Minimal faithful representations of reductive Lie algebras

Let G and H be Lie groups with Lie algebras and . Let G be connected. We prove that a Lie algebra homomorphism is exact if and only if it is completely positive. The main resource is a corresponding theorem about representations on Hilbert spaces. This article summarizes the main results of [1]. Received: 6 December 2005     

Minimal prime ideals in enveloping algebras of Lie superalgebras

Let be a finite dimensional Lie superalgebra over a field of characteristic zero. Let be the enveloping algebra of . We show that when , then is not semiprime, but it has a unique minimal prime ideal; it follows then that when is classically simple, has a unique minimal prime ideal. We further show that when is a finite dimensional nilpotent Lie superalgebra, then has a unique minimal prime ideal.

     

Differential algebras and simple Jordan superalgebras

In [14], a new example is constructed of a unital simple special Jordan superalgebra J over the field of reals. It turns out that J is a subsuperalgebra of a Jordan superalgebra of vector type but it cannot be isomorphic to a superalgebra of such a type. Moreover, the superalgebra of fractions of J is isomorphic to a Jordan superalgebra of vector type. In the present article, we find a similar example of a Jordan superalgebra. It is constructed over a field of characteristic 0 in which the equation t ² + 1 = 0 has no solutions.     

Jordan李超代数的一些性质

代数A称为不可分解的,如果A不能分解成理想的直和.证明了满足C(L_o)=C(L)={0}的Jordan李超代数L一些重要性质.     

Color Lie superalgebras and hopf algebras

We give a generalization of the well-known theorem stating that the category of primitively generated Hopf algebras is equivalent to the category of (restricted) Lie algebras. In so doing, instead of Lie algebras, we consider color Lie superalgebras, and instead of a primitively generated Hopf algebra, we take a Hopf algebra H whose semigroup elements form an Abelian group G =G(H), and H is generated by its relatively primitive elements which supercommute with the elements of G. Translated fromAlgebra i Logika, Vol. 34, No. 4, pp. 420–436, July-August, 1995.Supported by the Russian Foundation for Fundamental Research, grant No. 93-01-16171.     

Invariant ideals of abelian group algebras and representations of groups of Lie type

This paper contributes to the general study of ideal lattices in group algebras of infinite groups. In recent years, the second author has extensively studied this problem for an infinite locally finite simple group. It now appears that the next stage in the general problem is the case of abelian-by-simple groups. Some basic results reduce this problem to that of characterizing the ideals of abelian group algebras stable under certain (simple) automorphism groups. Here we begin the analysis in the case where the abelian group is the additive group of a finite-dimensional vector space over a locally finite field of prime characteristic , and the automorphism group is a simple infinite absolutely irreducible subgroup of . Thus is isomorphic to an infinite simple periodic group of Lie type, and is realized in via a twisted tensor product of infinitesimally irreducible representations. If is a Sylow -subgroup of and if is the unique line in stabilized by , then the approach here requires a precise understanding of the linear character associated with the action of a maximal torus on . At present, we are able to handle the case where is a rational representation with character field equal to .

     

Invariants of Lie superalgebras acting on associative algebras

LetR be a semiprime algebra over a fieldK acted on by a finite-dimensional Lie superalgebraL. The purpose of this paper is to prove a series of going-up results showing how the structure of the subalgebra of invariantsR ^Lis related to that ofR. Combining several of our main results we have: Theorem: Let R be a semiprime K-algebra acted on by a finite-dimensional nilpotent Lie superalgebra L such that if characteristic K=p then L is restricted and if characteristic, K=0 then L acts on R as algebraic derivations and algebraic superderivations.

1. If R^L is right Noetherian, then R is a Noetherian right R^L-module. In particular, R is right Noetherian and is a finitely generated right R^L-module.
2. If R^L is right Artinian, then R is an Artinian right R^L-module. In particular, R is right Artinian and is a finitely generated right R^L-module.
3. If R^L is finite-dimensional over K then R is also finite-dimensional over K.
4. If R^L has finite Goldie dimension as a right R^L-module, then R has finite Goldie dimension as a right R-module.
5. If R^L has Krull dimension α as a right R^L-module, then R has Krull dimension α as a right R^L-module. Thus R has Krull dimension at most α as a right R-module.
6. If R is prime and R^L is central, then R satisfies a polynomial identity.
7. If L is a Lie algebra and R^L is central, then R satisfies a polynomial identity.

We also provide counterexamples to many questions which arise in view of the results in this paper.     

Minimal codes in abelian group algebras

In this paper we show that two minimal codes $\text{M}$₁ and $\text{M}$₂ in the group algebra $\text{F}$₂[G] have the same (Hamming) weight distribution if and only if there exists an automorphism θ of G whose linear extension to $\text{F}$₂[G] maps $\text{M}$₁ onto $\text{M}$₂. If θ(M₁) = M₂, then $\text{M}$₁ and $\text{M}$₂ are called equivalent. We also show that there are exactly τ(l) inequivalent minimal codes in $\text{F}$₂[G], where ? is the exponent of G, and τ(?) is the number of divisors of ?.     

Toroidal Lie algebras and vertex representations

The paper describes the theory of the toroidal Lie algebra, i.e. the Lie algebra of polynomial maps of a complex torus ^××^× into a finite-dimensional simple Lie algebra g. We describe the universal central extension t of this algebra and give an abstract presentation for it in terms of generators and relations involving the extended Cartan matrix of g. Using this presentation and vertex operators we obtain a large class of integrable indecomposable representations of t in the case that g is of type A, D, or E. The submodule structure of these indecomposable modules is described in terms of the ideal structure of a suitable commutative associative algebra.To Professor J. Tits for his sixtieth birthday     

Some Lie superalgebras associated to the Weyl algebras

Let be the Lie superalgebra . We show that there is a surjective homomorphism from to the Weyl algebra , and we use this to construct an analog of the Joseph ideal. We also obtain a decomposition of the adjoint representation of on and use this to show that if is made into a Lie superalgebra using its natural -grading, then . In addition, we show that if and are isomorphic as Lie superalgebras, then . This answers a question of S. Montgomery.

     

Vertex operator algebras and representations of affine Lie algebras

We announce the construction of an explicit basis for all integrable highest weight modules over the Lie algebra A ₁ ⁽¹⁾. The construction uses representations of vertex operator algebras and leads to combinatorial identities of Rogers-Ramanujan-type.     

Irreducible representations of infinite-dimensional Lie algebras

Some results concerning the operatorially irreducible, topologically irreducible, operatorially and topologically irreducible representations of a class of infinite-dimensional Lie algebras by bounded operators in a complex Banach space are obtained.