UPPSALA DISSERTATIONS IN MATHEMATICS

94

Department of MathematicsUppsala University

UPPSALA 2016

Simple Modules over Lie Algebras

Jonathan Nilsson

Dissertation presented at Uppsala University to be publicly examined in Häggsalen,Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Wednesday, 1 June 2016 at 13:15 forthe degree of Doctor of Philosophy. The examination will be conducted in English. Facultyexaminer: Professor Erhard Neher (University of Ottawa).

AbstractNilsson, J. 2016. Simple Modules over Lie Algebras. Uppsala Dissertations in Mathematics94. 50 pp. Uppsala: Department of Mathematics. ISBN 978-91-506-2544-8.

Simple modules are the elemental components in representation theory for Lie algebras, andnumerous mathematicians have worked on their construction and classification over the lastcentury. This thesis consists of an introduction together with four research articles on the subjectof simple Lie algebra modules. In the introduction we give a light treatment of the basic structuretheory for simple finite dimensional complex Lie algebras and their representations. In particularwe give a brief overview of the most well-known classes of Lie algebra modules: highest weightmodules, cuspidal modules, Gelfand-Zetlin modules, Whittaker modules, and parabolicallyinduced modules.

The four papers contribute to the subject by construction and classification of new classes ofLie algebra modules. The first two papers focus on U(h)-free modules of rank 1 i.e. moduleswhich are free of rank 1 when restricted to the enveloping algebra of the Cartan subalgebra.In Paper I we classify all such modules for the special linear Lie algebras sln+1(C), and wedetermine which of these modules are simple. For sl2 we also obtain some additional results ontensor product decomposition. Paper II uses the theory of coherent families to obtain a similarclassification for U(h)-free modules over the symplectic Lie algebras sp2n(C). We also give aproof that U(h)-free modules do not exist for any other simple finite-dimensional algebras whichcompletes the classification. In Paper III we construct a new large family of simple generalizedWhittaker modules over the general linear Lie algebra gl2n(C). This family of modules isparametrized by non-singular nxn-matrices which makes it the second largest known family ofgl2n-modules after the Gelfand-Zetlin modules. In Paper IV we obtain a new class of sln+2(C)-modules by applying the techniques of parabolic induction to the U(h)-free sln+1-modules weconstructed in Paper I. We determine necessary and sufficient conditions for these parabolicallyinduced modules to be simple.

Keywords: Lie algebra, Representation, Simple module, Non-weight module, Classification,Construction

Jonathan Nilsson, Department of Mathematics, Algebra and Geometry, Box 480, UppsalaUniversity, SE-751 06 Uppsala, Sweden.

© Jonathan Nilsson 2016

ISSN 1401-2049ISBN 978-91-506-2544-8urn:nbn:se:uu:diva-283061 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-283061)

List of papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I Jonathan Nilsson;Simple sln+1-module structures on U (h).Journal of Algebra 424 (2015) 294–329.

II Jonathan Nilsson;U (h)-free modules and coherent families.Journal of Pure and Applied Algebra 220 (2016) 1475–1488.

III Jonathan Nilsson;A new family of simple gl2n(C)-modules.Pacific Journal of Mathematics 283, No. 1, (2016) 1–19.

IV Yan-an Cai, Genqiang Liu, Jonathan Nilsson, Kaiming Zhao;Generalized Verma modules over sln+2 induced from U (hn)-freesln+1-modules.Manuscript.

Reprints were made with permission from the publishers.

Also published by the Author but not included in this thesis:

Jonathan Nilsson;Enumeration of Basic Ideals in Type B Lie Algebras.Journal of Integer Sequences, Volume 15 2012, Article 12.9.5.

Foreword

This is a thesis on representation theory for Lie algebras. Thanks to con-nections throughout mathematics and physics, this subject has been activelyresearched for the last hundred years but many questions still remain unre-solved. In representation theory we study abstract mathematical objects calledmodules. To investigate these modules it is useful to break them down intosmaller, more simple components. To understand the principle, consider thefollowing analogy:

In basic chemistry we learn that there is a finite list of elements (the peri-odic table), and we know that the corresponding atoms can come together toform molecules. Moreover, some fundamental properties of a molecule, suchas its weight, can be deduced from knowing its atom constituents. Howeverthere also exist isomers – different molecules with the same atom components– which shows that molecules are not completely determined by their compo-nents.

The “atoms” of representation theory are called simple modules. Any mod-ule can analyzed in terms of its simple components – the simple modules whichit is made up of. Just as in chemistry however, truly different modules can con-sist of the same simple components, so we only get a rough picture by know-ing the simple constituents. While the regular periodic table contains 120 orso elements, the periodic table for the representation theory of a Lie algebra isunfortunately both incomplete and enormous: there are infinitely many simplemodules. This thesis is concerned with finding and classifying new classes ofsimple modules.

The thesis has two parts. In the first part we provide an overview of Lietheory and how it has developed over the years. In particular we discuss themost well-known classes of modules: highest weight modules, cuspidal mod-ules, Whittaker modules, Gelfand-Zetlin modules, and parabolically inducedmodules. The second part of the thesis is about my own work on constructingand classifying new families of modules. Here you will find summaries of thefour papers included at the end of the thesis, as well as a summary in Swedish.

Throughout this text all Lie algebras, modules, and vector spaces are as-sumed to be over the complex numbers C unless otherwise stated. We shallwrite ei, j for the matrix having a single 1 in position (i, j) and zeroes every-where else. N denotes the non-negative integers while N+ denotes the positiveintegers.

Contents

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Part I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3 Classification simple Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4 Some linear Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.5 The universal enveloping algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3 Simple modules and composition series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4 Categorical techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4.1 Characters and block decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4.2 Tensor functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4.3 Duality functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4.4 Functors related to subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.5 Twisting functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.6 Translation functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Known classes of modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1 Weight modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1.1 Highest weight modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1.2 Finite dimensional modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1.3 Category O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.1.4 Cuspidal modules and coherent families . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Whittaker modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.1 Kostant’s modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.2 Generalized Whittaker modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Gelfand-Zetlin modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.4 Parabolically induced modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Part II: Results of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 U (h)-free modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 On Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.1 Construction and classification of U (h)-free modules . . . . . . . . . . . . . . . 355.2 Submodules and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.3 Clebsch-Gordan problem for sl2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6 On Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.1 Weighting functor and coherent families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386.2 Classification in type C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.3 Applying translation functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

7 On Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407.1 Decomposition of gl2n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407.2 Module construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

8 On Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428.1 Generalized Verma modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428.2 Our construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428.3 Simplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

9 Summary in Swedish - Sammanfattning på svenska . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.1 Bakgrund . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.2 Kända klasser av enkla moduler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

9.2.1 Viktmoduler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449.2.2 Ändligtdimensionella moduler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.2.3 Whittakermoduler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459.2.4 Gelfand-Zetlinmoduler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

9.3 Avhandlingens resultat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469.3.1 U (h)-fria moduler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469.3.2 Generaliserade Whittakermoduler för gl2n . . . . . . . . . . . . . . . . . . . . 479.3.3 Generaliserade Vermamoduler från U (h)-fria

moduler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Part I:Introduction

1. Lie algebras

1.1 BackgroundSophus Lie was born in Norway in 1842. When he moved to Berlin for ascholarship he became close friends with the German mathematician FelixKlein and by reading the earlier works of Jacobi, they developed an interestfor transformations of differential equations together.

Their work and exchange of ideas resulted in an important paper by Lieabout what he called continuous transformation groups. Here a transformationgroup meant a set of transformations, closed under composition, from somesubset of n-dimensional complex space to itself. For Lie, the word continuous(as well as group) meant something different than it does today. Lie writes

"A group is called continuous when all of its transformations are generated byrepeating infinitesimal transformations infinitely often.“

Lie’s brilliant idea was to – instead of analyzing the transformation groupdirectly – study the space of such infinitesimal transformations.

The modern generalization of these transformation groups are called Liegroups. They are important in mathematics and physics since they describethe sets of symmetries of objects (think of the symmetries of a sphere forexample).

Although Lie viewed infinitesimal transformations as special elements inthe group itself, modern mathematics calls the space they generate the Liealgebra of the Lie group. The Lie algebra is a vector space and as such is easierto work with. Also, as Lie realized, the structure of the group can typically bereconstructed from the Lie algebra via an exponential map.

One of Lie’s great projects was to try and classify all transformation groups.The German mathematician Wilhelm Killing suggested that it would be nec-essary to first classify all possible Lie algebras, and he devoted much time tothis project himself during the late 19th century. Although Killing’s work wassometimes confusing, in the end he managed to show that aside from a fewexceptional cases, there can only exist Lie algebras of the special, symplecticand orthogonal kind. The young French mathmatician Élie Cartan removedany doubt about Killings result with his complete classification of simple com-plex Lie algebras in his doctoral thesis which was published in 1894. Cartancontinued on to also classify Lie algebras over the real numbers by realizingthem as real forms of the complex algebras. A review of Killing and Cartan’sapproach to Lie algebra classification is presented in Section 1.3. For moreinformation about the early developments of Lie theory, see [Ha].

