Algebraic groups and their representations springerlink. Pdf endomorphism algebras and representation theory. Acknowledgments many thanks go to hanspeter kraft, who introduced me to this interesting subject, for his constant support, for uncountable suggestions and ideas. This is a significantly revised edition of a modern classic. To analyze their structure, one seeks to build them up from algebraic groups of a specific geometric nature, such as smooth, connected. Standard constructions of representation theory restrictions to invariant. Topics of the workshop include globallocal conjectures in the representation theory of finite groups representations and cohomology of simple, algebraic and finite groups connections to lie theory and categorification, and applications to group theory, number theory, algebraic geometry, and combinatorics. Eric friedlander university of southern california 4. On the other hand, this example allows the reader to see the machinery. Algebras and representation theory features carefully refereed papers relating, in its broadest sense, to the structure and representation theory of algebras, including lie algebras and superalgebras, rings of differential operators, group rings and algebras, calgebras and hopf algebras, with particular emphasis on quantum groups. The theory of algebraic groups is central to various subjects in mathematics, ranging from number theory to mathematical physics. The kernel of this morphism is a ghomogeneous vector bundle b over x. Representations of algebraic groups american mathematical society.
Introduction to representation theory of lie algebras. In these notes we outline some aspects of the modular representation theories of. This approach enables us to obtain a number of new results in information theory. There are a veritable cornucopia of resources to study representations of algebraic groups. On the other hand, this example allows the reader to. Representation theory of algebraic groups inauguraldissertation zur. We say that a representation of a complex algebraic group kin a linear space v is algebraic if v is a union of. The following words then turn out to be independent x 1110, y 01010101. Representations of algebraic groups and quantum groups. Chapter 2 is devoted to the basics of representation theory.
Representation theory of finite abelian groups over c 17 5. For the representation theory of padic groups, the correct analogy is going to be representations of nite groups, rather than the theory for reductive or algebraic groups which makes heavy use of weights. Finite groups, lie groups, lie algebras, and representation. The theory of algebraic groups results from the interaction of various basic techniques from field theory, multilinear algebra, commutative ring theory, algebraic geometry and general algebraic representation theory of groups and lie algebras. In this introductory course, we exposit the rich theory of algebraic groups and supply useful tools to those interested in group theory, representation theory, number theory, algebraic geometry, and related subjects. Groups recur throughout mathematics, and the methods of group theory have influenced. Pdf ebooks can be used on all reading devices immediate ebook.
Algebraic transformation groups and representation theory ronald l. In this theory, one considers representations of the group algebra a cg of a. Besides being a subject of great intrinsic beauty, the theory enjoys the additional benefit of having applications in myriad contexts outside pure mathematics, including quantum field theory and the study of molecules in chemistry. Introduction algebraic groups have a relatively wellbehaved decomposition. Homology bases arising from reductive groups over a finite field. A group scheme g say over a field is called finite if the algebra of regular functions on g is finite dimensional. There are already many good books on representation theory for all kinds of groups. The workshop is held on the occasion of brians retirement. U g is a locally compact group, then the unitary representation theory of g is precisely that of its group c algebra c g. Researchers in mathematics and mathematical physics will also find this book useful.
It is thus an ideally suitable framework for exhibiting basic algebra in action. On the one hand, this example can be treated in an elementary way, simply by writing down a basis and calculating. The original purpose of representation theory was to serve as a powerful tool for obtaining information about nite groups via the methods of linear algebra, such as eigenvalues, inner product spaces and diagonalization. Representation theory university of california, berkeley. Group representations and algebraic information theory 1125 is called the mutual information of the words. Illustration with algebraic groups find, read and cite all the research you need on researchgate. An introduction revised and expanded version, under construction peter woit department of mathematics, columbia university. We have seen that the representation theory of sl2k in. Accordingly, our exposition emphasizes the discussion of illustrative examples and plausibility con.
As such, in the bonus exercises, we freely use the language of algebraic geometry and you are welcome to ask us for clari cation. The second part of the book is devoted to the representation theory of reductive algebraic groups and includes topics such as the description of simple modules, vanishing theorems, the borelbottweil theorem and weyls character formula, and schubert schemes and line bundles on them. Representation theory this is the theory of how groups act as groups of transformations on vector spaces. And if g contains a normal subgroup n, then c g may be identified with a twisted crossed product cg, cn, 7 of cn by gin. In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. Representation theory of algebraic groups and related topics. Besides gln, there are four other series of algebraic groups that are called classical. Buy this book isbn 9780817646974 digitally watermarked, drmfree included format. Representation theory of finite groups anupam singh. Preface dmodules, perverse sheaves, and representation theory is a greatly expanded translation of the japanese edition entitled d kagun to daisugun dmodules and algebraic groups which was published by springerverlag tokyo, 1995. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations. Some structure theorems for algebraic groups institut fourier. Two of the best in this authors opinion are the one by a. In mathematics, schurs lemma is an elementary but extremely useful statement in representation theory of groups and algebras.
