Graded G-structures

In this article, we examine the general space‐time fractional diffusion equation for left‐invariant hypoelliptic homogeneous operators on graded Lie groups. Our study covers important examples such as the time fractional diffusion equation, the space‐time fractional diffusion equation when diffusion is under the influence of sub‐Laplacian on the Heisenberg group, or general stratified Lie groups. We establish the global well‐posedness of the Cauchy problem for the general space‐time fractional diffusion equation of the Rockland operator on a graded Lie group in the associated Sobolev spaces and also develop some regularity estimates for it.

Owing to the existence of a dilatation generator of eigenvalues ± 2 , ± 1 , 0 \pm 2, \pm 1,0 the symplectic Lie algebra is considered as a | 2 | |2| -graded Lie algebra. The corresponding decomposition of the symplectic group Sp(2( n + 1 ), R ) {\text {Sp(2(}}n + 1{\text {),}}{\mathbf {R}}{\text {)}} makes the semidirect product (denoted L 0 {L^0} ) of the ( 2 n + 1 ) (2n + 1) -dimensional Weyl group by the conformal symplectic group CSp( 2 n , R ) {\text {CSp(}}2n,{\mathbf {R}}{\text {)}} appear as a privileged subgroup and permits one to construct a 2 n + 1 2n + 1 -dimensional homogeneous space possessing a natural contact form. Then Sp ( 2 ( n + 1 ) , R ) {\text {Sp}}(2(n + 1),{\mathbf {R}}) -valued Cartan connections on a L 0 {L^0} principal fibre bundle over a 2 n + 1 2n + 1 -dimensional manifold B 2 n + 1 {B_{2n + 1}} are constructed and called symplectic Cartan connections. The conditions for obtaining a unique symplectic Cartan connection are given. The existence of this unique Cartan connection is used to define the notion of contact structure over B 2 n + 1 {B_{2n + 1}} and it is shown that any L 0 {L^0} -structure of degree 2 2 over B 2 n + 1 {B_{2n + 1}} can be considered as a contact structure on it. Moreover it is shown that a contact structure can be associated in a canonical way to any contact manifold.

This paper presents the classification over the fields of real and complex numbers, of the minimal Z2×Z2-graded Lie algebras and Lie superalgebras spanned by four generators and with no empty graded sector. The inequivalent graded Lie (super)algebras are obtained by solving the constraints imposed by the respective graded Jacobi identities. A motivation for this mathematical result is to systematically investigate the properties of dynamical systems invariant under graded (super)algebras. Recent works only paid attention to the special case of the one-dimensional Z2×Z2-graded Poincaré superalgebra. As applications, we are able to extend certain constructions originally introduced for this special superalgebra to other listed Z2×Z2-graded (super)algebras. We mention, in particular, the notion of Z2×Z2-graded superspace and of invariant dynamical systems (both classical worldline sigma models and quantum Hamiltonians). As a further by-product, we point out that, contrary to Z2×Z2-graded superalgebras, a theory invariant under a Z2×Z2-graded algebra implies the presence of ordinary bosons and three different types of exotic bosons, with exotic bosons of different types anticommuting among themselves.

We propose a presymplectic BV-AKSZ sigma model encoding the ghost-free massive bigravity theory action as well as its Batalin-Vilkovisky extension in terms of the finite-dimensional graded geometry of the target space. A characteristic feature of the construction is that the target space is realised as a quasi-regular submanifold of a linear graded manifold which, in turn, is a direct product of two copies of the shifted Poincaré or (anti-)de Sitter Lie algebra. This graded manifold comes equipped with a natural presymplectcic structure and the compatible pre-Q structure which is a sum of the Chevalley-Eilenberg differentials of each copy of the Lie algebra and the interaction term. The constraints determining the submanifold are the supergeometrical realisation of the known Deser-van Nieuwenhuizen condition and its descendant.

