Lie–Yamaguti algebras related to g2

arXiv (Cornell University), 2023

This study aims to generalize the notion of compatible Lie algebras to the compatible Lie Yamaguti algebras. Along with describing the representation of the compatible Lie Yamaguti algebra in detail, we also introduce the Maurer-Cartan characterization and cohomology of Lie Yamaguti algebras. As a result of the obtained cohomology, we studied its deformation. We define Rota-Baxter operators on compatible Lie Yamaguti algebras as well as on compatible pre-Lie Yamaguti algebras. Using Rota-Baxter operators, we examine how compatible Lie (and compatible pre-Lie) Yamaguti algebras are related.

Journal of Algebra, 1999

Lie triple system T over a field F of characteristic zero. It turns out that it contains nontrivial elements if and only if T is related to a simple Jordan algebra.

Journal of Algebra, 2006

Symplectic (respectively orthogonal) triple systems provide constructions of Lie algebras (resp. superalgebras). However, in characteristic 3, it is shown that this role can be interchanged and that Lie superalgebras (resp. algebras) can be built out of symplectic triple systems (resp. orthogonal triple systems) with a different construction. As a consequence, new simple finite dimensional Lie superalgebras, as well as new models of some nonclassical simple Lie algebras, over fields of characteristic 3, will be obtained.

2020

Leibniz algebras E_n were introduced as algebraic structure underlying U-duality. Algebras E_3 derived from Bianchi three-dimensional Lie algebras are classified here. Two types of algebras are obtained: Six-dimensional Lie algebras that can be considered extension of semi-Abelian four-dimensional Drinfeld double and unique extensions of non-Abelian Bianchi algebras. For all of the algebras explicit forms of generalized frame fields are given.

Lecture Notes in Physics, 2015

Journal of the London Mathematical Society, 1973

A finite-dimensional Lie algebra L over a field F is called elementary if each of its subalgebras has trivial Frattini ideal; it is an A-algebra if every nilpotent subalgebra is abelian. The present paper is primarily concerned with the classification of elementary Lie algebras. In particular, we provide a complete list in the case when F is algebraically closed and of characteristic different from 2,3, reduce the classification over fields of characteristic 0 to the description of elementary semisimple Lie algebras, and identify the latter in the case when F is the real field. Additionally it is shown that over fields of characteristic 0 every elementary Lie algebra is almost algebraic; in fact, if L has no non-zero semisimple ideals, then it is elementary if and only if it is an almost algebraic A-algebra.

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

Progress of Theoretical and Experimental Physics, 2020

Leibniz algebras ${\mathcal E}_n$ were introduced as an algebraic structure underlying U-duality. Algebras ${\mathcal E}_3$ derived from Bianchi 3D Lie algebras are classified here. Two types of algebras are obtained: 6D Lie algebras that can be considered an extension of the semi-Abelian 4D Drinfel’d double and unique extensions of non-Abelian Bianchi algebras. For all of the algebras explicit forms of generalized frame fields are given.

Mathematische Nachrichten

The present paper is another step toward the classification of ℤ × ℤ-graded Lie algebras: we classify ℤ × ℤ-graded Lie algebras = ⊕i,j∈ℤi,j over a field of characteristic 0 with dimi,j ≤ 1 for i, j ∈ ℤ, satisfying:(1) 0 = ⊕i∈ℤi,0 ≃ Vir, the centerless Virasoro algebra,(2) 0,–1, 0,0, 0,1 span the 3-dimensional simple Lie algebra sl2,(3) is generated by the centerless Virasoro algebra and sl2.These algebras include some rank 2 centerless Virasoro algebras, and some rank 2 Block algebras and their extensions.

Revista De La Union Matematica Argentina, 2013

The problem of Lie algebras' classification, in their different varieties, has been dealt with by theory researchers since the early 20 th century. This problem has an intrinsically infinite nature since it can be inferred from the results obtained that there are features specific to each field and dimension. Despite the hundreds of attempts published, there are currently fields and dimensions in which only partial classifications of some families of algebras of low dimensions have been obtained. This article intends to bring some order to the achievements of this prolific line of research so far, in order to facilitate future research.