Related papers: Lie algebra theory without algebra
This is a survey paper on Alegbraic Geometry over Lie Algebras
In this paper we study Lie 2-algebras over an algebraically closed field of characteristic two, which have a triangulable Cartan subalgebra, and derive some general properties of centerless ones. These properties allow us to do an analysis…
We show that formal isomorphism of intransitive linear Lie equations along transversal to the orbits can be extended to neighborhoods of these transversal. In analytic cases, the word formal is dropped from theorems. Also, we associate an…
We adapt methods from the theory of complex semisimple Lie algebras to develop a root theory for a class of simple $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded (color) Lie algebras, which we call basic. As an application, assuming that the…
We establish some results about large restricted Lie algebras similar to those known in the Group Theory. As an application we use this group-theoretic approach to produce some examples of restricted as well as ordinary Lie algebras which…
We present a new proof of the classification of complex simple Lie algebras via the projective geometry of homogeneous varieties. Our proof proceeds by constructing homogeneous varieties using the ideals of the secant and tangential…
In the paper we complete a case by case proof of Reeder's Conjecture started in our previous work, proving the conjecture for simple Lie algebras of type $D$ and for the exceptional cases.
In this paper we show that a certain solvable Lie group constructed in a paper by Benson and Gordon has no lattices. This result answers (in the negative way) a question posed by several authors in the context of symplectic geometry. The…
By utilization of three elementary vector operators as position, angular momentum and their cross product, a simple realization of gl(2,c) Lie algebra on sphere are constructed. The coherent states based on this algebra can then be…
After a short review of the classical Lie theorem, a finite dimensional Lie algebra of vector fields is considered and the most general conditions under which the integral curves of one of the fields can be obtained by quadratures in a…
In this paper, we propose an abstract definition of dependent type theories as essentially algebraic theories. One of the main advantages of this definition is its composability: simple theories can be combined into more complex ones, and…
We explain that general differential calculus and Lie theory have a common foundation: Lie Calculus is differential calculus, seen from the point of view of Lie theory, by making use of the groupoid concept as link between them. Higher…
This is a survey work on Lie algebras with ad-invariant metrics. We summarize main features, notions and constructions, in the aim of bringing into consideration the main research on the topic. We also give some list of examples in low…
A transitive Lie algebra g of rational vector fields on a projective manifold which do not preserve any foliation determines a rational map to an algebraic homogenous space G/H which maps g to lie(G).
Associated to a symmetric space there is a canonical connection with zero torsion and parallel curvature. This connection acts as a binary operator on the vector space of smooth sections of the tangent bundle, and it is linear with respect…
Suppose that $R$ is an associative unital ring and that $E=(E^0,E^1,r,s)$ is a directed graph. Utilizing results from graded ring theory we show, that the associated Leavitt path algebra $L_R(E)$ is simple if and only if $R$ is simple,…
In present work, we find a class of Lie algebras, which are defined from the symmetrizable generalized intersection matrices. However, such algebras are different from generalized intersection matrix algebras and intersection matrix…
We consider the group algebra of the symmetric group as a superalgebra, and describe its Lie subsuperalgebra generated by the transpositions. The updated version corrects some of the arguments made in Sections 4.5 - 4.7. The statements of…
We prove a structure theorem for Lie n-algebras possessing an invariant inner product. We define the notion of a double extension of a metric Lie n-algebra by another Lie n-algebra and prove that all metric Lie n-algebras are obtained from…
We introduce the notion of a generalized representation of a Jordan algebra with unit. The greneralized representation has the following properties: (1) Usual representations and Jacobson representations correspond to special cases of…