Lie Algebras Of Cartan Type Research Articles

Automorphisms of the Lie Algebras W* in Characteristic 0

In a recent paper [2] we defined four classes of infinite dimensional simple Lie algebras over a field of characteristic 0 which we called W*, S*, H*, and K*. As the names suggest, these classes generalize the Lie algebras of Cartan type. A second paper [3] investigates the derivations of the algebras W* and S*, and the possible isomorphisms between these algebras and the algebras defined by Block [1]. In the present paper we investigate the automorphisms of the algebras of type W*.

The realizations of primitive p-envelopes and the support varieties for graded cartan type lie algebras

Let L be a graded Lie algebra of Cartan type over an algebraically closed field of characteristic p≧3, which has been proved to be generalized restricted in the sense of [Shul, Shu2]. For a generalized restricted L-module M, the homological support variety ‖L‖M is defined to be that of the primitive p-envelope P{L). A realization L of P(L) is given in Der(&(m : n)). Furthermore, a class of generalized restricted highest weight L-modules lift to Dist(Tx)V(p)-module structures and their support varieties can be computed by using algebraic group techniques developed in [LN].

Virasoro type Lie algebras and deformations

The first and second cohomologies of Cartan Type Lie algebras with coefficients in irreducible tensor modules are calculated. The spaceH 1(L, U) is interpreted as a space of deformations of (L, U)-modules.H 2(L, L)≠0 ifL=S 2,S 2 + orL=H n ,H + . Lie algebra of divergenceless vector fieldsS 2 + has only one nontrivial local deformation. The two-sided simple hamiltonian algebraH n has 2n 2+n new local deformations in addition to Moyal cocycle. The Lie algebrasL=W n (n>3),S n−1(n>2),H n (n>1),K n+1(n>1) have 3, 1, 1, 3 nonisomorphic tensor modules with irreducible bases and nonzero 1-cohomologies; respectively, the corresponding numbers for 2-cohomologies are 9, 6, 7 and 9.

Translative Isomorphisms and Left-Symmetric Structures onW(m, n)

LetW≔W(m,n) be the general Lie algebra of Cartan type and A≔A(m,n) the corresponding divided power algebra on whichWacts. Translative isomorphisms, i.e., isomorphisms ofWintoW∔gl(m,A) of the formA↦A+ϕ(A),A∈W, ϕ(A)∈gl(m,A), are investigated. The mixed product theorems of the author (1986,Sci. Sinica Ser. A29, No. 6, 570–581) are generalized. Under certain conditions, translative isomorphisms induce left-symmetric structures onW. Classes of left-symmetric structures onWare constructed, among which are restricted left-symmetric structures on the restricted Lie algebraW(m,1).

Algebraic Group Actions in the Cohomology Theory of Lie Algebras of Cartan Type

Given a Lie algebra g of Cartan type we construct an infinite-dimensional cocommutative Hopf algebraD(G)u(g) which is the analog of the distribution algebra of a connected reductive algebraic group. We show that the simpleu(g)-modules lift to aD(G)u(g)-structure. This additional structure is used to formulate and prove relative projectivity theorems for Lie algebras of Cartan type. The support varieties of certainD(G)u(g)-modules are computed by using these algebraic group techniques.

On the construction of indecomposable modules over restricted enveloping algebras

In this paper the authors provide a procedure for constructing indecomposable modules for restricted enveloping algebras of classical and Cartan type Lie algebras. It is shown that for dimensions which are divisible by multiples of a certain prime power order there are infinite families of indecomposable modules of that specified dimension.

The nilpotency class of the sandwich subalgebra of simple finite-dimensional Lie algebras

In this paper it is shown that the nilpotency class of sandwich subalgebras in Lie algebras of Cartan type and in Melikyan algebras over a field of characteristic is equal to , where is the sum of the heights of the variables. The only exception is the Zassenhaus algebra (and also the Hamiltonian and contact algebras for ).

Invariant filtrations of Lie algebras of Cartan type over commutative rings

The work of the classification of the simple Lie algebras over algebraical closed fieldF of prime characteristc p began nearly 50 years ago and the problem has been solved just recentlyunder the condition p>7,and the generalized Kostrikin-Safarevic conjecture that these al-gebras are all of classical or Cartan type is verified.For the classification of simple Liealgebras over non-closed fields,it is necessary to consider the forms.A central simple Lie

Non-splittable extensions of a general Lie algebra of Cartan type

In this paper we compute the extensions of a general Lie algebra of Cartan type in two variables by irreducible modules. Part of the results covers a still more general case.

ON SYMMETRIC INVARIANTS OF SOME MODULAR LIE ALGEBRAS

For the Lie algebras of Cartan type , , a simple condition is obtained that must be satisfied by every nontrivial invariant polynomial function on the coadjoint module for the algebra. Bibliography: 8 titles.