Lectures on the geometry and modular representation theory of. In this theory, one considers representations of the group algebra a cg of a finite group g the algebra with basis ag,g. Two contributions to the representation theory of algebraic groups. It is according to professor hermann a readable book, so it would be appropriate for this plannedtobe reading course. For the new english edition, the two authors of the original book, r. Group actions and group representations 1 passing to the lie algebra 5 a primer on the lie group lie algebra correspondence 8 chapter 2. Pdf on aug 6, 2014, aderemi kuku published k theory and representation theory. Representation theory investigates the different ways in which a given algebraic objectsuch as a group or a lie algebra can act on a vector space.
Later on, we shall study some examples of topological compact groups, such as u1 and su2. Representation theory of algebraic groups and quantum groups. Lectures on the geometry and modular representation theory. Connected components and homomorphisms 5 exercises to subsection 1. It is thus an ideally suitable framework for exhibiting basic algebra.
Combinatorial and geometric aspects of the representation. Pdf two contributions to the representation theory of. In general, it reflects the properties of g much better than lieg does. Algebraic groups, representations and invariant theory. Algebraic groups and invariant theory iii contents preface i 1. Induced representations in physics, probability and number theory ma1. The conventions of algebraic geometry that we use are as in hartshorne har. Actions and representations of algebraic groups 9 exercises to subsection 1. We suggest a new way of developing the representation theory of symmetric groups. Lang abstract algebra, third edition, by david dummit and richard foote. As to your second question the answer isit depends. Representation theory an overview sciencedirect topics.
The concept of a group is central to abstract algebra. The theory of g, kmodules first introduced by harishchandra for connected k and later defined in general by lepowsky is the connecting link between the algebraic results of chevalley and harishchandra and group representation theory. Ii the book begins with an introduction to schemes chapter 1. The theory of matrices and linear operators is wellunderstood, so representations of more abstract objects in terms of fa. Classify all representations of a given group g, up to isomorphism. Group representations and algebraic information theory v. 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. Algebraic dmodules and representation theory of semisimple. Xis the borel subalgebra bx which corresponds to the point x. General theory of lie algebras basic classes of lie algebras representations and the killing form 21 some basic results on semisimple lie algebras 29 chapter 3.
Two contributions to the representation theory of algebraic. For such g the representation theory is equiv alent. Introduction to actions of algebraic groups institut fourier. Examples and rst properties 1 exercises to subsection 1. Representation theory of algebraic groups and related topics abstract haruko nishi josai university, a pseudometric on moduli space of hyperelliptic curves troyanov s theorem says that the moduli space of riemann surfaces of genus g with p punctures is identi. Commutator subgroup and one dimensional representations 10 chapter 3. Thus readers who understand the theory of algebraic groups and their representations will. Representations of algebraic groups and their lie algebras jens. Representation theory people department of mathematics eth. Realizing a lie group as an algebraic group is the. Ithe action of kon vi induces a morphism of algebraic groups k.
Representation theory d 24 lectures, lent term linear algebra, and groups, rings and modules are essential. Lie groups arise as covering groups of algebraic groups. Representation theory representation theory and character theory of finite groups. A separate solutions manual is available for instructors. More generally let m be any algebraic representation of a group scheme k.
Msri representations of finite and algebraic groups. If representation theory is a black box, or a machine, then the input. Let me recall some basic facts on the lie algebra g lieg of g. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations for example. The adjective \combinatorial will refer to the way in which we answer representation theoretic questions. Later on, we shall study some examples of topological compact groups. Representation theory of finite groups and associative. Introduction to representation theory mit mathematics. Topics of the workshop include globallocal conjectures in the representation theory of finite groups representations and cohomology of simple, algebraic and finite groups connections to lie theory and categorification, and applications to group theory, number theory, algebraic.
Symmetry, representations, and invariants roe goodman. This is a rough preliminary version of the book published by cup in 2017, the final version is substantially rewritten, and the numbering has changed. Group representations and algebraic information theory. Calgebras and mackeys theory of group representations. For example, consider the natural 2dimensional representation3 of gl 2q p on v 2.
References for the representation theory of some algebraic groups. Lipsman 1 mathematische annalen volume 214, pages 149 157 1975 cite this article. In the group case it says that if m and n are two finitedimensional irreducible representations of a group g and. Using representation theory and invariant theory to analyze the symmetries that arise from group actions, and with strong emphasis on the geometry and basic theory of lie groups and lie algebras, symmetry, representations, and invariants is a significant reworking of an earlier highlyacclaimed work by the authors. This is the aim of part i of this book, whereas part ii then deals with the representations of reductive groups. The representation theory of anything else than groups. On the other hand, some authors use the term linear algebraic group in order to x the matrix realization of a group g.
391 734 878 968 971 7 1536 1181 796 68 161 683 1172 739 186 1446 1036 58 74 97 203 1607 625