In this paper, using a homological theory of graded Lie algebras and the representation theory of A(') , we compute the root multiplicities of the hyperbolic Kac-Moody Lie algebra HA(') up to level 4 and deduce some interesting combinatorial identities. INTRODUCTION The hyperbolic Kac-Moody Lie algebras have been considered as the next natural objects of study after the affine case. While the affine Kac-Moody Lie algebras have been extensively studied for their close connections to areas such as combinatorics, modular forms, and mathematical physics, many basic questions regarding the hyperbolic case are still unresolved. For example, the behavior of the root multiplicities is not well understood. Feingold and Frenkel [F-F] and Kac, Moody, and Wakimoto [K-M-W] made some progress in this area. They computed the level 2 root multiplicities for the hyperbolic Kac-Moody Lie algebras HA(1) and HE(1), respectively. In [Kangl] and [Kang2], we introduced an inductive program to study the higher level root multiplicities and the principally specialized affine characters of a certain class of Lorentzian Kac-Moody Lie algebras. More precisely, we realize these Lorentzian Kac-Moody Lie algebras. as the minimal graded Lie algebras L = efEz Ln with local part V + Lo + V*, where Lo is an affine Kac-Moody Lie algebra, V is the basic representation of Lo, and V* is the contragredient of V. Thus L = G/I, where G = ~nEZ Gn is the maximal graded Lie algebra with local part V+Lo+ V* and I = eDnEZ I' is the maximal graded ideal of G which intersects the local part trivially. By developing a homological theory, we determined the structure of homogeneous subspaces Ln = G/II, as modules over the affine Kac-Moody Lie algebra Lo for certain higher levels. The Hochschild-Serre spectral sequences played an important role in determining the structure of I,n. For the hyperbolic Kac-Moody Lie algebra HA(1), we were able to determine the structure of L4 for n = 1, 2, ... , 5, and computed the principally specialized affine characters up to level 5. But the root multiplicities were computed up to level 3 only. In this paper, using the representation theory of A(1) developed Received by the editors June 4, 1990 and, in revised form, September 12, 1991. 1991 Mathematics Subject Classification. Primary 1 7B67, 1 7B65; Secondary 1 7B10. This work was supported by NSF grant DMS 8806371. () 1994 American Mathematical Society 0002-9947/94 $1.00 + $.25 per page

The first part of this chapter treats Lie algebras, beginning with definitions and many examples. The notions of solvable, nilpotent, radical, semisimple, and simple are introduced, and these notions are followed by a discussion of the effect of a change of the underlying field. The idea of a semidirect product begins the development of the main structural theorems for real Lie algebras—the iterated construction of all solvable Lie algebras from derivations and semidirect products, Lie's Theorem for solvable Lie algebras, Engel's Theorem in connection with nilpotent Lie algebras, and Cartan's criteria for solvability and semisimplicity in terms of the Killing form. From Cartan's Criterion for Semisimplicity, it follows that semisimple Lie algebras are direct sums of simple Lie algebras. Cartan's Criterion for Semisimplicity is used also to provide a long list of classical examples of semisimple Lie algebras. Some of these examples are defined in terms of quaternion matrices. Quaternion matrices of size n-by-n may be related to complex matrices of size 2n-by-2n. The treatment of Lie algebras concludes with a study of the finite-dimensional complex-linear representations of sl[(2, ℂ). There is a classification theorem for the irreducible representations of this kind, and the general representations are direct sums of irreducible ones. Section 10 contains a review of the elementary theory of Lie groups and their Lie algebras. The abstract theory as in Chevalley [1946] is summarized, and the correspondence is made with the concrete theory of closed linear groups, where the Lie algebra is obtained as the space of derivatives at t = 0 of smooth curves in the group passing through the identity at t = 0. The section ends with a discussion of the adjoint representation. The remainder of the chapter explores some aspects of the connection between Lie groups and Lie algebras. One aspect is the relationship between automorphisms and derivations. The derivations of a semisimple Lie algebra are inner, and consequently the identity component of the group of automorphisms of a semisimple Lie algebra consists of inner automorphisms. In addition, simply connected solvable Lie groups may be built one dimension at a time as semidirect products with ℝ1, and consequently they are diffeomorphic to Euclidean space. For simply connected nilpotent groups the exponential map is itself a diffeomorphism. The earlier long list of classical semisimple Lie algebras corresponds to a list of the classical semisimple Lie groups. The issue that needs attention for these groups is their connectedness, and this is proved by using the polar decomposition of matrices.