11

1.2 DefinitionNowadays Lie algebras are often introduced abstractly without relying on aLie group.

A Lie algebra g is a vector space in which you can multiply vectors using aso called Lie bracket written [x,y]. Like regular multiplication of numbers, thebracket is required to be linear in each entry i.e. [x+y,z] = [x,z]+[y,z] etcetera.Unlike multiplication however, the bracket also has to satisfy [x,x] = 0 for allx which is equivalent to skew-symmetry:

[x,y] =−[y,x] for all x,y in g. (1.1)

The bracket is not associative either, instead it satisfies the Jacobi-identity:

[x, [y,z]]+ [z, [x,y]]+ [y, [z,x]] = 0 for all x,y,z in g. (1.2)

Any bilinear operation satisfying (1.1) and (1.2) gives rise to a Lie algebra sothere are many examples:• Let V be any vector space and define [x,y] = 0 for all x,y ∈V . This op-

eration is bilinear and satisfies, so this is a (trivial) Lie algebra structure.• Three-dimensional space R3 becomes a Lie algebra (over R) if we define

the bracket operation by [x,y] := x× y (cross product of vectors). ThisLie algebra structure is usually encountered by engineers and mathe-maticians in the first course in Linear algebra.• Let W be the infinite dimensional vector space with basis {ln|n ∈ Z}

equipped with the bracket [lm, ln] = (m− n)lm+n. This Lie algebra iscalled the Witt algebra. Together with the related Virasoro-algebra itplays a part in theoretical physics.• Let gln be the set of n×n-matrices and define [A,B] := AB−BA, where

the multiplication on the right side is usual matrix multiplication. Thisis called a general linear Lie algebra.• More generally, let g be any associative algebra with multiplication(x,y) 7→ x · y. By defining [x,y] := x · y− y · x we equip g with a Liealgebra structure.

1.3 Classification simple Lie algebrasThe elemental components of Lie algebras are called simple Lie algebras. Thetheory leading to the classification of simple finite dimensional Lie algebraswas developed by Killing-Cartan, and the results are very beautiful. Theirwork is summarized in what follows.

For a simple Lie algebra g, let h be a nilpotent self-normalizing subalgebraof g called a Cartan subalgebra. Now let h∗ = HomC(h,C) be the space oflinear maps h→C. For each nonzero α ∈ h∗ we define the corresponding root

12

space gα in g to be

gα := {x ∈ g | [h,x] = α(h)x for all h ∈ h}.

Then x ∈ gα just means that x is an eigenvector for the operator [h,−] forall h ∈ h simultaneously, where the corresponding eigenvalue is given by thescalar α(h). Most spaces gα will be zero, and the nonzero gα will have di-mension 1. It also turns out that every element of g can be written uniquely asa linear combination of root vectors and a Cartan element. In other words wehave

g= h⊕⊕α∈Φ

gα , (1.3)

where Φ is the set of roots i.e. nonzero α ∈ h∗ such that gα 6= 0. It is now easyto check that

[h,gα ]⊂ gα and that [gα ,gβ ]⊂ gα+β ,

so the decomposition above tells us much about the internal structure of g.We now turn our attention to the geometry of the set Φ in h∗. The vector

space g comes equipped with a symmetric bilinear form called the Killing formdefined by

κ(x,y) = tr([x, [y,−]]).

It turns out that the restriction of the Killing form to h×h is non-degeneratewhich allows us to for each α ∈ h∗ define tα ∈ h as the unique element satis-fying κ(tα ,h) = α(h) for all h ∈ h. This lets us define a symmetric bilinearform on h∗ by (α,β ) := κ(tα , tβ ). Using this form we define reflections in h∗

in the natural way: for each nonzero α ∈ h∗ we define

σα : h∗→ h∗ by σα(β ) = β −2(α,β )

(α,α)α.

Geometrically, σα reflect vectors in the hyperplane orthogonal to α .In fact, the set Φ appearing in our Lie algebra decomposition (1.3) always

satisfies the following symmetry properties.

• Φ is finite, does not contain 0, and spans h∗.• For α ∈Φ we also have −α ∈Φ, but no other multiples of α occur.• We have σα(Φ) = Φ for all α ∈Φ.• 2(β ,α)/(α,α) ∈ Z for all α,β ∈Φ.

In other words, Φ forms a so called root system in h∗. The rank of theroot system is defined as the dimension of h∗. The subgroup W of GL(h∗)generated by {σα}α∈Φ is called the Weyl group of g.

Here follows a picture of all possible root systems of rank 2.

13

A1×A1

//

OO

��

oo

A2

//

FF

��1111111111oo

��

XX1111111111

B2

//

OO

��

oo

??��������������

��??????????????

__??????????????

����������������

G2

FF

OO

XX111111

��111111

��

ffMMMMMMMMMM

&&MMMMMMMMMM

88qqqqqqqqqq

��

xxqqqqqqqqqq //oo

Instead of keeping track of all roots it is useful to introduce a base of theroot system. A base is defined as a subset ∆ ⊂ Φ such that each root β canbe uniquely expressed as β = ±∑α∈∆ cαα where all cα ∈ N+. Here, the setof roots for which the plus sign is chosen are denoted Φ+. Its elements arecalled positive roots. The set Φ− of negative roots is defined correspondingly.The elements of the base ∆ are called simple roots. For example, consider thefollowing root system:

α

β FF

OO

XX111111

��111111

��

ffMMMMMMMMMM

&&MMMMMMMMMM

88qqqqqqqqqq

��

xxqqqqqqqqqq //oo

Here ∆ = {α,β} is a base for the root system, which gives positive rootsΦ+ = {α, β , β +α, β +2α, β +3α, 2β +3α}, while Φ− =−Φ+. Letting

n+ :=⊕

α∈Φ+

gα and n− :=⊕

α∈Φ−gα ,

14

we have obtained a triangular decomposition of our Lie algebra:

g= n−⊕h⊕n+, (1.4)

separating it into a negative part, a Cartan subalgebra, and a positive part.Many aspects of the representation theory for Lie algebras relies on this de-composition.

In fact, a root system is almost completely determined by the angles be-tween its simple roots as illustrated in the figure of rank 2 root systems. More-over, the only possible such angles are π/2, 2π/3, 3π/4, and 5π/6. This letsus to encode the combinatorial data of a root system in so called Coxeter graphi.e. a graph with a vertex for each simple root, and 0,1,2, or 3 edges betweentwo vertices depending on whether the angle between the two correspondingroots are π/2, 2π/3, 3π/4, or 5π/6.

Thus the four root systems of rank 2 listed above has the following Coxetergraphs:

A1×A1 : ◦◦ ◦◦

A2 : ◦ ◦

B2 : ◦ ◦

G2 : ◦ ◦

In order to completely determine the root system however, it turns out thatwe need one more piece of information: for each double- and triple edge inthe Coxeter graph we need to specify which of the two involved roots is thelongest. We do this by for each such edge drawing an arrow pointing towardsthe longer root. Our new objects are called Dynkin diagrams, and they deter-mine the root system completely.

Not every Dynkin diagram we imagine can be derived from a root systemthough. The admissible Dynkin diagrams fall into four infinite families, to-gether with five special cases.

Theorem 1. Consider the following list of Dynkin diagrams where the sub-scripts indicate the number of vertices.

An (n≥ 1) ◦ ◦◦ ◦ ◦ ◦

Bn (n≥ 2) ◦ ◦ ◦ ◦◦ ◦//

Cn (n≥ 3) ◦ ◦ ◦ ◦◦ ◦oo

Dn (n≥ 4) ◦ ◦ ◦ ◦◦ ◦◦

◦

15

E6 ◦ ◦◦ ◦◦ ◦◦ ◦◦

◦

E7 ◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦◦

◦

E8 ◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦◦ ◦◦

◦

F4 ◦ ◦◦ ◦◦ ◦//

G2 ◦ ◦oo

Every simple finite dimensional complex Lie algebra has one of these listedDynkin diagrams, and conversely, each listed diagram determines the corre-sponding Lie algebra up to isomorphism.

One strength of having such a theorem is that many results can be provedby case-by-case investigations. This is illustrated in Paper I and Paper II ofthis thesis.

1.4 Some linear Lie algebrasIf V is a vector space, the space of linear maps V→V is a Lie algebra under theLie bracket [ f ,g] = f ◦g−g◦ f . This is called the general linear Lie algebraand it is denoted gl(V ). It is a basic object of study in Lie theory. When V isfinite dimensional we may choose a basis consisting of n elements and identifygl(V ) with a space of n× n matrices and we write gln for this algebra. Theimportance of this Lie algebra is displayed by Ado’s theorem which states thatevery finite dimensional Lie algebra can be embedded as a subalgebra of glnfor some n. The Lie algebra gln falls outside the classification in Theorem 1;it is not simple as the identity-matrix spans an ideal.

The four infinite-families of Lie algebras from Theorem 1 can be realizedas subalgebras of gln as follows.

Letsln+1 := {A ∈ gln+1 | tr(A) = 0}.

This is the special linear Lie algebra. It corresponds to the root system of typeAn. Of particular interest is the first Lie algebra in this series, sl2. It has a basis

x =( 0 1

0 0

)h =

( 1 00 −1

)y =

( 0 01 0

), (1.5)

16

with respect to which the bracket is given by

[h,x] = 2x, [h,y] =−2y, [x,y] = h.

Although this is an easy algebraic object, many deep Lie theoretic results canbe derived from its study, see for example [Maz2].

Next, the symplectic Lie algebra of rank n is defined by

sp2n ={( A B

C −AT

)∈ gl2n

∣∣∣ BT = B, CT =C}.

This is a simple Lie algebra of type Cn.The orthogonal Lie algebras fall into two classes with different properties

depending on whether the rank of the Lie algebra is even or odd – we define

o2n ={( A B

C −AT

)∈ gl2n

∣∣∣ BT =−B, CT =−C},

which is a simple Lie algebra of type Dn, and we define

o2n+1 =

{( 0 a b−bT A B−aT C −AT

)∈ gl2n+1

∣∣∣∣∣ BT =−B, CT =−C

},

which has type Bn.

