Split Lie Research Articles

LIE 3-ALGEBRA AND SUPER-AFFINIZATION OF SPLIT-OCTONIONS

The purpose of this study is to extend the concept of a generalized Lie 3-algebra, known to the divisional algebra of the octonions 𝕆, to split-octonions 𝕊𝕆, which is non-divisional. This is achieved through the unification of the product of both of the algebras in a single operation. Accordingly, a notational device is introduced to unify the product of both algebras. We verify that 𝕊𝕆 is a Malcev algebra and we recalculate known relations for the structure constants in terms of the introduced structure tensor. Finally we construct the manifestly supersymmetric [Formula: see text] affine superalgebra. An application of the split Lie 3-algebra for a Bagger and Lambert gauge theory is also discussed.

Coarse differentiation and quasi-isometries of a class of solvable Lie groups I

In this paper, we continue with the results in [12] and compute the group of quasiisometries for a subclass of split solvable unimodular Lie groups. Consequently, we show that any finitely generated group quasi-isometric to a member of the subclass has to be polycyclic and is virtually a lattice in an abelian-by-abelian solvable Lie group. We also give an example of a unimodular solvable Lie group that is not quasi-isometric to any finitely generated group, as well deduce some quasi-isometric rigidity results. 51F99; 22E40

Coarse differentiation and quasi-isometries of a class of solvable Lie groups II

In this paper, we continue with the results in \cite{Pg} and compute the group of quasi-isometries for a subclass of split solvable unimodular Lie groups. Consequently, we show that any finitely generated group quasi-isometric to a member of the subclass has to be polycyclic, and is virtually a lattice in an abelian-by-abelian solvable Lie group. We also give an example of a unimodular solvable Lie group that is not quasi-isometric to any finitely generated group, as well deduce some quasi-isometric rigidity results.

Primordials and primordial Lie algebras

Primordials \({d \in \mathcal{P}}\) are generalizations of ordinals \({\sigma \in \mathcal{O}}\) . Primordials are governed by their succession and precession. Primordials with their succession and precession are of interest in their own right. Remarkably, they also lead directly to certain primordial Lie algebras of set theory. Among these is the large primordial Lie algebra of set theory, whose basis is a class and not a set. The large primordial Lie algebra of set theory generalizes naturally to the large primordial Lie algebras of characteristic p ≥ 2. The simple primordial Lie algebras are the natural primordial Lie algebra \({\mathcal{L}^\natural}\) , the free primordial Lie algebras \({\mathcal{L}^c}\) for r ≥ 1 and r-tuples C of denumerable sequences Cj (1 ≤ j ≤ r) of elements of k, and, for p > 2, the normal sub Lie algebras of the \({\mathcal{L}^\natural,\mathcal{L}^c}\) as well. The split simple primordial Lie algebras are the Lie algebras L of type W—those which may be built directly from the natural primordial Lie algebra \({\mathcal{L}^\natural}\)—except when p = 2 and L is not free. Consequently, they are, up to isomorphism, the purely inseparable forms of the finite and infinite dimensional Lie algebras of type W. This sheds new light on, and adds interest to, the structure of these purely inseparable forms.

On split Leibniz algebras

In order to begin an approach to the structure of arbitrary Leibniz algebras, (with no restrictions neither on the dimension nor on the base field), we introduce the class of split Leibniz algebras as the natural extension of the class of split Lie algebras. By developing techniques of connections of roots for this kind of algebras, we show that any of such algebras (L,[·,·]) is of the form L=U+∑jIj with U a subspace of the abelian subalgebra H, (in the sense [H,H]=0), and any Ij a well described ideal of L, satisfying [Ij,Ik]=0 if j≠k. In the case of L being of maximal length we characterize the simplicity of the algebra in terms of connections of roots.

On the structure of split Lie color algebras

The class of Lie color algebras contains the one of Lie superalgebras and so the one of Lie algebras. In order to begin an approach to the structure of arbitrary Lie color algebras, (with no restrictions neither on the dimension nor on the base field), we introduce the class of split Lie color algebras as the natural extension of the classes of split Lie algebras and split Lie superalgebras. By developing techniques of connections of roots for this kind of algebra, we show that any such algebra L is of the form L = U + ∑ j I j with U a subspace of the abelian (graded) subalgebra H and eachx I j a well described (graded) ideal of L satisfying [ I j , I k ] = 0 if j ≠ k . Under certain conditions, the simplicity of L is characterized and it is shown that L is the direct sum of the family of its minimal (graded) ideals, each one being a simple split Lie color algebra.

$K$-finite solutions to conformally invariant systems of differential equations

Let $G$ be a connected semisimple linear real Lie group, and $Q$ (resp. $K$) a real parabolic subgroup (resp. maximal compact subgroup) of $G$. The space of $K$-finite solutions to a conformally invariant system of differential equations on a line bundle over the real flag manifold $G/Q$ is studied. The general theory is then applied to certain second order systems on the flag manifold that corresponds to the Heisenberg parabolic subgroup in a split simple Lie group.

On split Lie superalgebras

We study the structure of arbitrary split Lie superalgebras. We show that any of such superalgebras L is of the form L=U+∑jIj with U a subspace of the Abelian (graded) subalgebra H and any Ij, a well described (graded) ideal of L satisfying [Ij,Ik]=0 if j≠k. Under certain conditions, the simplicity of L is characterized and it is shown that L is the direct sum of the family of its minimal (graded) ideals, each one being a simple split Lie superalgebra.