It is proved that if a (ℤ/p ℤ)-graded Lie algebra L, where p is a prime, has exactly d nontrivial grading components and dim L 0 = m, then L has a nilpotent ideal of d-bounded nilpotency class and of finite (m,d)-bounded codimension. As a consequence, Jacobson's theorem on constant-free nilpotent Lie algebras of derivations is generalized to the almost constant-free case. Another application is for Lie algebras with almost fixed-point-free automorphisms.

Almost solubility of Lie algebras with almost regular automorphisms

Imbedding of Lie triple systems into Lie algebras

On a remarkable class of flat pseudo-Riemannian Lie groups

We prove that if a (ℤ/nℤ)-graded Lie algebra L = ↕i=0n−1Li has d nontrivial components Li and the null component L0 has finite dimension m, then L has a homogeneous solvable ideal of derived length bounded by a function of d and of codimension bounded by a function of m and d. An analogous result holds also for the (ℤ/nℤ)-graded Lie rings L = ↕i=0n−1 with few nontrivial components Li if the null component L0 has finite order m. These results generalize Kreknin’s theorem on the solvability of the (ℤ/nℤ)-graded Lie rings L = ↕i=0n−1Li with trivial component L0 and Shalev’s theorem on the solvability of such Lie rings with few nontrivial components Li. The proof is based on the method of generalized centralizers which was created by E. I. Khukhro for Lie rings and nilpotent groups with almost regular automorphisms of prime order [1], as well as on the technique developed in the work of N. Yu. Makarenko and E. I. Khukhro on the almost solvability of Lie algebras with an almost regular automorphism of finite order [2].

We define a class of infinite-dimensional Lie algebras that generalize the universal enveloping algebra of the algebra sl(2, ℂ) regarded as a Lie algebra. These algebras are a special case of ℤ-graded Lie algebras with a continuous root system, namely, their Cartan subalgebra is the algebra of polynomials in one variable. The continuous limit of these algebras defines new Poisson brackets on algebraic surfaces.

This chapter is devoted to studying some concepts that will be extensively used in the last chapters, namely the cohomology of Lie algebras with values in a vector space, the Whitehead lemmas and Lie algebra extensions (which are related to second cohomology groups). The same three different cases of extensions of chapter 5 as well as the ℱ( M )-valued version of cohomology will be considered. In fact, the relation between Lie group and Lie algebra cohomology will be explored here, first with the simple example of central extensions of groups and algebras (governed by twococycles), and then in the higher order case, providing explicit formulae for obtaining Lie algebra cocycles from Lie group ones and vice versa. The Lie algebra cohomology a la Chevalley-Eilenberg, which uses invariant forms on a Lie group, is also presented in this chapter. This will turn out to be specially useful in the construction of physical actions (chapter 8), i.e . in the process of relating cohomology and mechanics. The BRST formulation of Lie algebra cohomology will also be discussed here due to its importance in gauge theories. Cohomology of Lie algebras: general definitions We now discuss the cohomology of Lie algebras with values on a vector space V following the same pattern as used in chapter 5 for the cohomology of groups with values in an abelian group. Let G be a Lie algebra, and let the vector space V over the field K be a ρ(G)-module.

In this paper we establish the existence of a faithful matrix representation of finite type for every connected simply connected graded Lie group. We also show the 1-1 correspondence between finite-dimensional representations of a graded Lie algebra and the representations of finite type of the corresponding connected simply connected graded Lie group.

This article gives a local answer to the coquecigrue problem for Leibniz algebras, that is, the problem of finding a generalization of the (Lie) group structure such that Leibniz algebras are the corresponding tangent algebra structure. Using links between Leibniz algebra cohomology and Lie rack cohomology, we generalize the integration of a Lie algebra into a Lie group by proving that every Leibniz algebra is isomorphic to the tangent Leibniz algebra of a local Lie rack. This article ends with an example of a Leibniz algebra integration in dimension 5.