1.5 The universal enveloping algebraNon-associative structures are in general harder to handle than associativeones. The universal enveloping algebra of a Lie algebra is a unital associa-tive algebra which encodes the non-associative Lie bracket as the commutatoroperation. By passing to the universal enveloping algebra we obtain an al-gebraically easier object without loosing information about the representationtheory.

Recall that any algebra A becomes a Lie algebra by defining [x,y] = xy−yx. We denote this Lie algebra by A′. The universal enveloping algebra of aLie algebra g can be defined abstractly as an associative algebra U togetherwith a Lie algebra homomorphism i : g→U ′ such that for any algebra A andfor any Lie algebra homomorphism j : g→ A′ there exists a unique algebrahomomorphism ϕ : U→ A such that ϕ ◦ i = j. One can show that the universalenveloping algebra exists and is unique. It is denoted U (g).

More concretely, U (g) can be constructed as follows. We start with theTensor algebra T on g. This is the vector space spanned by elements of formx1⊗ x2⊗·· ·⊗ xr with xi ∈ g and r ≥ 0. The algebra product on this basis istensor product concatenation. Letting J be the two-sided ideal generated by all

17

elements of form x⊗y−y⊗x we define U (g) := T/J and we usually write itselements without tensor signs. A key observation here is that U (g) commutesup to terms of lower degree – this leads to the famous Poincaré-Birkhoff-Witttheorem which provides us with a basis for U (g).

Theorem 2. Let (x1,x2, . . . ,xn) be any ordered basis of the vector space g.Then the set

{xm11 xm2

2 · · ·xmnn |m1,m2, . . . ,mn ∈ N}

is a basis for U (g).

18

2. Representation theory

2.1 BackgroundParallell to the mathematical development of the Killing-Cartan theory in thebeginning of the 20th century, a movement started among the many greatphysicists at the university of Göttingen. The goal was to develop a soundmathematical formalization of physics. At this time interest grew for the rep-resentations of Lie algebras i.e. how Lie algebras can act on vector spacesin a coherent fashion. At this time the mathematicians Hermann Weyl, IssaiSchur, and Claude Chevalley made many contributions to representation the-ory for Lie algebras, such as the famous Schur’s lemma and Weyl’s characterformula. With some help from Weyl’s work, Cartan managed to classify allsimple finite dimensional representations in 1914.

When Weyl moved from Göttingen to Zürich in 1913 he met a young Ein-stein who was working on his theory of general relativity. Weyl was veryimpressed with Einsteins work. He wrote

For myself I can say that the wish to understand what really is the mathematicalsubstance behind the formal apparatus of relativity theory led me to the studyof representations and invariants of groups.

Since the early 1900’s, countless developments have been made in Lie al-gebra representation theory. Many new classes of modules were discoveredand I have tried to present most of them in Section 3.

2.2 DefinitionA representation of a Lie algebra g is a homomorphism g→ gl(V ) such thateach element of g is represented as an Lie algebra endomorphism on V . It isoften more useful to instead view a representation as a module – a vector spaceon which g acts.

A module for a Lie algebra g is a vector space M equipped with a bilinearaction g×M→M written (x,m) 7→ x ·m such that

x · (y ·m)− y · (x ·m) = [x,y] ·m

for all x,y ∈ g and m ∈M. Alternatively we can consider M as a module overthe universal enveloping algebra of g. The bilinear action of U (g) on M isthen instead required to satisfy

x · (y ·m) = (xy) ·m and 1 ·m = m

19

for all x,y ∈ g and m ∈M, which is the standard definition of a module over aring. Since xy− yx = [x,y] for all x,y ∈ U (g) the two definitions are equiva-lent. This lets us identify g-modules with U (g)-modules.

2.3 Simple modules and composition seriesModules can be broken down into their simple components in the followingsense. A submodule N of the g-module M is a subspace N ⊂ M closed un-der the g-action. For each submodule we also have a corresponding quotientmodule – a canonical module structure on the quotient space M/N. A modulecan be studied in terms of its submodules and quotients. The trivial module{0} and M itself are always submodules of M, and if these are the only twosubmodules, M is called a simple module.

A composition series for a module M is a chain of submodules

0 = M0 ⊂M1 ⊂M2 ⊂ ·· · ⊂Mr−1 ⊂Mr = M,

such that the quotient modules Li := Mi/Mi−1 are simple for all 1 ≤ i ≤ r. IfM admits a composition series, the integer r (called the length of M) and themulti-set of simple components Li are the same in all composition series of M.

To determine what kind of Lie algebra modules exist, it is therefore impor-tant to first determine what simple modules exist. For simple Lie algebras thereis however no complete classification of simple modules (except arguably forsl2, see [Bl, Maz2]). Moreover, such a classification seems close to hopelessto ever obtain. There are however various known classes of modules, and thepapers in this thesis contribute to Lie algebra representation theory by con-structing and analyzing some new such classes of simple modules.

2.4 Categorical techniquesFor a fixed Lie algebra g, the category of all modules over g is denoted g-Modor U (g)-Mod. Its objects are g-modules, and its morphisms are g-modulehomomorphisms. It is often useful to study g-Mod through subcategories andfunctors.

2.4.1 Characters and block decompositionA central character is a Lie algebra homomorphism χ : Z(g)→ C from thecenter of U (g) to the scalars. A module M is said to have central character χ

provided that

z · v = χ(z)v for all z ∈ Z(g) and v ∈M.

20

While not all modules have central character it is easy to prove that simplemodules always do. Moreover, there can exist no nontrivial homomorphismsbetween modules with different central characters.

We shall write χM for the central character of a module M. An importantresult is that every character can be realized as a character of a simple highestweight module L(λ ) (see Section 3.1.1). In other words, if M has centralcharacter we have χM = χL(λ ) for some λ ∈ h∗. One usually writes just χλ forχL(λ ).

More generally, a module is said to have generalized central character ifeach of its vectors is annihilated by a large enough power of (z− χ(z)) forevery central z. The full subcategory of U (g)-Mod consisting of moduleswith generalized central character χ is denoted U (g)-Mod(χ), and we haveprojection functors

Pr(χ) : g-Mod→ g-Mod(χ) M 7→M(χ),

where

M(χ) = {m ∈M|∀z ∈ Z(g),(z−χ(z))N = 0 for large enough N}.

The category of all U (g) modules on which the action of the center is lo-cally finite is denoted U (g)-ModZ f . This category decomposes completelyinto blocks depending on central characters – the so called block decomposi-tion:

U (g)-ModZ f =⊕χ∈Θ

U (g)-Mod(χ)Z f . (2.1)

Here Θ is the set space of central characters.

2.4.2 Tensor functorFor any two Lie algebra modules M and N there is a natural g-module structureon M⊗C N given by

x · (m⊗n) = (x ·m)⊗n+m⊗ (x ·n).

The assignment M 7→M⊗N is in fact functorial and the corresponding end-ofunctor on g-Mod is usually written −⊗N. The tensor functor interactsadditively with central character in the sense that χM⊗N = χM + χN wheneverM and N has central character.

2.4.3 Duality functorsIf M is a left g-module there is a natural right module structure on M∗ =HomC(M,C) given by ( f · x)(m) = f (x ·m). It is however desirable to stay

21

in the category of left g-modules so we can instead pick a Lie algebra anti-homomorphism τ and define a left g-structure on M∗ by

(x · f )(m) = f (τ(x) ·m) for all x ∈ g and f ∈M∗.

Here there are two natural choices for τ: either the involution mapping x to−xfor all x ∈ g, or the Chevalley involution, an involution exchanging root spacesgα → g−α .

2.4.4 Functors related to subalgebrasLet a be a fixed subalgebra of g.

First we have the induction functor

Indga : a-Mod→ g-Mod,

which maps an a-module M to U (g)⊗U (a) M (where the g-action is the nat-ural U (g)-multiplication on the left).

A functor going in the other direction is the restriction functor

Resga : g-Mod→ a-Mod,

which maps modules to themselves but with action restricted from g to a.Induction and restriction functors form an adjoint pair i.e. for each M ∈ a-

Mod and N ∈ g-Mod there is a natural vector space isomorphism

Homg(IndgaM,N)' Homa(M,ResgaN).

2.4.5 Twisting functorsFor each Lie algebra automorphism ϕ : g→ g we have a corresponding twist-ing functor

Fϕ : g-Mod→ g-Mod Fϕ : M 7→ ϕM.

Here ϕM is identified with M as a vector space but with a new twisted g-actiongiven by x•m = ϕ(x) ·m. It is clear that Fϕ ◦Fϕ−1 is isomorphic to the identityfunctor so Fϕ is an equivalence on g-Mod.

2.4.6 Translation functorsTranslation functors are compositions of the tensor functors and projectionfunctors discussed in 2.4.2 and 2.4.1 above. They provide categorical equiva-lences between blocks of different central character.

More precisely, let λ and µ be weights such that λ − µ is dominant andintegral (i.e. L(λ − µ) is finite dimensional, see Section 3.1.1). Then thefunctor

T (µ,λ ) : U (g)-Mod(χµ )→U (g)-Mod(χλ ) M 7→ (M⊗L(λ −µ))(χλ )

22

is an equivalence of categories. Translation functors were independently in-troduced Zuckerman and Jantzen see [Zu, Ja2]. This was based on previouswork by Bernstein, Gelfand, and Gelfand [BGG1]. See also [BG] which in-vestigates projective functors – a generalization of translation functors.

