Chapter, classification of complex semisimple lie algebras. We will use this to classify complex lie algebras, that is, lie algebras over c, so for the rest of the paper lwill denote a complex lie algebra. Simple lie subalgebras of locally finite associative algebras. Secondly, for each of these lie algebras, one needs to classify all possible finite dimensional imodules m of cm functions. Conversely, to any finite dimensional lie algebra over real or complex numbers, there is a corresponding connected lie group unique up to finite coverings lie s. Pdf chapter 2, first basic definitions on lie algebras. Are there nonreductive lie algebras with abelian cartan subalgebras. Locally finite lie algebras with root decomposition. Because xis semisimple, it is adsimisimple, so there is. The structure of locally finite split lie algebras core. Pdf cartan subalgebras, weight spaces, and criterion of. The topic of the present work is the algorithmic treatment of finite dimensional lie algebras. On the conjugacy of maximal toral subalgebras of certain. Conjugacy theorems for loop reductive group schemes and lie.
Contents 1 basic definitions and examples 2 2 theorems of engel and lie 4 3 the killing form and cartan s criteria 8 4 cartan subalgebras 12 5 semisimple lie algebras 15. Lie groups are smooth differentiable manifolds and as such can be studied using differential calculus, in contrast with the case of more general topological groups. Such a lie algebra is a split extension of an abelian lie algebra by a direct sum of copies of. This groupoid together with its twist is a complete isomorphism invariant of the cartan subalgebra. Chevalleyjordan decomposition for a class of locally. Locally finite lie algebras with root decomposition request pdf. We present a deterministic polynomial time algorithm for the case when the ground fieldk is sufficiently large. In this paper we essentially classify all locally finite lie algebras with an involution and a compatible root decomposition which permit a faithful unitary highest weight representation. Cartan subalgebras of rootreductive lie algebras sciencedirect. Inner ideals are closely related to adnilpotent elements which proved to be useful in classi cation of the modular simple lie algebras and are helpful in constructing gradings in lie algebras. Infinite dimensional lie algebras international workshop organising committee.
One way to construct a cartan subalgebra is by means of a regular element. Inner ideals of simple locally finite lie algebras a subspace i of a lie algebra lis called an inner ideal if i. Gradings on the simple cartan and melikyan type lie algebras 4. Regular subalgebras and nilpotent orbits of real graded lie.
A periodisation of semisimple lie algebras larsson, anna, homology, homotopy and applications, 2002. Pdf the frattini subalgebra of nlie algebras researchgate. Dec 19, 2014 this talk is about the notion of cartan subalgebras introduced by renault, based on work of kumjian. Among the kacmoody lie algebras the affine algebras stand out. In fact, the elements of a cartan subalgebra of a reductive lie algebra are semisimple, so a weaker question is. Borel and parabolic subalgebras of some locally finite lie algebras 3.
In 11, 12 the class under consideration comprised those algebras generated by a system of finite dimensional ascendant subalgebras. Totally geodesic subalgebras in 2step nilpotent lie algebras decoste, rachelle c. A parabolic subalgebra of is any subalgebra that contains a maximal locally solvable that is, borel subalgebra. Recall that a real semisimple lie algebra g is called a compact lie algebra if the killing form is negative definite. Lie algebras with abelian cartan subalgebras mathoverflow.
Let g be a lie algebra and h a cartan subalgebra of g. Lee is a group whose elements are organized continuously and smoothly, as opposed to discrete groups, where the elements are separatedthis makes lie groups differentiable manifolds. Following np, we say that a subalgebra h of g is a cartan subalgebra if h is locally nilpotent and h z gh ss. We apply this technique to sl3,o, a real form of e 6 with signature 52, 26, thereby identifying chains of real subalgebras and their corresponding cartan subalgebras within e 6. Cartan subalgebras, cayley transforms, and classi cation hannah m. Gein, modular subalgebras and projections of locally finite dimensional lie algebras of characteristic zero, russian ural. Lie groups are named after norwegian mathematician sophus lie, who laid the foundations of the theory of continuous transformation groups. In this worksheet we use the 15dimensional real lie algebra su2, 2 to illustrate some important points regarding the general structure theory and classification of real semisimple lie algebras.
