a simple complex Lie algebra .     

On Ext-transfer for Reductive Lie Algebras

Let G be a connected reductive algebraic group over an algebraically closed field of prime characteristic p and ?? be the Lie algebra of G. In this paper, we study the representations of ?? when p-character has standard Levi form. An Ext-transfer from the Ext-groups of induced ??-modules to its Levi subalgebras is obtained. Furthermore, we reduce the computation of the multiplicities of simple factors in baby Verma modules over ?? to its Levi subalgebras.     

Derivation Algebra of Quasi Rn-filiform Lie Algebra

In this paper we explicitly determine the derivation algebra of a quasi Rn-filiform Lie algebra and prove that a quasi Pn-filiform Lie algebra is a completable nilpotent Lie algebra.     

Derivation Algebra of Quasi $R_n$-Filiform Lie Algebra

In this paper we explicitly determine the derivation algebra of a quasi $R_n$-filiform Lie algebra and prove that a quasi $R_n$-filiform Lie algebra is a completable nilpotent Lie algebra.     

与一类unit型相联系的有限维单李代数的结构

Let n ≥ 4. The complex Lie algebra, which is attached to the unit form q(x1, x2,..., xn)■ and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type Dn, and realized by the Ringel-Hall Lie algebra of a Nakayama algebra. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.     

On the Irreducibility of Commuting Varieties Associated with Involutions of Simple Lie Algebras

Let be a reductive Lie algebra over an algebraically closed field of characteristic zero and an arbitrary -grading. We consider the variety , which is called the commuting variety associated with the -grading. Earlier it was proved by the author that is irreducible, if the -grading is of maximal rank. Now we show that is irreducible for and (E₆,F₄). In the case of symmetric pairs of rank one, we show that the number of irreducible components of is equal to that of nonzero non--regular nilpotent G ₀-orbits in . We also discuss a general problem of the irreducibility of commuting varieties.     

On Strongly Inert Subalgebras of an Infinite-Dimensional Lie Algebra

We study infinite-dimensional Lie algebras L over an arbitrary field that contain a subalgebra A such that dim(A + [A, L])/A < . We prove that if an algebra L is locally finite, then the subalgebra A is contained in a certain ideal I of the Lie algebra L such that dimI/A <. We show that the condition of local finiteness of L is essential in this statement.     

相应于一类对称自对偶李代数的顶点代数结构

设gsp是给定的对称自对偶李代数,给出了相应于gsp的一个顶点代数结构.