23

3. Known classes of modules

3.1 Weight modulesLet M be a module over a Lie algebra g with triangular decomposition n−⊕h⊕n+. For all λ ∈ h∗ we define the corresponding weight space

Mλ := {m ∈M|h ·m = λ (h)m for all h ∈ h}.

M is called a weight module if M =⊕

λ∈h∗Mλ , and the λ for which Mλ isnonzero are called the weights of M. The action of root vectors on weightspaces satisfies gα ·Mλ ⊂ Mλ+α which shows that if a module is simple, allits weights lie in a single coset of h∗/Φ. Many results have been proved aboutweight modules, see for example [Di, Hu, DMO, Ve].

3.1.1 Highest weight modulesChoosing a base ∆ for our root system also equips the space h∗ with a partialorder relation≺ generated by defining α ≺ β whenever β−α ∈∆. M is calleda highest weight module provided that M is generated by a weight vector vλ

of weight λ such that n+vλ = 0. This means that M is a weight module whereλ is maximal among its weights with respect to the order ≺.

Verma modules are universal objects among the highest weight modules.They were originally studied by Verma [Ve] and can be constructed as follows:For each λ ∈ h∗ let Cλ be the one dimensional h⊕n+ module, where h ∈ hacts by the scalar λ (h), and all of n+ acts by 0. Now let

M(λ ) := U (g)⊗U (h⊕n+)Cλ .

This is a U (g)-module under the natural left action. Note that M(λ ) is isomor-phic to U (n−) as a vector space (and as a n−-module). Verma modules havea unique maximal submodule and a corresponding unique simple quotient de-noted L(λ ). One can show that any highest weight module is a quotient of aVerma module, and so the assignment

λ 7→ L(λ )

parametrizes the simple highest weight modules.

24

3.1.2 Finite dimensional modulesIn fact one can show that every finite dimensional module is a highest weightmodule with weights invariant under the Weyl group in the sense that

dimMλ = dimMw(λ ),

for all λ ∈ h∗, w ∈W . Moreover, every simple finite dimensional module iscompletely determined by its highest weight.

This situation becomes especially easy for g = sl2 where we can easilyidentify h∗ with C via λ 7→ λ (h) with notation as in (1.5). In this case it turnsout that the simple module L(λ ) is finite dimensional if and only if λ ∈ N.Moreover, for n ∈ N, L(n) is (n+1)-dimensional and its set of weights are

{n,n−2,n−4, . . . ,−(n−2),−n}.

Moreover, x and y acts on weight vectors by increasing the weight by ±2.For example, the module L(4) can be visualized as follows:

-4

h�� x

++-2

h�� x

++

ykk 0

h�� x

++

ykk 2

h�� x

++

ykk 4

h��

ykk

Here the boxes indicate one-dimensional weight spaces of weight λ (h) indi-cated by the number.

The following diagram illustrates an example of the occurring weights spacesof a simple finite dimensional weight module when the Lie algebra is of typeA2. Here the positions of the circles indicate the occurring weights in h∗, whilethe numbers indicates the dimension of the corresponding weight space.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

×

¬

¬

¬

­

¬

¬

­

¬

­

­

¬

­

¬

­

¬

¬

¬

¬

25

3.1.3 Category OA natural generalization of the category of highest weight modules is categoryO . This is the full subcategory of U (g)-Mod consisting of finitely generatedweight modules V on which the action of n+ is locally finite:

dimU (n+)v < ∞ for all v in V.

Category O enjoys many nice properties which makes it attractive to workin. All its modules are Noetherian and have finite dimensional weight spaces,the category is abelian, and has enough projectives and injectives. Moreoverevery module in O is Z(g)-finite which makes category O subject to the blockdecomposition (2.1):

O =⊕χ∈Θ

O(χ).

Category O was first defined in the paper [BGG2] where the authors extendedmuch of the former work by Verma and Jantzen [Ve, Ja1]. See also [Hu] for acomprehensive summary.

3.1.4 Cuspidal modules and coherent familiesA weight module M is called cuspidal provided that all root vectors xα actsbijectively on weight spaces:

xα : Mλ →Mλ+α is an isomorphism for all λ ∈ h∗.

The set of weights of a simple cuspidal module thus have form λ +ZΦ.The classification of cuspidal weight modules with finite dimensional weight

spaces was completed by Mathieu in 2000 [Ma]. His approach used coherentfamilies - another type of module important to this thesis.

A coherent family of degree d is a weight module M such that• dimMλ = d for all λ ∈ h∗.• The function λ 7→ tr(u ·−|Mλ

) is polynomial in λ for each fixed u com-muting with h.

Here u ·− is the linear map M→ M mapping m to u ·m while u ·−|Mλis its

restriction to the weight space Mλ . It can be shown that each cuspidal moduleoccurs as a submodule of a coherent family. Since one can show that coherentfamilies only exist for simple finite dimensional Lie algebras of type A and C,the same is true for cuspidal modules, see [Fe, Fu1] for details.

3.2 Whittaker modules3.2.1 Kostant’s modulesWhittaker modules were originally defined by Kostant in 1978 [Ko]. They aredefined in terms of a triangular decomposition as in (1.4), and can be abstractly

26

described as modules M on which the algebra of positive roots n+ acts locallyfinitely. This just means that that dimU (n+) ·m < ∞ for all m ∈M.

To construct such modules we let η : n+→ C be a nonsingular Lie algebrahomomorphism. This means that η is determined by the nonzero scalars η(α)for α ∈ ∆. For each such η we have a corresponding one dimensional n+-module Cη where each x ∈ n+ acts by the scalar η(x).

We may now define the corresponding Whittaker module Wη by

Wη = U (g)⊗U (n+)Cη .

For each central character χ : Z(g)→ C we have the corresponding centralideal Iχ generated by z−χ(z), z ∈ Z(g). We may now define

W χ

η =Wη/IχWη

A result due to Kostant in 1978 states that W χ

η is simple for all χ .

3.2.2 Generalized Whittaker modulesWhittaker modules are generated by eigenvectors for the subalgebra n+. Gen-eralized Whittaker modules arise by instead looking at generalized eigenvec-tors for a larger class of subalgebras.

More precisely, let (n,g) be a pair of Lie algebras such that• n is a subalgebra of g• n is quasi-nilpotent• the adjoint n-module g/n is locally nilpotent

A module W is said to be a generalized Whittaker module for the Whittakerpair (n,g) if the action of n on M is locally finite in the sense that

dimU (n) ·w < ∞ for all w ∈W.

This type of generalized Whittaker modules were originally studied in [BM].

3.3 Gelfand-Zetlin modulesThe Lie algebras gln of n×n complex matrices can be sequentially embeddedin each other with respect to the upper left corner:

gl1 ⊂ gl2 ⊂ ·· · ⊂ gln−1 ⊂ gln.

Let Γ be the subalgebra of U (gln) generated by all the centers Z(gli) for 1≤i ≤ n. Gelfand-Zetlin modules arose out of an attempt to construct bases forsimple finite dimensional gln-modules that were eigenbases with respect to thealgebra Γ.

27

The combinatorial data of such Gelfand-Zetlin basis elements can be en-coded as a doubly indexed complex vector t = {ti j | 1≤ j≤ i≤ n}, which alsocan be visualized as a tableau:

[t] =

tn,1 tn,2 · · · tn,n−1 tn,ntn−1,1 · · · · · · tn−1,n−1· · · · · · · · ·

t2,1 t2,2t1,1

A tableau [t] is called standard if all its entries ti, j lie in the same coset of C/Z

and if for each sub-tableau of forma c

b we have a≤ b < c. For example,

1 4 7 91 5 8

3 76

is standard.

It is well known that simple finite dimensional gln-modules are parametrizedby their highest weight vectors λ = (λ1, . . . ,λn) where λi+1−λi ∈N. Gelfandand Zetlins original result in [GZ] states the corresponding finite dimensionalmodule V (λ ) has an eigenbasis consisting of all standard tableaux which hastop row

λ1 λ2−1 · · · · · · λn−n+1

and where the action of the generators of gln upon this basis is given by theformulas

ek,k+1 · [t] =−∑ki=1

(∏

k−1j=1(tk,i−tk−1, j)

∏kj 6=i(tk,i−tk, j)

)[t+ εk,i]

ek+1,k · [t] = ∑ki=1

(∏

k+1j=1(tk,i−tk+1, j)

∏kj 6=i(tk,i−tk, j)

)[t− εk,i]

ek,k · [t] =

(∑

ki=1(tk,i + i−1)−∑

k−1i=1 (tk−1,i + i−1)

)[t],

(3.1)

where εk,i is the unit vector having a 1 in position (k, i) and zeroes elsewhere.For example: The simple gl3-module V (1,4,7) is 8-dimensional with basis

1 3 51 3

1,

1 3 51 3

2,

1 3 51 4

1,

1 3 51 4

2,

1 3 51 4

3,

1 3 52 3

2,

1 3 52 4

2,

1 3 52 4

3.

28

About 40 years after Gelfand and Zetlins original paper, Drozd, Ovsienko,and Futorny had the idea that the formulas (3.1) could be applied on a widerrange of tableaux to obtain many new simple infinite dimensional modules,see [DFO1, DFO2, DFO3].

A tableau [t] is called generic if each row has non-integer differences, i.e.ti, j− ti,k /∈ Z for all 1≤ j < k ≤ i≤ n−1.

Theorem 3. For each fixed generic tableau [t], let V ([t]) be the vector spacespanned by all tableaux [l] that satisfies li, j − ti, j ∈ Z. Then V ([t]) is a gln-module under the formulas (3.1).

