Abelian Lie Algebra Research Articles

Cohomology and weight systems for nilpotent Lie algebras

1. This paper announces results concerning the cohomology groups H*(N, N) where A* is in a certain class of finite-dimensional nilpotent Lie algebras over a field k and T is an abelian Lie algebra faithfully represented as a maximal diagonalizable algebra of derivations of N; we shall refer to such an iV as a T-algebra. The additional hypotheses to be placed on the pair N, T are inspired by the case when J is a Cartan subalgebra and T+N=B is a Borel subalgebra of a complex semisimple Lie algebra. In that case Kostant has shown [2] that H%N, N)=0 for i^.2 and the authors applied this result in [3] to conclude that H*(B, B) = 0. (A similar argument shows H*(P9 P ) = 0 for P parabolic.) Here we are concerned with the relations between the vanishing ofH%N, N), especially for i=2 , and the structure of the algebras N. Let W denote the set of weights of T in N. If dirn(r)=dim(7\T/A)=m then the subset of W arising from the induced representation of T on N/N has precisely m elements, say {a1? • • • , am}. Every a e Wthen has a unique representation a = 2 ^a* with each c? a nonnegative integer and ct 0. For such an a we call the sum (in Z) 2 i the height of a and denote it by |a|. For a in W, denote by Aa the weight space for a in N. DEFINITION. A T-algebra is called positive if (i) dim(70=dim(A7iV), (ii) N is graded by the heights of the weights, i.e., if N(j) = Q)\al=j Na then [N(j)9N(k)]cN(j+k). REMARK. Condition (ii) is superfluous in characteristic 0. However, in characteristic p>0 it has such consequences as N=0 for r > ( / > l ) d i m ( r ) .

On an infinite-dimensional group

We present here an infinite-dimensional Lie algebra, semi-direct product of the Poincaré Lie algebra ℘ by an infinite-dimensional abelian Lie algebra. It gives rise to Schur-irreducible subgroups of the unitary group of the (separable) Hilbert space, with a discrete mass-spectrum (real positive isolated mass-eigenvalues). Some related mathematical problems are also examined.

Note on the global dimension of a certain ring

If M is a module over a ring S, we denote by ds(M) the projective dimension of M as an 5-module, and by gl.d.(S) the global dimension of S; i.e., the sup of ds(M) as M runs over all left 5-modules [2, VI ]. Let 5 be a commutative ring with identity element 1, and let F be a subring of 5 containing 1. Let T be the left i?-module of all F-derivations of S. T may be regarded as a Lie algebra over F in the natural fashion. Regarding S as an abelian Lie algebra over F, we construct a Lie algebra L over F, containing 5 as an ideal and T as a subalgebra, by letting L be the semidirect sum 5+7, the commutation being given by

XXII.—Some Nilpotent Algebras with Centres of Given Dimension

SYNOPSISAlgebras which are nilpotent and anti-commutative are studied. Canonical forms are found for all such algebras of dimension n whose centres have dimension n−r (r < 3), and characters are given which enable any two non-isomorphic algebras to be distinguished.A metrisable Lie algebra is a Lie algebra for which there is a non-singular, symmetric, adjoint-invariant bilinear form a(λ, μ), and such an algebra is reduced if its centre is contained in its derived algebra. The importance of the reduced algebras follows from the fact that every metrisable Lie algebra is the direct sum of a reduced metrisable Lie algebra and an abelian Lie algebra. Tsou (Thesis 1955) introduced metrisable Lie algebras, and obtained canonical forms for all real reduced metrisable Lie algebras whose derived algebras have dimension 3. We conclude this paper by providing an alternative derivation, two of the algebras being nilpotent.

On torsion-free abelian groups and Lie algebras

It is known that many of the classes of simple Lie algebras of prime characteristic of nonclassical type have simple infinite-dimensional analogues of characteristic zero (see, for example, [4, p. 518]). We consider here analogues of those algebras which are defined by a modification of the definition of a group algebra. Thus we consider analogues of the Zassenhaus algebras as generalized by Albert and Frank in [1]. The algebras considered are defined as follows. Let G be a nonzero abelian group, F a field, g an additive mapping from G to F, and f an alternate biadditive mapping from G X G to F. We index a basis of an algebra over F by G, denoting by ua the basis element corresponding to a in G, and define multiplication by

The Representations of Lie Algebras of Prime Characteristic

There are some simple facts which distinguish Lie-algebras over fields of prime characteristic from Lie-algebras over fields of characteristic zero. These are(1) The degrees of the absolutely irreducible representations of a Lie-algebra of prime characteristic are bounded whereas, according to a theorem of H. Weyl, the degrees of the absolutely irreducible representations of a semi-simple Lie-algebra over a field of characteristic zero can be arbitrarily high.(2) For each Lie-algebra of prime characteristic there are indecomposable representations which are not irreducible, whereas every indecomposable representation of a semi-simple Liealgebra over a field of characteristic zero is irreducible (cf.[4]).(3) The quotient ring of the embedding algebra of a Lie-algebra over a field of prime characteristic is a division algebra of finite dimension over its center, whereas this is not the case for characteristic zero. (cf.[4]).(4) There are faithful fully reducible representations of every Lie-algebra of prime characteristic, whereas for characteristic zero only ring sums of semi-simple Lie-algebras and abelian Lie-algebras admit faithful fully reducible representations (cf.[6], [2], [4]).

A theorem on the derivations of Jordan algebras

The restriction on the dimensionality of the simple components arises from the fact that the (3-dimensional) central simple Jordan algebra of all 2 X 2 symmetric matrices has for its derivation algebra the abelian Lie algebra of dimension 1. However, most simple Jordan algebras over F have simple derivation algebras, and all except those of dimension 3 over their centers have derivation algebras which are semisimple or {0 }, as may be seen from the listing by N. Jacobson [3, ?4] of these derivation algebras.2 The part of the theorem then follows from the direct sum relationship. To demonstrate the converse it is sufficient to show that, if Z is semisimple or {0}, then 2( is semisimple. For then, if any simple component of 2t had dimension 3 over its center, it would have an abelian derivation algebra not equal to {0 } [3, ?4], which would give rise to a nonzero abelian ideal in Z, a contradiction. To show that 2t is semisimple whenever Z is semisimple or {0}, we use the so-called Wedderburn Principal Theorem for Jordan algebras, proved recently by A. J. Penico [5]. Also Lemma 1 below is taken from the proof of that theorem [5, ?2]. Revisions have been made in the proof of our theorem in accordance with helpful suggestions of Professor Jacobson.