On split Lie triple systems II

On split Lie triple systems II

Polynomial bases of split simple Lie algebras

Polynomial bases of split simple Lie algebras

On Integrable roots in split Lie triple systems

We focus on the notion of an integrable root in the framework of split Lie triple systems T with a coherent 0-root space. As a main result, it is shown that if T has all its nonzero roots integrable, then its standard embedding is a split Lie algebra having all its nonzero roots integrable. As a consequence, a local finiteness theorem for split Lie triple systems, saying that whenever all nonzero roots of T are integrable then T is locally finite, is stated. Finally, a classification theorem for split simple Lie triple systems having all its nonzero roots integrable is given.

Universal central extensions of twisted forms of split simple Lie algebras over rings

We give sufficient conditions for the descent construction to be the universal central extension of a twisted form of a split simple Lie algebra over a ring. In particular, the universal central extensions of twisted multiloop Lie tori are obtained by the descent construction.

An algorithm for Lang's Theorem

We give an efficient Las Vegas type algorithm for Lang's Theorem in split connected reductive groups defined over finite fields of characteristic greater than 3. This algorithm can be used to construct many important structures in finite groups of Lie type. We use an algorithm for computing a Chevalley basis for a split reductive Lie algebra, which is of independent interest. For our time analysis we derive that the proportion of reflection derangements in a Weyl group is less than 2/3.

On split Lie triple systems

In [4] it is studied that the structure of split Lie triple systems with a coherent 0-root space, that is, satisfying [T0, T0, T] = 0 and [T0, Tα, T0] ≠ 0 for any nonzero root α and where T0 denotes the 0-root space and Tα the α-root space, by showing that any of such triple systems T with a symmetric root system is of the form T = U + ΣjIj with U a subspace of the 0-root space T0 and any Ij a well described ideal of T, satisfying [Ij, T, Ik] = 0 if j ≠ k. It is also shown in [4] that under certain conditions, a split Lie triple system with a coherent 0-root space is the direct sum of the family of its minimal ideals, each one being a simple split Lie triple system, and the simplicity of T is characterized. In the present paper we extend these results to arbitrary split Lie triple systems with no restrictions on their 0-root spaces.

On Simple Split Lie Triple Systems

It is shown that simple split Lie triple systems with a coherent 0-root space and satisfying finiteness conditions are isomorphic to a direct limit of (well-known) finite dimensional simple Lie triple systems of a same type. The key tool in this job is the notion of connection of roots in the framework of split Lie triple systems.

Small principal series and exceptional duality for two simply laced exceptional groups

We use the notion of rank defined in an earlier paper (2007) to introduce and study two correspondences between small irreducible unitary representations of the split real simple Lie groups of types E n , where n ∈ {6, 7}, and two reductive classical groups. We show that these correspondences classify all of the unitary representations of rank two (in the sense of our earlier paper) of these exceptional groups. We study our correspondences for a specific family of degenerate principal series representations in detail.

A path model for geodesics in Euclidean buildings and its applications to representation theory

In this paper we give a combinatorial characterization of projections of geodesics in Euclidean buildings to Weyl chambers. We apply these results to the representation theory of complex reductive Lie groups and to spherical Hecke rings associated with split nonarchimedean reductive Lie groups. Our main application is a generalization of the saturation theorem of Knutson and Tao for SLn to other complex semisimple Lie groups.

On split Lie algebras with symmetric root systems

We develop techniques of connections of roots for split Lie algebras with symmetric root systems. We show that any of such algebras L is of the form L = \( \mathcal{U} \) + ΣjIj with \( \mathcal{U} \) a subspace of the abelian Lie algebra H and any Ij a well described ideal of L, satisfying [Ij, Ik] = 0 if j ≠ k. Under certain conditions, the simplicity of L is characterized and it is shown that L is the direct sum of the family of its minimal ideals, each one being a simple split Lie algebra with a symmetric root system and having all its nonzero roots connected.

Constructing simply laced Lie algebras from extremal elements

For any finite graph Gamma and any field K of characteristic unequal to 2 we construct an algebraic variety X over K whose K-points parameterise K-Lie algebras generated by extremal elements, corresponding to the vertices of the graph, with prescribed commutation relations, corresponding to the non-edges. After that, we study the case where Gamma is a connected, simply laced Dynkin diagram of finite or affine type. We prove that X is then an affine space, and that all points in an open dense subset of X parameterise Lie algebras isomorphic to a single fixed Lie algebra. If Gamma is of affine type, then this fixed Lie algebra is the split finite-dimensional simple Lie algebra corresponding to the associated finite-type Dynkin diagram. This gives a new construction of these Lie algebras, in which they come together with interesting degenerations, corresponding to points outside the open dense subset. Our results may prove useful for recognising these Lie algebras.

Infinite-dimensional super Lie groups

A super Lie group is a group whose operations are G ∞ mappings in the sense of Rogers. Thus the underlying supermanifold possesses an atlas whose transition functions are G ∞ functions. Moreover the images of our charts are open subsets of a graded infinite-dimensional Banach space since our space of supernumbers is a Banach Grassmann algebra with a countably infinite set of generators. In this context, we prove that if h is a closed, split sub-super Lie algebra of the super Lie algebra of a super Lie group G , then h is the super Lie algebra of a sub-super Lie group of G . Additionally, we show that if g is a Banach super Lie algebra satisfying certain natural conditions, then there is a super Lie group G such that the super Lie algebra g is in fact the super Lie algebra of G . We also show that if H is a closed sub-super Lie group of a super Lie group G , then G → G / H is a principal fiber bundle. We emphasize that some of these theorems are known when one works in the super-analytic category and also when the space of supernumbers is finitely generated in which case, one can use finite-dimensional techniques. The issues dealt with here are that our supermanifolds are modeled on graded Banach spaces and that all mappings must be morphisms in the G ∞ category.