The theorem gives rise to the largest known family of modules. This familyof generic Gelfand-Zetlin modules is parametrized the set of generic tableaux,i.e. by n(n+1)

2 generic complex parameters. The simple generic Gelfand-Zetlinmodules were recently classified in [FGR]. Although the stated theorem isfor gln, it translates easily to the special Lie algebras of type A. An analo-gous construction for the orthogonal algebras (type B and D) were consideredin [Maz1].

A more abstract approach is to define Gelfand-Zetlin modules as moduleson which the subalgebra Γ⊂U (g) acts locally finitely. By this definition thereare also many other Gelfand-Zetlin modules than the ones discussed above.

3.4 Parabolically induced modulesLet g be a (semi-)simple Lie algebra with triangular decomposition g= n−⊕h⊕ n+. A subalgebra p of g is said to be parabolic if it contains the Borelsubalgebra h⊕n+. Parabolic subalgebras admit a Levi decomposition

p= (a⊕ha)⊕n,

where n is nilpotent, a′ = a⊕ ha is reductive, a is semisimple and ha ⊂ h isabelian and central in a′.

A generalized Verma module over g is an induced module

Mp(V ) = U (g)⊗U (p)V,

where V is a simple a′-module and nV = 0. A parabolically induced moduleis a module of form U (g)⊗U (p)V where p is a parabolic subalgebra of g andV is a finitely generated p-module.

Many general results have been proved about generalized Verma modulesand parabolically induced modules, in particular, for g = sln one can givegeneral conditions for when a parabolically induced module is simple. See[Fu2, FM, KM1, KM2, MS, SM] for details.

29

Part II:Results of the Thesis

4. U (h)-free modules

The majority of papers in this thesis are concerned with construction and clas-sification of so called U (h)-free modules and related objects.

A weight modules V can be can be characterized as a module on whichU (h) acts locally finitely:

dimU (h) · v < ∞

for all v ∈ V . By contrast, a U (h)-free module is a module on which U (h)acts freely, so in this sense U (h)-free modules are the opposite of weightmodules.

We define F to be the full subcategory of U (g)-Mod with objects

ob(F ) = {M ∈U (g)-Mod | ResU (g)U (h)

M is free of finite rank}.

Such modules are isomorphic to U (h)⊕k when restricted to U (h), where kis the rank of the module. The easiest case is when k= 1; then as a vector space(and as a U (h)-module) M is isomorphic to U (h) – a polynomial algebra inrank g variables.

As an example, let g = sl2 with basis as in (1.5). In this case U (h) isisomorphic to C[h].

Denote by Mc the vector space C[h] equipped with the following sl2-modulestructure. For any polynomial f (h) ∈ C[h] we define

h · f (h) = h f (h),x · f (h) = f (h−2),y · f (h) = −1

4(h+ c+2)(h− c) f (h+2).

Then ResU (g)U (h)

Mc is free of rank 1. Moreover, for all non-constant f (h) wehave

deg(

f (h−2)− f (h))= deg f (h)−1,

and we note that the element (x−1)∈U (g) can be used to reduce any elementf (h) to a nonzero constant which shows that the module Mc is simple for all c.Moreover, since the action of x does not increases the degree of a polynomial,we have

dimU (n+) · f (h)≤ deg f (h)+1,

which shows that the modules Mc in fact all Whittaker modules in the sense ofSection 3.2.

33

This is however not the typical case as the following exemplifies.Denote by Nc the vector space C[h] equipped with the following sl2-module

structure.h · f (h) = h f ,x · f (h) = 1

2(h+ c) f (h−2),y · f (h) = −1

2(h− c) f (h+2).

The modules Nc are still U (h)-free but are no longer Whittaker modules. Fur-thermore, the simplicity of the modules Nc is no longer obvious. For example,N0 is clearly not simple: hC[h] is a submodule. In fact Nc will be simpleprecisely for c /∈ N.

This last point also illustrates a problem inherent to the category F. Thequotient N0/hN0 is not U (h)-free, so the category F is not closed under takingquotients. This suggests that it might be categorically more interesting to studya larger category which contains F and is abelian, i.e. closed under formingdirect sums, submodules, and quotients.

Here the natural choice is to define M to be the full subcategory of U (g)-Mod having objects

ob(M ) = {M ∈U (g)-Mod | ResU (g)U (h)

M is finitely generated}.

This category is abelian and contains all objects of F, all finite dimensionalmodules, and even some non-free infinite dimensional modules.

U (h)-free modules were introduced in Paper I which also classified allrank 1 U (h)-free modules for Lie algebras of type A. This classification wasextended to all other types of Lie algebras (see Section 1) in Paper II. Similarmodules were also investigated for the Virasoro algebra, see [CC, CG].

34

5. On Paper I

The first paper investigates sln+1-modules whose restriction to U (h) is free ofrank 1. The main result of the paper is the classification of such modules. Forsl2 we also give a formula for tensor product decomposition.

5.1 Construction and classification of U (h)-freemodules

As a vector space, any rank 1 U (h)-free module can be identified with the leftregular U (h)-module U (h)U (h), which itself can be identified with C[h1,h2, . . . ,hn]where {h1, . . . ,hn} is a basis of h. In fact the paper uses the explicit choice ofbasis

hk := ek,k−1

n+1

n+1

∑i=1

ei,i.

For any root vector xα we have the following relations in U (g):

xαh = (h−α(h))xα for all h ∈ h.

Thus we observe that any module structure on U (h) is determined up to iso-morphism by the action of generators of sln+1 on the generator 1 of C[h1,h2, . . . ,hn].We now choose the generating set {e1,n+1, . . . ,en,n+1,en+1,1, . . . ,en+1,n} of sln+1and define

pi := ei,n+1 ·1 and qi := ei,n+1 ·1.Any rank 1 U (h)-free module is then determined by the the 2n polynomials

p1, p2, . . . , pn,q1,q2, . . . ,qn.Introducing algebra homomorphisms σi : U (h)→U (h) defined by

σi(

f (h1, . . . ,hn))= f (h1, . . . ,hi−1, . . . ,hn),

we are able to write down the action of sln+1 on U (h) explicitly as

hi · f = hi f ,ei,n+1 · f = piσi( f ),en+1,i · f = qiσ

−1i ( f ),

ei, j · f =(

piσi(q j)−q jσ−1j (pi)

)σiσ

−1j f ,

(5.1)

for 1≤ i, j ≤ n, i 6= j.

35

Conversely however, it is not immediately clear which 2n-tuples

(p1, p2, . . . , pn,q1,q2, . . . ,qn)

give rise to a sln+1 module under the formulas (5.1). This is the main questionanswered in the paper.

The paper proceeds by constructing a set of modules.For all b ∈ C and for all S ⊂ {1, . . . ,n} we denote by MS

b the rank 1 U (h)-free module determined by the 2n-tuple (p1, p2, . . . , pn,q1,q2, . . . ,qn) where

pi =

{(h1 +h2 + · · ·hn +b) if i ∈ S(h1 +h2 + · · ·hn +b)(hi−b−1) if i /∈ S

and

qi =

{−(hi−b) if i ∈ S−1 if i /∈ S

The main result of the paper is that any rank 1 U (h)-free module can beobtained from a module MS

b by twisting it by an automorphism.More precisely, for each (n+1)-tuple a = (a1,a2, . . . ,an+1) ∈ (C∗)n+1 we

let ϕa be the automorphism of sln+1 defined by

ϕa(ei, j) =ai

a jei, j for i 6= j and ϕa(h) = h for all h ∈ h.

Furthermore, let τ be the automorphism of sln+1 defined by

τ(ei, j) = e j,i for i 6= j and τ(h) =−h for all h ∈ h.

Theorem 30 then states that a complete set of isomorphism classes of rank1 U (h)-free modules are given by

{Fa(MSb)|a ∈ (C∗)×{1},S⊂ {1, . . . ,n},b ∈ C}

∪{Fa ◦ τ(MSb)|a ∈ (C∗)×{1},S⊂ {1, . . . ,n},b ∈ C}.

Moreover, this list of modules is irreduntant for n > 1. See Theorem 29 and30 of Paper I for details.

5.2 Submodules and propertiesHaving classified the rank 1 U (h)-free modules, we turn to the problem ofdetermining which of them are simple. The conclusion is that MS

b is simpleunless both (n+1)b ∈N and S = n := {1, . . . ,n}. This determines the simplessince the functors τ and Fa are equivalences.

36

When (n+1)b ∈ N and S = n we show that there is a short exact sequence

0→W →Mnb → Q→ 0

where W is simple and finitely generated (but not free) over U (h), and Q isa simple finite dimensional module of dimension

((n+1)b+nn

). This shows that

all rank 1 U (h)-free modules has length ≤ 2.Worth noting is that when g= sl2, and 2b ∈N the above sequence becomes

0→Mn−b−1→Mb→ L(2b)→ 0,

showing that all simple finite dimensional modules can be constructed as quo-tients of U (h)-free modules. This is analogous to the known fact that anysuch simple finite dimensional module can be expressed as a quotient of Vermamodules: with notation as in Section 3.1.1 we have

0→M(−2b−2)→M(2b)→ L(2b)→ 0.

5.3 Clebsch-Gordan problem for sl2The classical Clebsch-Gordan theorem gives a formula for the decompositionof the tensor product simple finite dimensional sl2 modules. Namely, for non-negative integers m≥ n we have

L(m)⊗C L(n)' L(m+n)⊕L(m+n−2)⊕·· ·⊕L(m−n+2)⊕L(m−n).

In the paper, it is established that an analogous formula holds for the de-composition of then tensor product of a simple finite dimensional module anda U (h)-free module.

We show that for all 2b ∈ C\N, we have

MS2b⊗C L(k)'

k⊕i=0

MS2b+ k−2i

2.

