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Cohomology of Lie Algebras

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Cohomology of Lie Algebras FACULTY OF SCIENCE MASARYK UNIVERSITY COHOMOLOGY OF LIE ALGEBRAS Ph.D. thesis Auckland, December 2003 Josef Šilhán Acknowledgement. I would like to thank my supervisor Jan Slovák for di­ recting my study and research and providing me resources which helped me significantly during a work on this dissertation. I thank him also for an encour- agemenet which I really needed. Further I thank Arkadiy Onishchik for his lectures in Masaryk University in Brno, 2001. They provided a background for main results involved in the dissertation. The dissertation was finished during my study at University of Auckland under the supervision of Rod Gover. I also thank him for his hospitality. Contents Introduction 2 1 Cohomology of Lie algebras, the complex case 4 1.1 Definition of the cohomology 4 1.2 The complex case 4 2 The real case 8 2.1 Real semisimple Lie algebras and their representations 9 2.2 Symmetries of diagrams 12 2.3 Formulation of the problem 13 2.4 Conjugated representations of reductive algebras 15 2.5 Indices 17 2.6 Relation between real and complex cohomologies 21 3 Algorithmic computations 25 3.1 Cohomology H(f_;V) 25 3.2 Algorithm 26 A Duals of reductive subalgebras 29 B Satake diagrams and representations of real semisimple Lie al­ gebras 33 1 Introduction The theory of Lie algebras and their representations offers important tools in the study of differential geometry. This algebraic theory can be used, for example to give a classification of geometric objects (see below for details about the classification of invariant operators and the curvature of Cartan geometries). Moreover, algebraic properties of the geometric objects can be described by an algorithm, which makes use of the theory effective. In next three paragraphs, we are going to remind a geometry background, an algebraic setting and to show some possible applications of results involved in this dissertation. Cartan geometries of a type (G, P) are "curved analogs" of homogeneuos spaces G/P where P is a closed subgroup of a Lie group G. This means that the bundle G —»• G/P and a Maurer-Cartan form ui G í)1 (G, g) are replaced by a principal P-bundle Q —• M equipped with a one-form u G fí1(^,g) which satisfies most of properties of the Maurer-Cartan form. Here g denotes the Lie 2 algebra of G. The two-form K = du + |[W,ÍU] G Í2 (í/;g) is called curvature. It meausures how "curved" the analog is, that is K vanishes if and only if (Q,ui) is locally isomorphic to (G,u). It can be shown that we can view K as an equivariant function on Q with values in f\ (g/p)* <g>g where p is the Lie algebra of P. Henceforth we will suppose that P is a parabolic subgroup of a semisimple Lie group G. Then, we have a natural decomposition g = p © g' where g' is a Lie algebra isomorphic to g/p as a vector space. In this case, Lie algebraic theory is effective in the study of Cartan geometries. If A : g —> gt(^) is a represen­ tation we can define a differential d : Horn (/\n g';V) —^ Horn (/\ g'',V), n G N U {0} such that d2 = 0. The corresponding cohomology space will be de­ noted by i?(q'; V). The structure of the parabolic subalgebra induces a natural representation of p on i?(q'; V). It can be shown that putting A = ad|g', the curvature K determines an equiv­ ariant function on Q with values in H2(g';g). Moreover, after an appropriate normalization of Cartan geometries, the structure of the cohomology determines completely the curvature of the Cartan geometry. The cohomology is also im­ portant in the study of invariant differential operators of Cartan geometries. In the flat case (i.e. K = 0), there is a bijective correspondence between in­ variant operators and homomorphisms of appropriate modules induced by rep­ resentation A and a parabolic subalgebra p C g. These modules are highest 2 weight modules. It can be shown that such homomorphisms can appear only between modules whose highest weights correspond to irreducible components in H(g'; V) up to some exceptions. Nevertheless, the same technique can be used for the classification of the exceptional homomorphisms. The structure of the cohomology is well-known in the complex case, that is if p C g are complex Lie algebras. This is described in a famous paper [Ko] by Bertram Kostant. Using the Dynkin diagrams, his results leads directly to an algorithm. A formulation of Kostant's result is the main subject of Chapter 1. Furthermore, while recounting some of the necessary facts from the standard theory of complex semisimple