2-recognizeable classes of Leibniz algebras

Burch, Tiffany; Harris, Meredith; McAlister, Allison; Rogers, Elyse; Stitzinger, Ernest L.; Sullivan, S. McKay

doi:10.1016/j.jalgebra.2014.10.039

Cited by 5 publications

(11 citation statements)

References 11 publications

(25 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since N is triangulable, N 2 is nil on N and, thus, N is strongly solvable. Therefore A is strongly solvable by [5] and A 2 is nil on A. Then, using Theorem 3, A is triangulable.…”

mentioning

confidence: 90%

“…The concept when n = 2 has been considered in groups, Lie algebras and Leibniz algebras for classes that are solvable, supersolvable, nilpotent and strongly solvable (see [4], [5], [8]). A version of this has also been considered for triangulable in the Lie case in [3].…”

mentioning

confidence: 99%

“…The result can be extended from solvable to all algebras in characteristic 0, again using a result in [5].…”

mentioning

confidence: 99%

“…Simple Lie algebras are 2-generated, hence A is triangulable. If I is a proper ideal in A, then all 2-generated subalgebras of I and A/I are strongly solvable by Theorem 3 and I and A/I are strongly solvable by Theorem 3 of [5] and A is solvable. The result holds by Theorem 4.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Triangulable Leibniz Algebras

Burch

Stitzinger

2016

Communications in Algebra

Self Cite

View full text Add to dashboard Cite

A converse to Lie's theorem for Leibniz algebras is found and generalized. The result is used to find cases in which the generalized property, called triangulable, is 2-recognizeable; that is, if all 2-generated subalgebras are triangulable, then the algebra is also. Triangulability joins solvability, supersolvability, strong solvability, and nilpotentcy as a 2-recognizeable property for classes of Leibniz algebras.Simultaneous triangulation of a set of linear transformations related to Leibniz algebras will be considered in this work. A necessary condition is that the minimum polynomials of the linear transformations are the products of linear factors. If the matrices commute, then this condition is sufficient. Lie's theorem is a generalization of this result, a result that is field dependent even in the algebraically closed case. Lie's theorem extends to Leibniz algebras. It can also be useful to have a condition on the set which guarantees simultaneous triangulation when extending to the algebraic closure, K, of the field of scalars, F . We will call this property, borrowing from Bowman, Towers and Varea [3], triangulable. Thus a Leibniz algebra A over a field F is triangulable on module M if the induced representation of K A on K M admits a basis such that the representing matrices are upper triangular. We find a condition on the Leibniz algebra that is equivalent to being triangulable. The condition is independent of the field and includes Lie's theorem. The condition is on A although triangulable refers to the action in the algebra that is over K. As an application, we consider to what extent an algebra is triangulable if all 2-generated subalgebras have the property. Such a property is called 2-recognizable, a concept investigated in groups, Lie algebras and Leibniz algebras for classes such as solvable, supersolvable, strongly solvable and, trivially, nilpotent and abelian. For an introduction to Leibniz algebras, see [1], [6] and [7].Let M be a module for the Leibniz algebra A. Let left and right multiplication by x ∈ A be denoted by T x and S x , respectively. If T x is nilpotent, then S x is also nilpotent and an Engle theorem holds; that is, A acts nilpotently on M [2]. Define A to be nil on M if this property holds. If I is an ideal of a Leibniz algebra A, then I is nil on A if and only if I is nilpotent. We will show that A is triangulable on itself if and only if A 2 is nilpotent. When A 2 is nilpotent, then A is called strongly solvable, a term introduced in [3].

show abstract

“…Since N is triangulable, N 2 is nil on N and, thus, N is strongly solvable. Therefore A is strongly solvable by [5] and A 2 is nil on A. Then, using Theorem 3, A is triangulable.…”

mentioning

confidence: 90%

mentioning

confidence: 99%

“…The result can be extended from solvable to all algebras in characteristic 0, again using a result in [5].…”

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Triangulable Leibniz Algebras

Burch

Stitzinger

2016

Communications in Algebra

Self Cite

View full text Add to dashboard Cite

show abstract

“…A general theory can be found in [14] and there are many works on special classes of algebras, especially Lie algebras. Leibniz algebras, as a generalization of Lie algebras, is a natural class to investigate and [2] - [9] contain results on Frattini subalgebras and ideals. Frattini theory for groups goes back to the nineteenth century and there have been many results that are similar in groups and Lie algebras.…”

Section: Introductionmentioning

confidence: 99%