Three-dimensional Simple Lie Algebra Research Articles

Classification of finite-dimensional Lie superalgebras whose even part is a three-dimensional simple Lie algebra over a field of characteristic not two or three

Let k be a field of characteristic not two or three. We classify up to isomorphism all finite-dimensional Lie superalgebras over k, where is a three-dimensional simple Lie algebra. If denotes the center of the result is the following: either or or

Colength of the variety generated by a three-dimensional simple Lie algebra

The variety generated by a three-dimensional simple Lie algebra over a field of characteristic zero was studied rather well. The study of this variety is continued in the paper and a formula for calculation of its colengths is presented.

Description of some classes of Leibniz algebras

In this paper we describe the isomorphism classes of finite-dimensional complex Leibniz algebras whose quotient algebra with respect to the ideal generated by squares is isomorphic to the direct sum of three-dimensional simple Lie algebra sl2 and a three-dimensional solvable ideal. We choose a basis of the isomorphism classes’ representatives and give explicit multiplication tables.

Nice parabolic subalgebras of reductive Lie algebras

This paper gives a classification of parabolic subalgebras of simple Lie algebras over C \mathbb {C} that are complexifications of parabolic subalgebras of real forms for which Lynch’s vanishing theorem for generalized Whittaker modules is non-vacuous. The paper also describes normal forms for the admissible characters in the sense of Lynch (at least in the quasi-split cases) and analyzes the important special case when the parabolic is defined by an even embedded TDS (three-dimensional simple Lie algebra).

Groups of automorphisms of Lie algebras such that the fixed-point subalgebra of each non-identity element is solvable*1

Let G be a finite group of automorphisms of a non-solvable Lie algebra L of finite dimension over a field of characteristic zero. This paper is concerned with the relationship between the structure of G and that of L . By imposing the condition that the fixed-point subalgebra of each non-identity element of G is solvable, we are able to determine the structure of the group G and that of the quotient algebra of L by its solvable radical, Solv( L ), except when G is a group of odd order in which every Sylow subgroup is cyclic and the center is non-trivial. In this case there exists an element x ∈ L , x ∉Solv( L ), which is fixed by every element of G of prime order. We note that the imposed condition is satisfied by every finite group of automorphisms of a three-dimensional simple Lie algebra, regular groups of automorphisms of a non-solvable Lie algebra which are abelian of type ( p , p ), p prime, and by the center of a finite group of automorphisms of a semisimple Lie algebra without proper semisimple invariant subalgebra. We exhibit several examples of groups satisfying that condition. Also, we give some results relating the structures of G and L in more general cases.

Groups of automorphisms of Lie algebras such that the fixed-point subalgebra of each non-identity element is solvable

Let G be a finite group of automorphisms of a non-solvable Lie algebra L of finite dimension over a field of characteristic zero. This paper is concerned with the relationship between the structure of G and that of L. By imposing the condition that the fixed-point subalgebra of each non-identity element of G is solvable, we are able to determine the structure of the group G and that of the quotient algebra of L by its solvable radical, Solv( L), except when G is a group of odd order in which every Sylow subgroup is cyclic and the center is non-trivial. In this case there exists an element x∈ L, x∉Solv( L), which is fixed by every element of G of prime order. We note that the imposed condition is satisfied by every finite group of automorphisms of a three-dimensional simple Lie algebra, regular groups of automorphisms of a non-solvable Lie algebra which are abelian of type ( p, p), p prime, and by the center of a finite group of automorphisms of a semisimple Lie algebra without proper semisimple invariant subalgebra. We exhibit several examples of groups satisfying that condition. Also, we give some results relating the structures of G and L in more general cases.

The Forms and Representations of the Lie algebra sl 2 (ℤ)

We study the structure of integral p-adic forms of the splitting three-dimensional simple Lie algebra over the field of p-adic numbers. We discuss the questions of diagonalizability of such forms and description for maximal diagonal ideals. We consider torsion-free finite-dimensional modules over the splitting three-dimensional simple Lie algebra with integral and p-adic integral coefficients. We describe diagonal modules, demonstrate finiteness of the number of modules in each dimension, and prove a local-global principle for irreducible modules.

Lie algebras satisfying identities of degree 5

Let Φbe an associative ring with unity, containing 1/6.We prove that every prime Lie Φ-algebra satisfying the identity [(yx)(zx)]x = 0is embedded as a subring of a special form in a three-dimensional simple Lie algebra over some field A. It follows that there exists no central simple Lie algebra which is not three-dimensional and the cube of every inner derivation in which is a derivation. It is proved that if a semiprime Lie algebra over a field Φsatisfies an arbitrary identity of degree 5 (not following from the anticommutativity and Jacobi identities), then it also satisfies the standard identity of degree 5. Essentially used in the proof is the notion of antiderivation. In passing we show that every prime Lie algebra having a nonzero antiderivation satisfies the standard identity of degree 5.

Some forms of Kac-Moody algebras

Let k be an algebraically closed field of char 0. The (unique up to isomorphism) three-dimensional simple Lie algebra s1(2, k) plays an important role in the theory of the semisimple Lie algebras over k. For example, Chevalley, Harish Chandra, and Serre’s construction of the semisimple Lie algebras g over k corresponding to a Dynkin diagram 9 can be viewed as “glueing together” copies of sl(2, k) in “Lf but we can not choose the distinct TDS in an arbitrary way. We need some compatibility conditions between the TDS attached to neighbouring vertices: they are described by some scalars sij. And for simplicity, we restrict our attention in this definition to Dynkin diagrams without cycles. For both definitions, we also make the additional hypothesis that the

A construction of Lie algebras from a class of ternary algebras

A class of algebras with a ternary composition and alternating bilinear form is defined. The construction of a Lie algebra from a member of this class is given, and the Lie algebra is shown to be simple if the form is nondegenerate. A characterization of the Lie algebras so constructed in terms of their structure as modules for the three-dimensional simple Lie algebra is obtained in the case the base ring contains 1/2. Finally, some of the Lie algebras are identified; in particular, Lie algebras of type E8 are obtained. A construction of Lie algebras from Jordan algebras discovered independently by J. Tits [7] and M. Koecher [4] has been useful in the study of both kinds of algebras. In this paper, we give a similar construction of Lie algebras from a ternary algebra with a skew bilinear form satisfying certain axioms. These ternary algebras are a variation on the Freudenthal triple systems considered in [1]. Most of the results we obtain for our construction are parallel to those for the TitsKoecher construction (see [3, Chapter VIII]). In ?1, we define the ternary algebras, derive some basic results about them, and give two examples of such algebras. In ?2, the Lie algebras are constructed and shown to be simple if and only if the skew bilinear form is nondegenerate. In ?3, we give a characterization, in the case the base ring contains 1/2, of the Lie algebras obtained by our construction in terms of their structure as modules for the threedimensional simple Lie algebra. Finally, in ?4, we identify some of the simple Lie algebras obtained by our construction from the examples of ?1. In particular, we show that we can construct a Lie algebra of type E8 from a 56-dimensional space which is a module for a Lie algebra of type E7. A similar construction was given by H. Freudenthal in [2]. 1. A class of ternary algebras. We shall be interested in a module TM over an arbitrary commutative associative ring D with 1 which possesses an alternating bilinear form and a ternary product which satisfy (TI) = + z for x,y, ze M; (T2) = + x for x, y, z e 9; (T3) , w> = , z> + for x, y, z, w e 9; Received by the editors March 12, 1970 and, in revised form, June 19, 1970. AMS 1969 subject classifications. Primary 1730.