Ernest Stitzinger

ON CHARACTERIZING NILPOTENT LIE ALGEBRAS BY THEIR MULTIPLIERS

In recent work, groups of order pj whose multiplier has order , have been classified when t(G) = 0 or 1 in [2] and when t(G) = 2 in [6]. It is the purpose of this paper to obtain similar results for Lie algebras.

On covers of lie algebras

In dealing with the central extensions of a finite group G one finds that although covers need not be isomorphic, for each such H there exists a cover for which H is a. homomorphic image [1]. For finite dimensional Lie algebras, covers are isomorphic. We shall show that the second property also holds for Lie algebras. Thus to find all such extensions one needs to compute the cover and consider ideals contained in the multiplier (kernel of the homomorphism). Several examples are constructed. Our Lie algebras are taken over a field.

On Lie algebras with only inner derivations

It is the purpose of this article to give a characterization of abelian by nilpotent Lie algebras which have only inner derivations. The motivation has been provided by two recent papers [3, 41 which deal with a similar problem in group theory. The Lie algebra case allows some modifications and simplifications as compared to the group theory case. We begin with a brief summary about Lie algebras which have only inner derivations and also about Lie algebras which must possess outer derivations. If the algebra has nondegenerate Killing form, then all derivations are inner [6, p. 741, while if the algebra is nilpotent, then there exist outer derivations [S]. This latter result has been extended in a variety of ways (see [ 131 and the bibliography given there). In particular, we men- tion that if the algebra is solvable and has no outer derivations, then its center must be 0 [2, 121. On the other hand there are examples of Lie algebras with nonzero center and only inner derivations (see [9]). Also a general method for constructing algebras with only inner derivations is described in [7]. In another direction, Schenkman’s famous derivation tower theorem [lo] asserts that an algebra with zero center can be sub- invariantly embedded in an algebra with only inner derivations. Schenkman’s paper also provides bounds on this embedding which are actually met as is shown in [8]. All Lie algebras considered here are finite dimensional over a field. Let

A Frattini Theory for Leibniz Algebras

A Frattini theory for non-associative algebras was developed in [13] and results for particular classes of algebras have appeared in various articles. Especially plentiful are results on Lie algebras. It is the purpose of this paper to extend some of the Lie algebra results to Leibniz algebras.

Minimal non-elementary Lie algebras

Minimal non-elementarty finite groups must be nilpotent. The Lie algebra analogue admits non-nilpotent examples. We classify them for complex solvable Lie algebras.

CODES AND SHIFTED CODES OF PARTITIONS

In a recent paper, Carrell and Goulden found a combinatorial identity of the Bernstein operators that they then used to prove Bernsteins theorem. We show that this identity is a straightforward consequence of the classical result. We also show how a similar approach using the codes of partitions can be generalized from Schur functions to also include Schur Q-functions and derive the combinatorial formulation for both cases. We then apply them by examining the Littlewood–Richardson and Pieri rules.

Concerning derivations of lie algebras

Schur showed that if the center of a group has finite index, then the commutator subgroup is finite. By replacing inner automorphisms by automorphisms, Hegarty obtained a variation of Schurs result. The Lie algebra analogue of Schurs result is well-known. We obtain a strong Lie algebra version of Hegartys result.

Supersolvable Malcev algebras

Recall that a group G is supersolvable if it possesses a finite increasing maximal chain of normal subgroups of G such that each factor is cyclic. This concept was introduced with the hope that it would be more tractable than solvability in general. Indeed many results on this class have appeared (see [9]). Motivated at least in part by this success, an analogous concept was introduced in Lie algebras where normal subgroups with cyclic factors were replaced by ideals with one dimensional factors. Once again the investigations proved fruitful [l, 2, 31. Adding to the importance of the concept in the Lie setting is the famous theorem of S. Lie which yields that over an algebraically closed field of characteristic 0, all solvable Lie algebras are supersolvable. This fundamental theorem had influence on the development of the structure and representation theory of these algebras. On the other hand, Malcev algebras are a generlization of Lie algebras in which results on the latter often find extensions to the former. Because of all the foregoing, supersolvability shows much promise as an object of study in Malcev algebras. Barnes [ 11 used cohomology theory to find Lie algebra analogues to theorems of Baer, Gaschtitz, and Huppert. These results were used to show various structure theorems and to construct a theory of formations [a]. Eventually, simpler proofs of the Barnes’ theorems were given by Barnes and Newell [3]. Still later the analogues of the theorems of Baer and Gas- chiitz were extended to Malcev algebras [S] and [S]. The final result con- cerns supersolvability and we extend this result to Maicev algebras in this paper where the algebra M is called supersolvable if there exists an increas- ing maximal chain of ideals of M each codimension one in the next. Although this result and related concepts extend to the present case, com- plications arise in the proofs due to the weaker defining identity for Malcev 69

Some Finite Varieties of Lie Algebras

in the class whenever all n-generated subalgebras of L are in the class. Thus the classes JV of nilpotent Lie algebras and d of abelian Lie algebras are 2-recognizable. The problems are to find if a given class is n-recognizable and to find the smallest n. We consider the classes &‘JV of Lie algebras which have an abelian ideal with nilpotent quotient, JV&, of Lie algebras which have a nilpotent ideal with abelian quotient, % of supersolvable Lie algebras, and Y of solvable Lie algebras. The same notation is used for the corresponding classes of finite groups. Most notation is standard except where indicated.

CODES AND SHIFTED CODES

The action of the Bernstein operators on Schur functions was given in terms of codes by Carrell and Goulden (2011) and extended to the analog in Schur Q-functions in our previous work. We define a new combinatorial model of extended codes and show that both of these results follow from a natural combinatorial relation induced on codes. The new algebraic structure provides a natural setting for Schur functions indexed by compositions.