ON THETA PAIRS FOR A MAXIMAL SUBALGEBRA

Given a maximal subalgebra M of a finite-dimensional Lie algebra L, a θ-pair for M is a pair (A, B) of subalgebras such that A ≰ M, B is an ideal of L contained in A ∩ M, and A/B includes properly no nonzero ideal of L/B. This is analogous to the concept of θ-pairs associated to maximal subgroups of a finite group, which has been studied by a number of authors. A θ-pair (A, B) for M is said to be maximal if M has no θ-pair (C, D) such that A < C. In this paper, we obtain some properties of maximal θ-pairs and use them to give some characterizations of solvable, supersolvable and nilpotent Lie algebras.

IRREDUCIBLE REPRESENTATIONS OF THE HAMILTONIAN ALGEBRAH(2r;n)

LetL=H(2r;n) be a graded Lie algebra of Hamiltonian type in the Cartan type series over an algebraically closed field of characteristicp>2. In the generalized restricted Lie algebra setup, any irreducible representation ofLcorresponds uniquely to a (generalized)p-characterχ. When the height ofχis no more than min {pni−pni−1∣i=1,2,…,2r}−2, the corresponding irreducible representations are proved to be induced from irreducible representations of the distinguished maximal subalgebraL0with the aid of an analogy of Skryabin’s category ℭ for the generalized Jacobson–Witt algebras and modulo finitely many exceptional cases. Since the exceptional simple modules have been classified, we can then give a full description of the irreducible representations withp-characters of height below this number.

The index complex of a maximal subalgebra of a Lie algebra

Let M be a maximal subalgebra of the Lie algebra L. A subalgebra C of L is said to be a completion for M if C is not contained in M but every proper subalgebra of C that is an ideal of L is contained in M. The set I(M) of all completions of M is called the index complex of M in L. We use this concept to investigate the influence of the maximal subalgebras on the structure of a Lie algebra, in particular, finding new characterizations of solvable and supersolvable Lie algebras.

Cartan Subalgebras of

Let V be a vector space over a field of characteristic zero and V* be a space of linear functionals on V which separate the points of V. We consider V ⊗ V* as a Lie algebra of finite rank operators on V, and set (V, V*) := V ⊗ V*. We define a Cartan subalgebra of (V, V*) as the centralizer of a maximal subalgebra every element of which is semisimple, and then give the following description of all Cartan subalgebras of (V;V*) under the assumption that is algebraically closed. A subalgebra of (V, V*) is a Cartan subalgebra if and only if it equals for some one-dimensional subspaces Vj ⊆ V and (Vj)* ⊆ V* with (Vi)* (Vj) = δij and such that the spaces . We then discuss explicit constructions of subspaces Vj and (Vj)* as above. Our second main result claims that a Cartan subalgebra of (V, V*) can be described alternatively as a locally nilpotent self-normalizing subalgebra whose adjoint representation is locally finite, or as a subalgebra h which coincides with the maximal locally nilpotent h-submodule of (V, V*), and such that the adjoint representation of is locally finite.

Douglas algebras without maximal subalgebra and without minimal superalgebra

We give several examples of Douglas algebras that do not have any maximal subalgebra. We find a condition on these algebras that guarantees that some do not have any minimal superalgebra. We also show that ifAis the only maximal subalgebra of a Douglas algebraB, then the algebraAdoes not have any maximal subalgebra.

Corrigendum et addendum: the Frattini subalgebra of a Bernstein algebra

In a previous paper it is supposed that if A is a Bernstein algebra, every maximal subalgebra, M, verifies that dim M = dim A − 1. This is not true in general. Therefore Proposition 2 in this paper is not correct. However other results there, where this assertion was used, are correct but their proofs need some modifications now.

Maximal subalgebra of Douglas algebra

Whenqis an interpolating Blaschke product, we find necessary and sufficient conditions for a subalgebraBofH∞[q¯]to be a maximal subalgebra in terms of the nonanalytic points of the noninvertible interpolating Blaschke products inB. If the setM(B)⋂Z(q)is not open inZ(q), we also find a condition that guarantees the existence of a factorq0ofqinH∞such thatBis maximal inH∞[q¯]. We also give conditions that show when two arbitrary Douglas algebrasAandB, withA⫅Bhave property thatAis maximal inB.