37

6. On Paper II

The second paper continues the classification started in Paper I, but it takes adifferent approach. The idea for the first part of this paper was suggested byOlivier Mathieu.

6.1 Weighting functor and coherent familiesLet M be any U (g)-module and for each λ ∈ h∗ let mλ be the correspondingmaximal ideal in U (h) i.e. the ideal generated by all elements of form h−λ (h). Extending λ : h→ C to an algebra homomorphism λ : U (h)→ C, wehave mλ = ker(λ ).

We may now defineM =

⊕λ∈h∗

Mλ ,

where Mλ = M/mλ M. M then has the natural structure of a U (h)-module.However, we may further extend this to a g-module structure as follow: foreach root vector xα define

xα · (m+mλ M) = (xα ·m)+mλ+αM.

This lets us define a functor W : U (g)-Mod→U (g)-Mod called the weight-ing functor. It maps M to M , and maps a morphism f : M → N to W ( f )defined by

W ( f )(m+mλ M

)= f (m)+mλ N.

It can now be shown that

• W (M) is always a weight module.• W (M)'M if M is already a weight module.• W maps U (h)-free modules of rank d to coherent families of degree d.

Since we know that coherent families only exists when the Lie algebra isof type A or type C (see the discussion in Section 3.1.4), it now easily followsthat U (h)-free modules also only exist in type A and C. So to complete theclassification of rank 1 U (h)-free modules for all simple complex finite di-mensional Lie algebras as started in Paper I, only the classification in type Cremains.

38

6.2 Classification in type CWe proceed by explicitly constructing an object M0 which is U (h)-free or rank1 in U (sp2n)-mod. We use the matrix realization of sp2n from Section 1.4.

Letting hi := ei,i−en+i,n+i we have U (h)'C[h1, . . . ,hn]. We let M0 be thisvector space equipped with the sp2n-action

hi · f = hi f8ei,n+i · f = (2hi−1)(2hi−3)σ2

i ( f )−2en+i,i · f = σ

−2i ( f )

4(ei,n+ j + e j,n+i) · f = (2hi−1)(2h j−1)σiσ j( f )−(en+i, j + en+ j,i) · f = σ

−1i σ

−1j ( f )

2(ei, j− en+ j,n+i) · f = (2hi−1)σiσ−1j ( f )

for all 1 ≤ i, j ≤ n, i 6= j. Here the matrix elements on the left side togetherform a basis for sp2n. The algebra-homomorphisms σi are defined by σi(hk) =hk−δi,k as before.

Now if M is U (h)-free of rank 1 we know that W (M) is a coherent familyof degree 1. Since there is a unique such coherent family we know that W (M)is isomorphic to W (M0) and this allows us to prove that any rank 1 U (h)-freemodule can be obtained by twisting M0 by an (explicitly given) automorphism.See Section (3.4) of Paper II for details.

This completes the classification of rank 1 U (h)-free modules for all simplecomplex finite dimensional Lie algebras.

6.3 Applying translation functorsIn the final section of the paper it is shown that there exist simple U (h)-freemodules of rank higher than 1. This is done by using the translation functorsT (µ,λ ) discussed in Section 2.4.6.

It is known that when λ −µ is dominant and integral, T (µ,λ ) is an equiv-alence. In the paper we show that the restriction of T (µ,λ ) to the category ofU (h)-free modules is still an equivalence. This is used to show that if M isrank 1 U (h)-free with character χµ , then for λ 6= µ , the module T (µ,λ )(M)is a simple U (h)-free of rank higher than 1.

39

7. On Paper III

Paper III deals with construction of a new large family of simple gl2n-modules.This family of modules is parametrized by the set of nonsingular complexn× n-matrices i.e. by n2 generic parameters. The only larger known fam-ily of gl2n-modules are the Gelfand-Zetlin modules which for comparison areparametrized by n(2n+1) generic complex parameters (see Section 3.3).

7.1 Decomposition of gl2nThe Lie algebra gl2n has a natural decomposition into subalgebras as indicatedin the diagram (

A BC D

).

Explicitly, we have

A = {ei, j|1≤ i, j ≤ n}, B = {ei,n+ j|1≤ i, j ≤ n},

C = {en+i, j|1≤ i, j ≤ n}, D = {en+i,n+ j|1≤ i, j ≤ n},which gives gl2n =A ⊕B⊕C ⊕D . Note that A 'D ' sln, while B and Care commutative.

7.2 Module constructionWe proceed to construct our modules in steps.

Let Q = (qi, j) be any complex n× n-matrix. We denote by LQ the onedimensional B-module where the action is given by Q:

ei,n+ j · v = qi, jv for all 1≤ i, j ≤ n, and v ∈ LQ.

Since B is commutative, this is indeed a Lie algebra module. We also notethat this action can be reformulated using the trace function:(

0 B0 0

)· v = tr(QBT )v.

Next we define

MQ := IndA⊕BB LQ = U (A ⊕B)⊗U (B) LQ.

40

MQ is then an A ⊕B-module which is isomorphic to U (A ) as a vectorspace (and A -module).

We now consider the simplicity of the module MQ. The first results of thepaper can be summarized as follows.

Theorem 4. In the above setting the following are equivalent:1. The matrix Q is nonsingular.2. The module MQ is injective in B-Mod.3. The module MQ is simple.

The remainder of the paper uses techniques of induction and taking quo-tients to extend the module MQ to a gl2n-module. Finally an explicit formulafor the action of gl2n on this module is given. The results can be summarizedas follows:

Theorem 5. Let A.B denote the matrix product while AB means the product inU (A ). Let ϕ and ψ be the algebra homomorphisms U (A )→U (A )⊗Aacting on generators by ϕ(A) = A⊗ I + I ⊗ A and ψ(A) = A⊗ I − I ⊗ AT

respectively, and let F := (e ji)i j ∈ U (A )⊗A . Define an action of gl2n onMQ 'U (A ) as follows: for any a ∈U (A ), let(

A BC D

)·a = Aa−aD+ tr(ψ(a).Q.BT )

− tr(ϕ(a).F2.Q−T .C)− tr(ϕ(a).Q−T .C)tr(F).

This is a simple gl2n-module structure.

This formula was obtained by first determining the formula for Q = I, andthen twisting the module by the automorphisms ϕS : gl2n→ gl2n defined by

ϕS :(

A BC D

)7→(

A B.S−1

S.C S.D.S−1

).

for all nonsingular S. See Theorem 17 and Theorem 18 of Paper III for all thedetails.

41

8. On Paper IV

About 30 years ago, McDowell used parabolic induction to construct new sim-ple modules from Whittaker modules, see [McD1, McD2]. This paper com-bines the techniques used by McDowell with the results of Paper I to obtain anew family of simple generalized Verma modules.

8.1 Generalized Verma modulesParabolic induction was discussed in Section 3.4. Recall a parabolic subalge-bra p of g has a Levi-decomposition p = (a⊕ ha)⊕ n, where n is nilpotent,a′ = a⊕ha is reductive, a is semisimple and ha ⊂ h is abelian and central ina′, and that a generalized Verma module over g is an induced module

Mp(V ) = U (g)⊗U (p)V,

where V is a simple a′-module and nV = 0.

8.2 Our constructionConsider sln+1 as a subalgebra of sln+2 with respect to the upper left cornerand let

p := sln+1 +h+n+.

The paper studies the modules Mp(V ) where Respsln+1V is a rank 1 U (h)-free

sln+1-module as in Paper I. The main result is the determination of necessaryand sufficient conditions for simplicity of Mp(V ).

Let hi = ek,k− 1n+1 ∑

n+1i=1 ei,i for 1≤ i≤ n+2 so that {h1, . . . ,hn} is a basis for

the Cartan subalgebra of sln+1 and {h1, . . . ,hn,hn+2} is a basis for the Cartansubalgebra of sln+1.

Each p-module V as above can be parametrized as V (a,S,b,λ ) for a =(a1, . . . ,an,1)∈ (C∗)n+1, S⊂{1, . . . ,n+1}, b∈C and λ ∈C. Here a,S, and bcorresponds to the classification parameters in Paper I and λ is the eigenvalueof hn+2 acting on V .

As vector spaces we now have

Mp(V (a,S,b,λ ))' C[en+2,1,en+2,2, . . . ,en+2,n+1]⊗C[h1, . . . ,hn],

42

and we consider its elements as polynomials in en+2,1,en+2,2, . . . ,en+2,n+1 withcoefficients in U (h) = C[h1, . . . ,hn].

For m = (m1, . . . ,mn+1) ∈ Nn+1 we write Em = em1n+2,1em1

n+2,2 · · ·emn+1n+2,n+1.

This lets us express every element of v of Mp(V (a,S,b,λ )) uniquely as

v = ∑m∈Nn+1

EmPm,

where finitely many coefficients Pm ∈U (h) are nonzero. This can be refinedinto homogeneous components: with |m| := m1 + · · ·+mn+1 we may write

v =N

∑k=0

∑|m|=k

EmPm.

We define the degree of v by degv = N.

8.3 SimplicityIt is shown in the paper that degei,n+2 · v < degv for all 1 ≤ i ≤ n+ 1. Thusif v is a homogeneous nonzero element of a proper submodule S ⊂Mp(V ) ofminimal degree, v must be annihilated by ei,n+2 for all 1 ≤ i ≤ n+1. It turnsout that this implies that for each m with |m|= N−1, the vector

(a−11 (m1 +1)σ−1

1 (Pm+ε1), . . . ,a−1n (mn +1)σ−1

n (Pm+εn),(mn+1 +1)Pm+εn+1)

is a solution to the linear system

A(λ ,b,S,N)x = 0

where A(λ ,b,S,N) is a (n + 1)× (n + 1)-matrix with coefficients in U (h)explicitly defined in the paper.