The main result of this paper is a classification of the separable simple p algebras which have cartan decompositions see 2 and it is shown that this class coincides with the simple selfadjoint lie subalgebras of a separa. We end with examples of cartan subalgebras in c algebras. Chapter 8 begins with the structure of split semisimple lie. It is a subalgebra that is nilpotent and its own normaliser. Throughout this paper denotes a field of characteristic zero and a. Cartan subalgebras exist for finitedimensional lie algebras whenever the base field is infinite. Positive energy representations for locally finite split lie.
The cartan decomposition of a complex semisimple lie algebra. In this paper the irreducible representations of infinitedimensional filtered lie algebras are studied. Let be a locally reductive complex lie algebra that admits a faithful countabledimensional finitary representation v. Lewis the di erential geometry software package in maple has the necessary tools and commands to automate the classi cation process for complex simple lie algebras. Suppose that g is the lie algebra of a lie group g. Readings introduction to lie groups mathematics mit. Regular subalgebras and nilpotent orbits of real graded lie algebras 3 given a carrier algebra, we. Cartan subalgebras of rootreductive lie algebras request pdf. Automorphisms of finite order of semisimple lie algebras.
The structure of tensor representations of the classical nitedimensional lie algebras was described by h. Chevalleyjordan decomposition for a class of locally finite lie. We consider the algorithmic problem of computing cartan subalgebras in lie algebras over finite fields and algebraic number fields. We call an abelian subalgebra of a lie algebra a splitting cartan. The killing form and cartans criterion the killing form is a symmetric bilinear form on lie algebras that will allow us to determine when lie algebras are semisimple or solvable. Following np, we say that a subalgebra h of g is a cartan subalgebra if h. First, one needs to classify the finite dimensional lie algebras of vector fields b on the manifold up to diffeomorphism. The simple lie algebras over c and their compact real forms. Our method is based on a solution of a linear algebra problem. Chevalleyjordan decomposition for a class of locally finite. Since h is nilpotent, the adjoint action of h on g induces a weight space decomposition g l. Conjugacy theorems for loop reductive group schemes and. Locally finite basic classical simple lie superalgebras.
Locally finite lie algebras ivan penkov and konstantin styrkas abstract. Being infinite objects, locally finite lie algebras are very complicated in comparison with lie algebras of finite. Conjugacy of cartan subalgebras the simple finite dimensional algebras. Lie algebras of differential operators over a given manifold. Chapter 12, classification of connected coxeter graphs. Pdf in this work the properties of cartan subalgebras and weight spaces of finite dimensional lie algebras are extended to the case of leibniz. The concept of the height of a representation is introduced, and it is proved that the representations of height greater than one of the lie algebras w, s. We also solve the problem of describing the set of conjugacy classes of cartan subalgebras of the three simple rootreductive lie algebras sl. Lecture 11 cartan subalgebras october 11, 2012 1 maximal toral subalgebras a toral subalgebra of a lie algebra g is any subalgebra consisting entirely of abstractly semisimple elements. Contents 1 basic definitions and examples 2 2 theorems of engel and lie 4 3 the killing form and cartans criteria 8 4 cartan subalgebras 12 5 semisimple lie algebras 15. Rootreductive lie algebras are a specific class of locally finite. It is also worth to mention that solvable lie algebras always have cartan subalgebras. Over a finite field, the question of the existence is still open. The purpose of this thesis is to write the programs to complete the classi cation for.
A cartan subalgebra of a lie algebra g is a subalgebra h, satisfying the following two conditions. In the study of the complex finitedimensional semisimple lie algebras a crucial role is played by the fundamental bilinear form x, y tradxady. The vector space together with this operation is a nonassociative algebra, meaning that the lie bracket is not necessarily associative lie algebras are closely related to lie groups. This talk is about the notion of cartan subalgebras introduced by renault, based on work of kumjian. A subalgebra s of l is stable if 5 remains a subalgebra under small deformations of l. We assume henceforth that kis algebraically closed, but the reader should keep in mind that our results are more akin to the setting of chevalleys theorem for general k than to conjugacy of cartan subalgebras in nitedimensional simple lie. In this paper we extend weyls results to the classical in nitedimensional locally nite lie algebras gl 1, sl, sp 1 and so, and study important. Chapter 7 deals with cartan subalgebras of lie algebras, regular elements and.