Automorphisms and forms of lie algebras of cartan type

Let L be any of simple Lie algebras of Cartan type L = X(2)(m : n : Φ), X = 5, H or K, n≠1, over a field F of characteristic p > 3, and R a commutative ring extension of F. Then AutR(R⊗L)≌ Aut R(R⊗21 (m : n) : R⊗L). It follows that, all forms of L are standard and thereby are determined

Left-Symmetric Structures on Simple Modular Lie Algebras

This paper investigates left-symmetric structures on finite-dimensional simple Lie algebras g over a field k. If k is of characteristic 0, then g does not admit any left-symmetric structure. This is known in the theory of affine manifolds. In the modular case, however, such structures may exist. The main purpose of this paper is to show that classical simple Lie algebras of characteristic p > 3 admit left-symmetric structures only in case p divides dim( g ). The proof involves the computation of the first cohomology groups of classical Lie algebras for certain g -modules of small dimension. Here g is regarded as the Lie algebra of a connected semisimple algebraic group over an algebraically closed field of characteristic p > 0. Most of these computations are due to Jantzen. For nonrestricted simple Lie algebras of Cartan type it is shown that many more left-symmetric structures can be found. One studies so-called adjoint structures, induced by nonsingular derivations of g . The simple algebra L ( G, δ, ƒ) of Block of dimension p n − 1, for example, admits adjoint structures for every p > 0. If p = 2, the results are more complicated.

Block Degeneracy and Cartan Invariants for Graded Lie Algebras of Cartan Type

Block Degeneracy and Cartan Invariants for Graded Lie Algebras of Cartan Type

NEW SERIES OF SIMPLE LIE ALGEBRAS OF CHARACTERISTIC 3

Three new series of simple finite-dimensional Lie algebras over a field of characteristic 3 are constructed. For explicit realization, these algebras can be represented as a sum of a Lie algebra of general or special Cartan type and of certain of its tensor modules. The algebras have gradings of depth 2 or 4, with a classical zero component; and the author gives a characterization for them in the class of all graded Lie algebras. In order to prove nonisomorphism of these algebras with the Lie algebras of Cartan type, the subalgebras containing a certain invariant set are studied. With a view of systematizing the known examples of simple finite-dimensional Lie algebras of characteristic 3, analogous realizations of two previously known series are also presented. Bibliography: 20 titles.

Principal Indecomposable Representations for Restricted Lie Algebras of Cartan Type

Let L be any one of W( n, 1), S( n, 1) H( n, 1), and K( n, 1) over an algebraically closed held F of characteristic p > 3. In this paper, we extend the results concerning modular representations of classical Lie algebras and semisimple groups to the case of L and obtain some properties of principal indecomposable modules of u( L) which parallel closely those of classical Lie algebras.

Non-alternating hamiltonian algebraP(n,m)of characteristic two

Over a field F of characteristic p = 2, a class of Lie algebras P(n,m), called non-alternating Hamiltonian algebras, is constructed, where n is a Positive integer and m=(m1, ⋯,mn) is an n-tuple of positive integers. P(n,m) is a graded and filtered subalgebra of the generalized Jacobson-Witt algebra W(n,m) and bears resemblance to the Lie algebras of Cartan type. P(n,m) is shown to be simple unless m=1 and n < 4. The dimension of P(n,m) is Different from the Lie algebras of Cartan type, all P(n,m) are nonrestrictable. The derivation algebra of P(n,m) is determined, and the natural filtration of P(n,m) is proved to be invariant. It is then determined that P(n,m) is a new class of simple Lie algebras if (n,m) satisfies some condition.

Universal graded Lie algebras

For a given negatively graded Lie algebra K −, the universal graded Lie algebra U of type K − is defined and its existence proved. By posing additional conditions on K −, other types of universal graded Lie algebras are defined and discussed. The concept of universal graded Lie algebra leads naturally to the graded Cartan type Lie algebras. In fact, all graded Cartan type Lie algebras can be characterized as certain types of universal graded Lie algebras.

Ad-NILPOTENT ELEMENTS, QUASI-NILPOTENT ELEMENTS AND INVARIANT FILTRATIONS OF INFINITE DIMENSIONAL LIE ALGEBRAS OF CARTAN TYPE

Let F be an arbitrary field of characteristic p≠2, and L be an infinite Lie algebra of Cartan type (graded or complete). When p3 (or p is arbitrary), the set of ad-nilpotent (or quasi-nilpotent) elements of L is determined. Consequently, it is proved that the natural filtration and the noncontractible filtration of L are invariant.

Central extensions and [formula omitted] of the graded lie algebras of cartan type

Let F be an algebraically closed field of characteristic p > 0, L= ⊕ i = − r s L ( i) a graded Lie algebra of Cartan type over F. We give a description of the dual adjoint module L ∗ by means of the mixed product and the coinduced module such that the computation of H 1(L, L ∗) can be reduced to that of the first cohomology of L (0). Then we determine the central extensions and H 2( L, F) by computing H 1(L, L ∗) .

TRUNCATED INDUCED MODULES OVER TRANSITIVE LIE ALGEBRAS OF CHARACTERISTIC $ p$

By constructing truncated coinduced modules a theorem is proved on the minimal imbedding of a transitive Lie algebra over a perfect field into the Lie algebra . A formula for cohomology with coefficients in a truncated induced module is obtained. A description is given of filtered Lie algebras over a perfect field associated with graded Lie algebras of Cartan type and their derivations. Cartan prolongations of truncated induced modules are investigated. Bibliography: 29 titles.