We proceed by proving that

det A(λ ,b,S,N) = (−nb−λ −N +1)(b−λ −N +2)n.

This immediately shows that Mp(V (a,S,b,λ )) is simple whenever (−nb−λ−N +1) and (b−λ −N +2) both are nonzero.

We also prove the converse statement that Mp(V (a,S,b,λ )) is reduciblewhenever either (−nb−λ −N + 1) or (b−λ −N + 2) are zero by explicitlyconstructing proper nontrivial submodules in both cases.

We have thus proved the following theorem.

Theorem 6. Let V (a,S,b,λ ) be a simple p-module as above. Then the corre-sponding generalized Verma module Mp(V (a,S,b,λ )) is simple if and only if(−nb−λ −N +1) 6= 0 and (b−λ −N +2) 6= 0.

43

9. Summary in Swedish - Sammanfattning påsvenska

9.1 BakgrundInom algebra studerar man algebraiska objekt så som grupper, ringar, krop-par, och algebror. Ett sätt att analysera dessa objekt är att representera deraselement som linjära avbildningar i ett vektorrum, det vill säga att till varje el-ement i den algebraiska strukturen tillskriva en linjär avbildning på så vis attmultiplikationen i den algebraiska strukturen motsvaras av sammansättning avlinjära avbildningar i vektorrummet. Vektorrummet tillsammans med dennaextra struktur kallas för en modul.

Representationsteori kan därmed ses som ett sätt att förstå kompliceradealgebraiska strukturer genom att översätta dem till linjär algebra - ett välkäntmatematiskt område som är enkelt att arbeta med.

Genom att undersöka delmoduler och kvotmoduler kan en godtycklig modulbrytas ned i sina enklaste komponenter. Dessa komponenter kallas enkla mod-uler och de kan karakteriseras som moduler vilka saknar äkta delmoduler.

För vissa algebraiska strukturer – till exempel för ändliga grupper – är dengrundläggande representationsteorin redan väl utredd. Här har vi för varjeändlig grupp en ändlig lista med alla enkla moduler, och vi vet att varje modulkan brytas ned som en direkt summa vars komponenter tillhör denna lista.

Representationsteori för Lie algebror är speciellt tillämpbart inom fysiken.Här kan till exempel olika partikeltillstånd beskrivas som element i moduler.För Lie algebror finns det oändligt många enkla moduler, vilket gör det svårareatt få en komplett bild från ett representationsteoretiskt perspektiv. Trots dettaär vissa klasser av enkla moduler välutforskade.

9.2 Kända klasser av enkla modulerI det här avsnittet låter vi g vara en enkel ändligtdimensionell Lie algebra överde komplexa talen. Det är välkänt att g har en triangulär uppdelning

g= n−⊕h⊕n+.

9.2.1 ViktmodulerViktmoduler är en fundamental klass av g-moduler som har studerats sedanbörjan av 1900-talet. En viktmodul är en modul i vilken element från Cartan-algebran h verkar genom skalär multiplikation på basvektorer i modulen.

44

Mer exakt; om M är en modul kan vi för varje funktional λ : h→C definieramotsvarande viktrum

Mλ = {v ∈M|h · v = λ (h)v för alla h ∈ h}.

Om M = ⊕λ∈h∗Mλ så är M en viktmodul, och de λ för vilka Mλ 6= 0 utgörvikter för M.

Det visar sig att det finns en naturlig partiell ordning på h∗ vilket gör att vikan vi kan introducera så kallade högstaviktmoduler. Dessa är moduler somgenereras av en maximal viktvektor med avseende på den partiella ordningen.

Ett fundamentalt resultat är att det för varje λ ∈ h∗ finns en unik enkelhögstaviktmodul L(λ ) med högsta vikt λ . Enkla högstaviktmoduler kan såledesparametriseras av h∗.

År 2000 färdigställde Olivier Mathieu klassifikationen av enkla viktmod-uler vars viktrum är ändligtdimensionella genom att klassificera så kallade”cuspidal modules“ – moduler vars mängd vikter är tät i h∗, se [Ma].

9.2.2 Ändligtdimensionella modulerMan kan visa att varje enkel modul av ändlig dimension är en högstavikt-modul, och därmed av form L(λ ). Omvänt kan man vidare visa att L(λ ) ärändligt dimensionell endast om λ kan skrivas som en summa av så kalladefundamentala vikter. Geometriskt kan detta ses som att enkla ändligtdimen-sionella moduler klassificeras av ett koniskt gitter i h∗.

9.2.3 WhittakermodulerModuler där verkan av n+ är lokalt ändlig, det vill säga dimU (n+) · v < ∞

för alla vektorer v i modulen kallas för Whittakermoduler. Whittakermodulerbeskrevs ursprungligen av Kostant år 1978, se [Ko]. Det enklaste sättet attkonstuera Whittakermoduler är att välja en ickesingulär Lie algebra homo-morfi η : n+ → C med vilken man definierar en endimensionell modul Cη

genomx · v = η(x)v för alla x ∈ n+ och v ∈ Cη .

Genom att inducera till g erhålls en Whittakermodul Wη = U (g)⊗U (n+)Cη .När man slutligen tar en kvot med en verkan av Z(g) erhålls en enkel Whit-takermodul.

Generaliserade Whittakermoduler kan också definieras för godtyckliga såkallade Whittakerpar (n,g) där n⊂ g uppfyller vissa ändlighets-villkor, se [BM].

9.2.4 Gelfand-ZetlinmodulerGelfand-Zetlinmoduler är en stor klass moduler som existerar för Lie algebrorav typ A, B, och D. I Gelfand och Zetlins ursprungliga konstruktion [GZ]

45

studerades serien av inbäddningar

gl1 ⊂ gl2 ⊂ ·· · ⊂ gln,

med avseende på över vänster hörn. Låt Γ vara delalgebran i U (gln) somgenereras av Z(gl1), . . . ,Z(gln). Vi kan nu definiera Gelfand-Zetlinmodulersom de moduler för vilka verkan av Γ är lokalt ändlig.

Man kan visa att man i enkla ändligtdimensionella gln-moduler kan välja enGelfand-Zetlin-bas bestående triangulära scheman av tal, så kallade generiskaGelfand-Zetlin-tablåer. Med avseende på denna bas är det enkelt att skriva neren explicit verkan av generatorer i gln. Genom att använda samma formler påen utökad klass av tablåer erhålls en stor klass av enkla gln-moduler. Dessamoduler kan parametriseras av n(n+1)

2 komplexa tal.

9.3 Avhandlingens resultatAvhandlingen bidrar till representationsteori för Lie algebror genom att kon-struera och klassificera nya klasser av moduler över Lie algebror. Modulernasom konstrueras är alla oändligtdimensionella icke-vikt moduler. Specielltfokus ligger på så kallade U (h)-fria moduler som behandlas i Paper I, PaperII och Paper IV. Dessutom konstrueras en stor klass enkla gl2n-moduler i PaperIII.

9.3.1 U (h)-fria modulerLåt g vara en enkel ändligtdimensionell Lie algebra. De g-moduler M somuppfyller

ResU (g)U (h)

'U (h)⊕k,

det vill säga moduler vars restriktioner till U (h) är fria av ändlig rang kallasför U (h)-fria moduler. Dessa är i en mening motsatsen till viktmoduler -Cartanalgebran verkar fritt istället för diagonalt.

U (h)-fria moduler av rang 1 undersöks i Paper I och Paper II. Som h-moduler är dessa isomorfa med U (h), vilket innebär att deras element kanidentifieras med polynom i rank g variabler. Deras modulstrukturer bestämsentydigt av de ändligt många polynomen {xα · 1|α är en rot till g} vilket gördirekta beräkningar relativt enkla att genomföra.

Paper I fokuserar på typ A. Dess huvudresultat är en klassifikation av mod-uler som är U (h)-fria av av rang 1 för Lie algebran sln+1. Vi visar att dennaklass av moduler kan parametriseras av delmängder av {1, . . . ,n} tillsammansmed n+ 2 generiska komplexa tal. I artikeln ges även explicita formler försln+1-verkan på dessa moduler. Dessutom visar vi att de flesta moduler i klas-sifikationen är enkla, och för resterande moduler bestämmer vi deras komposi-tionsserier. Speciellt ger detta upphov till exempel på nya enkla moduler som

46

inte är fria, men ändligt genererade över U (h). För sl2 bevisas dessutom enversion av Clebsch-Gordans formel: vi ger en formel för uppdelning av ten-sorprodukten mellan enkla U (h)-fria moduler och enkla ändligtdimensionellamoduler.

Paper II fortsätter med att utvidga klassifikationen av U (h)-fria moduler avrang 1 till typ C. Dessutom etableras ett samband mellan U (h)-fria moduleroch koherenta familjer. Vi konstruerar en viktningsfunktor W på g-Mod ochvisar att om M är U (h)-fri av rang n så är W (M) en koherent familj av gradn. Vi kan därigenom använda den sedan tidigare kända klassifikationen avkoherenta familjer för att analysera U (h)-fria moduler. Med dessa metodervisar vi att U (h)-fria moduler endast kan existera i typ A och C, vilket därmedfärdigställer klassifikationen av U (h)-fria moduler av rank 1 för enkla ändligt-dimensionell Lie algebror. Vi använder även translationsfunktorer för att visaatt det existerar enkla U (h)-fria moduler av rang högre än 1.