Lie algebras and their representations and its notation. Chapter 3 involves details about the algorithmic computation. If p is a parabolic subalgebra of a real semisimple Lie algebra g, the situation is more involved. The structure of representations of real semisimple Lie algebras is known. We shall use results from [On] though this approach is not widely known (see references therin for details). Another possibility in the study of real semisimple Lie algebras is given by Satake diagrams. We will use them for a diagramatic description. A summary of both these approaches is presented in the beginning of Chapter 2. A description of the real cohomology is a core of the dissertation. This is based on an observation that the complexification of the real cohomology is the cohomology of the complexified algebra and its complexified representation. First, a relation between Satake diagrams and the approach from [On] is estab­ lished. It turns out that it is sufficient to consider a representation of a reductive part of p on the cohomology. To exploit this, we need to generalize the neccesary results from [On] about semisimple algebras to reductive cases. This is done by combining the latter approach with the structure of Satake diagrams. The re­ sult is a description of the real cohomology in terms of the complexification and symmetries of Satake diagrams derived from its arrows. The usual setting in differential geometry leads to a cohomology dual to the case discussed in [Ko] and in Chapter 2. Required duals can be easily computed using tables from Appendix A, for details see 3.1.1. Appendix B involves a table of Satake diagrams with the complete information which is necessary for the description of representations of real semisimple Lie algebras as presented in [On]. Finally, a list of symbols is appended. 3 Chapter 1 Cohomology of Lie algebras, the complex case 1.1 Definition of the cohomology Let a be an arbitrary Lie algebra and TT : a —> gi(U) a representation on a (real or complex) vector space U. We define the differential d : Horn (/\n a; U) —> Horn {/\n+ a; U) by the formula i+ (dp)(X0, ...,Xn) ='52(-l) ip([Xi,Xj]1Xo1 ...Xi...Xj...1Xn) i<j + $3(-1)Í7r(^i)p(^0,...^i...,^n). i Since d? = 0, the differential d induces the cohomology Hn(a; U), called coho­ mology of a with the coefficients in U. We set Horn (/\n a; U) = 0 for n < 0 and n > dim a. We will be interested only in the special case where a is a nilradical of a parabolic subalgebra (see below) of a semisimple algebra a', TT is a restriction of a representation of a' and U is a finite dimensional vector space. In this case, we can define a representation of the parabolic subalgebra on Hn(a;U)', its description gives a required description of the cohomology. Let us note that H (a; U\ © U2) = H (a; U\) © H (a; U2) which follows directly from the definition of d. Therefore it is sufficient to consider only irreducible representations of 0. 1.2 The complex case In the rest of this chapter, we will suppose that all algebras are complex ones and A : f —> fitfOO is a representation on a complex vector space V. This case (where IT is a restriction of A) is completele solved in the paper [Ko] by B. Kostant. First, we remind some facts from the standard theory of complex 4 semisimple Lie algebras [Sam, FH]. For more detailed description of the complex cohomology see also Appendix of [SI], a monograph [CS] or [Sil]. 1.2.1. Weyl group and weights. Let us consider a complex semisimple Lie algebra f with a Cartan subalgebra f), sets of simple roots, positive roots and roots íl C A+ C A and Weyl group W. The group W is generated by simple reflections, i.e. the reflections corresponding to the simple roots. The number of positive roots a G A+ which are transformed to w(a) € A_ = — A+ is called the length of w for which we write \w\. Equivalently (see [FH]), the length of w is the minimal number of simple reflections in any expression for w in terms of simple reflections. The weights of f can be described by labelling the nodes of the Dynkin diagram by the integer coefficients referring to the linear combination of funda­ mental weights. The weight is dominant for f if and only if all the coefficients are nonnegative (such a labeled Dynkin diagram describes an irreducible repre­ sentation of f). The affine action of the Weyl group is defined by w.A = w(A 4- R) - R for the weight A where R = | ]Ca€A a *s *ne l°west strictly dominant weight of f. It means (in the terms of the Dynkin diagram) to add one over each node, then act with w and finally subtract one over each node.
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