In addition, when gis a nonsimplylaced group, we show that there are three. In mathematics, specifically in representation theory, a semisimple representation also called a completely reducible representation is a linear representation of a group or an algebra that is a direct sum of simple representations also called irreducible representations. This book is written for students who are studying nite group representation theory beyond the level of a rst course in abstract algebra. Representation theory of compact groups and complex reductive. Our study of real semisimple lie groups and algebras is based on the theory of complex semisimple lie groups developed in ch.
Xis the borel subalgebra bx which corresponds to the point x. Non semisimple group representation mathematics stack. Glv, where v is an ndimensional vector space over kand glv denotes the group of invertible linear maps v. We develop a concrete fourier transform on a compact lie group by means of a symbol calculus, or product, on each integral coadjoint orbit. Ill by harishchandra the main object of this paper is to define the character of an irreducible quasisimple1 representation ir of a connected semisimple lie group g on a hubert space this will be done as follows.
This is possible because the complexification of a real semisimple lie algebra is also semisimple see 1. This book is a model of exposition and an invaluable resource for both graduate students and researchers. The main sources for these notes are the books 6 and 8. On the fourier transform of a compact semisimple lie group. C an unconventional definition of semisimplicity that allows for a rapid development of the structure theory of semisimple lie algebras. A semisimple introduction would serve as a textbook for graduate and some advanced undergraduate courses in mathematics.
Some exponential inequalities for semisimple lie group. We discuss the classification of discrete series representations, their characters, their relevance to representation theory, and some explicit realisations of their. Lie group module associated to an integrable gmodule. Knapp offers a survey of representation theory of semisimple lie groups in a way that reflects the spirit of the subject and corresponds to the natural learning process. Semisimple lie algebra an overview sciencedirect topics. Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and its algebraic operations for example, matrix. Let gbe a group either a lie group or a discrete group. Representations of semisimple lie algebras in this chapter we study a special type of representations of semisimple lie algberas. Determining a semisimple group from its representation degrees.
Representation theory of compact groups and complex. On representations of complex semisimple lie groups school of. An ndimensional representation of gover k n 1 is a group homomorphism g. Real representations of semisimple lie algebras have q. Of these i follows from the fact that h appears as a subrepresentation of an induced representation see the simple proof by casselman 9. Weyl group is developed in chapter xiii and exploited in chapter xiv in the proof of weyls dimension formula. The discrete series of semisimple groups peter hochs september 5, 2019 abstract these notes contain some basic facts about discrete series representations of semisimple lie groups. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. This expository paper represents an introduction to some aspects of the current research in representation theory of semisimple lie groups. We let g be its lie algebra, regarded as a real lie algebra of real or. A group is an abstract mathematical object, a set with elements and an operation satisfying certain axioms.
Let a be a complex commutative associative algebra with unity and g be a complex finite dimensional semisimple lie algebra. For example, when g is a semisimple lie algebra, the representations of g form a tannakian category rep. It is called reductive if its lie algebra is a direct sum of simple and trivial onedimensional lie algebras. These products are constructed by means of a moment map defined for each irreducible representation.
If j is a nonzero nilpotent ideal in b, then its preimage under the natural projection map is a nilpotent ideal in. Representations of semisimple lie groups u of u math. W e are going to be studying group representations, and w e begin with some. A representation of g consists of a finitedimensional complex vector space v along with. Suppose that the killing form of l is nondegenerate. Determining a semisimple group from its representation degrees, international mathematics research notices, volume 2004, issue 38.
Representation theory of semisimple groups princeton. The lie algebra of a compact semisimple lie group g is determined by the degrees of the irreducible representations of g. Discuss the basics of the representation theory of a nite group g over an algebraically closed eld kof characteristic 0. Let g be a group either a lie group or a discrete group. This work was triggered by a letter to frobenius by r. Pdf representations of semisimple lie groups researchgate. Representation theory princeton math princeton university.
In this letter dedekind made the following observation. The quotient group of a connected nonsolvable linear group by its radical is semisimple. Algebraic dmodules and representation theory of semisimple lie groups dragan mili. This is an integrable algebra over c, because one can choose ad locally nilpotent operators for g. This is the approach followed in bourbaki 2005, for instance, which classifies representations of split semisimple reductive lie algebras.