If g is a semisimple finitedimensional lie algebra, the. Lie algebras of differential operators in two complex. Are there nonreductive lie algebras with abelian cartan subalgebras all of whose elements are. Isotropy subalgebras of elliptic orbits in semisimple lie algebras, and the canonical representatives of pseudohermitian symmetric elliptic orbits. As part of a structure theory program for rootreductive lie algebras, cartan subalgebras of the lie algebra gl. The irreducible riemannian globally symmetric spaces of type ii and type iv. Cartan subalgebras of locally reductive lie algebras let g be a locally reductive lie algebra. Regular subalgebras and nilpotent orbits of real graded. We explain how cartan algebras build a bridge between dynamical systems and operator algebras. A locally finite lie algebra is said to be finitary if there exists a faithful finitary representation of any finitary lie algebra is isomorphic to a subalgebra of a locally finite lie algebra is said to be locally semisimple if it admits an exhaustion by finitedimensional semisimple subalgebras.
The conjugacy of split cartan subalgebras in the finitedimensional simple case chevalley and in the symmetrizable kacmoody case petersonkac are fundamental results of the theory of lie algebras. Abstract we will extend the conjugacy problem of maximal toral subalgebras for lie algebras of the form g. Alexander zalesskii classification of simple infinite dimensional lie subalgebras of locally finite. Computing cartan subalgebras of lie algebras springerlink.
Cartan subalgebras exist for finite dimensional lie algebras whenever the base field is infinite. We assume henceforth that kis algebraically closed, but the reader should keep in mind that our results are more akin to the setting of chevalleys theorem for general k than to conjugacy of cartan subalgebras in nitedimensional simple lie algebras over algebraically closed elds. In the case at hand, it is useful to introduce the following concepts. The lie algebra g is compact if and only if all the root vectors for any. In this paper we study the concept of a cartan subalgebra in the context of locally finite and borcherdskacmoody lie algebras. One of the key ideas in the theory of lie groups is to replace the global object, the group, with its local or linearized version, which lie himself called its infinitesimal group and which has since become known as its lie algebra. If h is a subalgebra of g, let h ss denote the set of semisimple jordan components of the elements of h. Moreover, a general definition of a splitting cartan subalgebra h. Since the definition is meaningless when the restriction of. Lie algebras are closely related to lie groups, which are groups that are also smooth manifolds. Discovering real lie subalgebras of e6 using cartan.
Locally finite lie algebras with root decomposition springerlink. On the other hand, the existence of cartan subalgebras of lie algebras defined over small fields still remains an open problem. Zhilinskii has also provided a classification of coherent local systems zh1, zh2. Let g be a reductive lie algebra and let h be a cartan subalgebra of g.
If the field is algebraically closed of characteristic 0 and the algebra is finite dimensional then all cartan subalgebras are conjugate under automorphisms of the lie. Cyclic highest weight modules serre s theorem clifford algebras and spin representations the kostant dirac operator. Theorem any two cartan subalgebras of g are conjugate. The cartan subalgebras of a reductive lie algebra are abelian. Simple lie subalgebras of locally finite associative algebras article pdf available in journal of algebra 2811 november 2004 with 35 reads how we measure reads. Cartan subalgebras, compact roots and the satake diagram. If the field is algebraically closed of characteristic 0 and the algebra is finitedimensional then all cartan subalgebras are conjugate under automorphisms of the lie. Chapter 1 deals with the correspondence between lie groups and their lie algebras, subalgebras and ideals, the functorial relationship determined by the exponential map, the topology of the classical groups, the iwasawa decomposition in certain key cases, andthe baker campbell. Any cartan subalgebra of g is a maximal nilpotent subalgebra proof. Parabolic and levi subalgebras of finitary lie algebras.
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