Commutative restricted Lie algebras (original) (raw)

The restricted simple Lie algebras with a two-dimensional Cartan subalgebra

Bulletin of the American Mathematical Society, 1979

The classification of the finite-dimensional simple Lie algebras over an algebraically closed field of prime characteristic p is still an open problem after 40 years of investigation (going back to the pioneering work of Jacobson and Zassenhaus). The problem can be made more tractable by considering only restricted Lie algebras (i.e., Lie p-algebras), these being Lie algebras with an extra mapping x t-> x p satisfying certain hypotheses, in particular (ad x) p = ad x p (see [3, p. 187]); throughout the subject's history this restriction has proved fruitful, being a good indicator of the general shape of nonrestricted results as well. It is also customary to exclude some small p. But even with these two hypotheses the problem remains very much open, although progress to date (some of it mentioned below) and the new contribution announced here, namely, the classification of the algebras of the title, give cause for optimism. From now on let L denote a finite-dimensional simple restricted Lie algebra over an algebraically closed field F of characteristic p > 7 (some of the results cited below hold in more generality). The known L are of two kinds: either classical, i.e., algebras of the usual types A n ,. .. , G 2 , these being analogues of the finite-dimensional simple Lie algebras over C, or of Cartan type, i.e., one of the algebras W n , S n , H n or K n , these being analogues (previously known but first described in this way by Kostrikin and Safarevic [7]) of the infinite-dimensional simple Lie algebras over C corresponding to Lie pseudogroups. In particular, W n is the np n-dimensional Jacobson-Witt algebra

Classification of the restricted simple Lie algebras with toral Cartan subalgebras

Journal of Algebra, 1983

All rights of reproduction in any form reserved. ROBERT LEE WILSON to find maximal subalgebras L, for which the hypotheses of the theorem hold. Most of this paper is devoted to describing an appropriate choice of L,, and to verifying that the hypotheses of the recognition theorem are indeed satisfied. Our definition of an appropriate L, depends upon the notion of a section of a Lie algebra M. Let M have Cartan decomposition M = H + C&f,. For roots a and /I define Mea) = ZyEZaMy and M(a,4' = CYEZa+ ZBMy. Define the rank one section M[a] = M'"'/solv(M'"') and the rank two section M[ a, p] = M'"~4'/solv(M'"*D') (w h ere solv(M) denotes the solvable radical of M). Assume H is a torus. Then M[a] is a restricted semisimple Lie algebra with a Cartan subalgebra of dimension <l, hence by Kaplansky's Theorem [ 14, Theorem 21 must be (0), sI(2), or IV,. Also, M[a, /I] is a semisimple restricted Lie algebra with a Cartan subalgebra of dimension <2. Hence it must be (0), 41(2), W, or one of the 23 restricted semisimple Lie algebras of rank two determined by Block and the author in 161. We use this knowledge of the possible M[a] and M[a, /I] in the following manner. For any Lie algebra M containing a subalgebra N let g(M, N) denote the set of all (ad N)-invariant subalgebras of M in which every composition factor is abelian or classical simple. Now g(Ltm'), H) may or may not contain a unique element of maximum dimension. If such an element does exist we say a is proper and we denote the element by Qcu) = H + ZQi,. (This is a restatement of the definition of proper from [ 6, Definition 2.6.1 I.) Our technique is to describe L, in terms of the Q,. However, to have all the Q, available we must know that all roots are proper. A technique, due to Winter [28], exists for switching Cartan subalgebras in such a way that an improper root can be replaced by a proper root. This alone is not sufficient to show that all roots can be made proper, since it is possible a priori for Winter's technique, while making an improper root a proper, to simultaneously make a proper root /l improper. However, if this were to occur in the algebra M the same behavior would occur in the rank two section M[a,/.I]. We show that this is impossible in any rank two semisimple algebra, and hence that if a restricted Lie algebra M contains a toral Cartan subalgebra then it contains a toral Cartan subalgebra with respect to which all roots are proper. Once we know that all roots of a restricted simple Lie algebra L are proper we define Q = H + CQ,. We show that Q is a subalgebra and that, for each a and /3, dim L'"*4'/Q (rr~4) < 2. It is again suflicient to check this in each L[a, p]. Here, however, the situation is more subtle than above. The result we need is not true in all 23 rank two restricted semisimple Lie algebras. Hence we must first study (using a result of Schue [ 191) the structure of a rank two section of a restricted simple Lie algebra L. We then show that our result is true for all such sections.

The restricted simple Lie algebras are of classical or Cartan type

Proceedings of the National Academy of Sciences, 1984

The classification of the finite-dimensional restricted simple Lie algebras over an algebraically closed field F of prime characteristic p > 7 is announced. Such an algebra is either of classical type (an analogue over F of a finite-dimensional simple Lie algebra over the complex numbers) or of Cartan type (an analogue over F of one of the infinite Lie algebras of Cartan over the complex numbers).

On powerful and p-central restricted Lie algebras

BULLETIN OF THE AUSTRALIAN MATHEMATICAL …, 2007

In this note we analyse the analogy between m-potent and p-central restricted Lie algebras and p-groups. For restricted Lie algebras the notion of m-potency has stronger implications than for p-groups (Theorem A). Every finite-dimensional restricted Lie algebra L is ...

c ○ 2003 Heldermann Verlag On the Cartan Subalgebras of Lie Algebras

2008

Abstract. In this note we study Cartan subalgebras of Lie algebras defined over finite fields. We prove that a possible Lie algebra of minimal dimension without Cartan subalgebras is semisimple. Subsequently, we study Cartan subalgebras of gl(n, F). AMS classification: 17B50

A new definition of restricted Lie superalgebras

Chinese Science Bulletin, 2000

A new definition of restricted Lie superalgebras has been given; some important results on the representation theory of restricted Lie superalgebra have been obtained.

On Semi-Modular Subalgebras of Lie Algebras Over Fields of Arbitrary Characteristic

Asian-European Journal of Mathematics, 2008

This paper is a further contribution to the extensive study by a number of authors of the subalgebra lattice of a Lie algebra. It is shown that, in certain circumstances, including for all solvable algebras, for all Lie algebras over algebraically closed fields of characteristic p > 0 that have absolute toral rank ≤ 1 or are restricted, and for all Lie algebras having the one-and-a-half generation property, the conditions of modularity and semi-modularity are equivalent, but that the same is not true for all Lie algebras over a perfect field of characteristic three. Semi-modular subalgebras of dimensions one and two are characterised over (perfect, in the case of two-dimensional subalgebras) fields of characteristic different from 2, 3.