As another example consider the representation theory of quivers. The kernel of this morphism is a ghomogeneous vector bundle b over x. Introduction a compact semisimple lie group ghas a. Algebraic dmodules and representation theory of semisimple. The following is a semisimple algebra that appears not to be of this form. This makes the representation theory of nite groups elementary. In chapter 4 our attention is turned to the unitary representation theory of real semisimple lie groups. The weyl group operates naturally in a and a and the isomorphism exp. May 31, 2000 these notes give an elementary introduction to lie groups, lie algebras, and their representations. Semisimple algebraic group encyclopedia of mathematics. Complex semisimple quantum groups and representation theory 3 in the second part of the notes, consisting of chapters 3, 4 and 5, we restrict ourselves to the case k c and assume that q eh6 1 is a positive real number. Topics include definitions and examples of lie groups and lie algebras, the relationship between lie groups and lie algebras via the exponential mapping, the basics of representations.
Throughout these notes, gdenotes a group and kdenotes a eld. Introduction to representation theory of nite groups. We give complete proofs of all statements with the exception of the conjugacy of cartan subgroups, the uniqueness theorem for semisimple lie algebras, and the. A representation of a group realizes the elements of the group concretely as geometric symmetries. The same group will generally have many di erent such representations. Herb, fourier inversion and the plancherel theorem for semisimple real lie groups, amer. Algebraic dmodules and representation theory 5 consider the adjoint action of gon g, the trivial bundle x.
It is an example of the general mathematical notion of semisimplicity. In the case where gis a lie group, then we ask that the map g glv be a smooth map. Then l must be semisimple since any abelian ideal is contained in the kernel of. It is an example of the general mathematical notion of semisimplicity many representations that appear in. Topics include definitions and examples of lie groups and lie algebras, the relationship between lie groups and lie algebras via the exponential mapping. If j is a nonzero nilpotent ideal in b, then its preimage under the natural projection map is a nilpotent ideal in a which is strictly larger than rad a, a contradiction. A connected lie group is called semisimple if its lie algebra is a semisimple lie algebra, i. H that span a subalgebra k of l isomorphic to s l 2 f. These notes give an elementary introduction to lie groups, lie algebras, and their representations.
For a large part, they summarise relevant material from knapps book 12. Lecture speciali es to the representation theory of. The representation theory of semisimple lie groups has its roots in invariant theory and the strong links between representation theory and algebraic geometry have many parallels in differential geometry, beginning with felix kleins erlangen program and elie cartans connections, which place groups and symmetry at the heart of geometry. Lie groups, lie algebras, and representations springerlink. The group algebra is not semisimple if characteristic divides group order. Conversely, suppose that the killing form of l has a nonzero kernel s. G glw are said to be equivalent or isomorphic if there exists a vector space isomorphism. A faithful representation is one in which the homomorphism g glv is injective. If gis compact, then it has a complexi cation g c, which is a complex semisimple lie group, and the irre. If there is torsion in the homology these representations require something other than ordinary character theory to be understood. The ultimate goal is to be able to understand all the irreducible unitary representations of any such group gup to unitary equivalence. From now on, all rings will be associative rings with 1 6 0 which. Representation theory was born in 1896 in the work of the german mathematician f. If gis a nite group then a representation of gover c is the same as a module over the group ring cg also written cg.
Determining a semisimple group from its representation. Although theorems are always stated precisely, many illustrative examples or classes of. Prerequisites include acquaintance with elementary group theory and some familiarity with rings and modules. In particular, we discuss the theory of localization of modules overthe envelop. A geometric construction of the discrete series for. In other words, a representation is a rule, how to assign a linear transfor. Let v be an irreducible representation of a of dimension n. Designed to be accessible to graduate students in mathematics or physics, they have a minimum of prerequisites. Trapa, representations of semisimple groups lecture 3 develops the abstract representation theory of compact groups and the theory of induced representations. Connectedness of centralizers of semisimple liealgebra elements under the action of a semisimple algebraic group 4 existence of regular semisimple elements of reductive groups in characteristic 0.
On the representation theory of semisimple lie groups. Then y endv, with action of a by left multiplication, is a semisimple representation of a, isomorphic to nv the. A semisimple or completely reducible representation of a is a direct sum of irreducible representations. Representtionsa of semisimple lie groups zhaoting wei jonathan block this thesis consists of three chapters.
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