9.3.2 Generaliserade Whittakermoduler för gl2nI paper III konstrueras en stor familj av enkla moduler för gl2n. Konstruktionenbygger på följande blockuppdelning av gl2n i fyra n×n-matrisalgebror:(

A BC D

).

Modulerna konstrueras genom att börja med en endimensionell B-modul ochdärefter stegvis inducera först till A ⊕B, sedan till A ⊕B⊕D , och till sisttill hela gl2n.

De resulterande modulerna kan karakteriseras som enkla moduler vilka ärisomorfa med U (A ) som A -moduler och där verkan av B är lokalt ändlig.Modulerna är därmed exempel på generaliserade Whittakermoduler för Whit-takerparet (B,gl2n). Modulerna från konstruktionen parametriseras av invert-erbara n× n-matriser, det vill säga av n2 generiska parametrar. Således gerdetta den näst största kända klassen av gl2n-moduler efter Gelfand-Zetlinmodulerna.

9.3.3 Generaliserade Vermamoduler från U (h)-fria modulerI Paper IV använder vi parabolisk induktion för att konstruera en ny klass avsln+2-moduler från U (h)-fria sln+1-moduler. Vi låter p vara summan av stan-dard Borel-algebran i sln+2 och sln+1 inbäddad i övre vänster hörn. Modulernavi undersöker är av form

U (g)⊗U (p)V,

där ResU (p)U (sln+1)

V är en av modulerna klassificerade i Paper I. Vi ger nöd-vändiga och tillräckliga villkor för att en den resulterande modulen ska varaenkel.

47

Acknowledgements

During my years as a doctoral student I have come to think that doing mathe-matics is less about writing proofs and more about developing ideas and waysof thinking. I’m thankful to the many people who helped inspire the ideasunderlying this work. First I want to thank my doctoral advisor VolodymyrMazorchuk whose advice and support made it possible for me to write mydissertation. Throughout our many discussions I’ve been impressed with hisencouragement, optimism, and excitement for mathematics. I’m also gratefulfor the work he’s doing with keeping the algebra group in Uppsala active byorganizing speeches, seminars, and conferences. Indeed, it is through thesemeetings with other algebraists from around the world that I have made con-nections with people that were important to writing the papers of the thesis.I’m especially grateful to Olivier Mathieu for his friendliness and hospitalityduring my visit to Institut Camille Jordan in Lyon 2015. The second paper inthis thesis is heavily inspired by his ideas. The mathematics in Paper IV wereessentially done in one intense week of work during my visit to Wilfrid Laurierin Canada. Here I want to thank Kaiming Zhao as well as our co-authors Gen-qiang and Yan-an for their hospitality and friendliness. Additionally I want tothank Arne Meurman in Lund who got me interested in the subjects of alge-bra and representation theory in the first place. Thanks also to my family forthe continual encouragement over the almost 10 years that I’ve been studyingmathematics.

I didn’t know anybody in Uppsala when I first moved here in 2011, soI’m particularly grateful to the friends that I’ve made since then. Among myfriends at the mathematics department I’m especially thankful to Katja, Sei-don, and Djalal for our many interesting discussions about mathematics andlife. I also appreciate to have had so many friendly office mates over the years- thank you Martin, Marta, Reza, Viktoria, and Anna. With risk of forget-ting people I intend to keep the list short so I will finish by just thanking allmy other friends – thank you for making life as a doctoral student in Uppsalamore enjoyable!

48

References

[BG] J. N. Bernstein, S. I. Gelfand. Tensor products of finite- andinfinite-dimensional representations of semisimple Lie algebras. CompositioMath. 41 (1980), 245–285.

[BGG1] I. N. Bernstein, I. M. Gelfand, S. I. Gelfand; Structure of representationsthat are generated by vectors of highest weight. (Russian) Funckcional. Anal.i Priložen. 5 (1971), no. 1, 1–9.

[BGG2] I. N. Bernstein, I. M. Gelfand, S. I. Gelfand; A certain category ofg-modules. Funkcional. Anal. i Priložen. 10, (1976) 1–8.

[Bl] R. Block; The irreducible representations of the Lie algebra sl(2) and of theWeyl algebra. Adv. in Math. 139 (1981), no. 1, 69–110.

[BM] P. Batra, V. Mazorchuk; Blocks and modules for Whittaker pairs. J. PureAppl. Algebra 215 (2011), no. 7, 1552–1568.

[Ca] E. Cartan; Les groupes projectifs qui ne laissent invariante aucunemultiplicité planet. Bull. Soc. Math. France vol. 41 (1913), 53–96.

[CC] Q. Chen, Y. Cai; Modules over algebras related to the Virasoro algebra.International Journal of Mathematics 26.09 (2015): 1550070.

[CG] H. Chen, X. Guo; A new family of modules over the Virasoro algebra. Journalof Algebra, 457 (2016), 73–105.

[DFO1] Yu. Drozd, S. Ovsienko, V. Futorny. Irreducible weighted sl3-modules.Funktsional. Anal. i Prilozhen 23 (1989), 57–58.

[DFO2] Yu. Drozd, S. Ovsienko, V. Futorny. On Gelfand-Zetlin modules.Proceedings of the Winter School on Geometry and Physics (Srni, 1990).Rend. Circ. Mat. Palermo (2) Suppl. No. 26 (1991), 143–147.

[DFO3] Yu. Drozd, S. Ovsienko, V. Futorny. Harish-Chandra subalgebras andGel’fand-Zetlin modules. In: Finite-Dimensional Algebras and RelatedTopics (Ottawa, ON, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci.,Vol. 424, Kluwer Acad. Publ., Dordrecht, 1994, 79–93.

[Di] J. Dixmier; Enveloping Algebras. American Mathematical Society, 1977.[DMO] I. Dimitrov, O. Mathieu, I. Penkov; On the structure of weight modules.

Trans. Amer. Math. Soc. 352 (2000), no. 6, 2857–2869.[Fe] S. Fernando; Lie algebra modules with finite dimensional weight spaces, I.

Trans. Amer. Mas. Soc., 322 (1990), 757–781.[FGR] V. Futorny, D. Grantcharov, L.E. Ramirez; Irreducible Generic

Gelfand–Tsetlin Modules of gl(n). SIGMA Symmetry Integrability Geom.Methods Appl. 11 (2015), Paper 018, 13 pp.

[Fu1] V. Futorny; Weight representations of semisimple finite dimensional Liealgebras. Ph. D. Thesis, Kiev University, 1987.

[Fu2] V. Futorny; Weighted sl(3)-modules generated by semiprimitive elements.(Russian) Ukrain. Mat. Zh. 43 (1991), no. 2, 281–285; translation inUkrainian Math. J. 43 (1991), no. 2, 250–254.

49

[FM] V: Futorny, V. Mazorchuk; Structure of α-stratified modules forfinite-dimensional Lie algebras. I. J. Algebra 183 (1996), no. 2, 456–482.

[GZ] I.M. Gelfand, M.L. Zetlin; Finite-dimensional representations of the group ofunimodular matrices. Doklady Akad. Nauk SSSR (N.S.), 71 (1950),825–828.

[Ha] T. Hawkins; Emergence of the theory of Lie groups, An essay in the history ofmathematics 1869–1926. Sources and Studies in the History of Mathematicsand Physical Sciences. Springer-Verlag, New York, 2000.

[Hu] J. E. Humphreys; Representations of Semisimple Lie Algebras in the BGGCategory O . American Mathematical Society, 2008.

[Ja1] J. C. Jantzen; Zur Charakterformel gewisser Darstellungen halbeinfacherGruppen und Lie-Algebren. Math. Z. 140 (1974), 127–149.

[Ja2] J. C. Jantzen; Moduln mit einem höchsten Gewicht. Lecture Notes inMathematics, 750. Springer, Berlin, 1979. ii+195 pp.

[KM1] A. Khomenko, V. Mazorchuk; On the determinant of Shapovalov form forgeneralized Verma modules. J. Algebra 215 (1999), no. 1, 318–329.

[KM2] A. Khomenko, V. Mazorchuk; Structure of modules induced from simplemodules with minimal annihilator. Canad. J. Math. 56 (2004), no. 2,293–309.

[Ko] B. Kostant. On Whittaker vectors and representation theory. Invent. Math. 481978, no. 2, 101–184.

[Ma] O. Mathieu. Classification of irreducible weight modules, Ann. Inst. Fourier(Grenoble) 50 (2000), no. 2, 537–592.

[Maz1] V. Mazorchuk; On Gelfand-Zetlin modules over orthogonal Lie algebras.Algebra Colloq. 8 (2001), no. 3, 345–360.

[Maz2] V. Mazorchuk. Lectures on sl2(C)-modules. Imperial College Press, London,2010.

[McD1] E. McDowell; On modules induced from Whittaker modules. J. Algebra 96(1985), no. 1, 161–177.

[McD2] E. McDowell; A module induced from a Whittaker module. Proc. Amer.Math. Soc. 118 (1993), no. 2, 349–354.

[MS] V. Mazorchuk, C. Stroppel; Categorification of (induced) cell modules andthe rough structure of generalised Verma modules. Adv. Math. 219 (2008),no. 4, 1363–1426.

[SM] D. Milicic, W. Soergel; The composition series of modules induced fromWhittaker modules. Comment. Math. Helv. 72 (1997), no. 4, 503–520.

[Ve] N. Verma; Structure of certain induced representations of complexsemisimple Lie algebras. Bull. Amer. Math. Soc. 74 (1968), 160–166.

[Zu] G. Zuckerman; Tensor products of finite and infinite dimensionalrepresentations of semisimple Lie groups. Ann. of Math. (2) 106 (1977), no.2, 295–308.

50