Introduction to lie algebras and representation theory. Thus, enveloping algebras is a vibrant area of research, with many mathematicians around the globe working and hundreds of papers published. A ne niltemperleylieb algebras and generalized weyl algebras. The representation theory of the temperleylieb algebras. These are the notes of a topics in representation theory class i taught in.
If g is a lie algebra, there is a universal associative algebra. 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. Valued graphs and the representation theory of lie algebras. A ne niltemperleylieb algebras and generalized weyl. In representation theory of such algebras, the enveloping algebra. More precisely, we shall investigate the representation theory of the following finite dimensional k algebras. A representation of an associative algebra a also called a left amodule is a vector space. Part ii of the text covers semisimple lie algebras and their representations. The interplay of representation theory and combinatorics builds on a long tradition. In representation theory they appear as the images of the associated representations of the lie algebras or of the enveloping algebras on the garding domain and in quantum field theory they occur as the vector space of field operators or the algebra generated by them.
The latter freeness property is essential in a treatment of maassshimura operators. These algebras play an important role in the representation theory of restricted lie algebras since their module categories approximate to a certain degree to be made precise below the category of modules for the correspondinglie algebra, and methods. Enveloping algebras and geometric representation theory 3 abstracts towards a riemannhilbert correspondence for domodules konstantin ardakov joint work with simon wadsley 1. Study spectral decomposition of l2x under hecke operators 2. The representation theory of the temperleylieb algebras christopher ryba supervisor. Highest weight representationsunitary representations of the virasoro algebra examples. Recall that, for an associative algebra a with unity 1, a lie algebra structure on a. Enveloping algebra an overview sciencedirect topics. In particular, their seminal work shows that primitive ideals can be characterised both topologically and algebraically among the prime ideals, as. For the tensor product of vector spaces v and wover f, we use the notation v w v fw. Restricted cohomology and representation theory 5 is a restricted lie algebra. Skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites.
Proposition 4 proposition 5 cmu2 let c and d be pointed hopf algebras with the same coradical a. The ramification behaviour of the maximal ideals of zh with respect to the subalgebra r is studied. Restricted enveloping algebras with minimal lie derived. Lie theory, universal enveloping algebras, and the poincar. Representation theory of complex semisimple lie algebras 59 the theorem of the highest weight 59 some multilinear algebra 63 existence of irreducible representations 67 the universal enveloping algebra and verma modules 72 chapter 5.
Representation theory of quantized enveloping algebras with. I begin with an entire chapter on the representation theory of sl. Ji1 independently showed that the universal enveloping algebra of a semisim ple lie algebra. The classi cation of root systems and complex simple lie algebras 54 chapter 4. The tensor algebra or free algebra of the vector space v over the eld f is the vector space tv 1 k. May 16, 2015 enveloping algebras and geometric representation theory. Universal enveloping algebras and some applications in physics. Please approve edit to the sentence the sets of completely prime. An algebra aover a eld f is a vector space aover f together with a multiplication map. Coorganizergeometric representation theory, ams central sectional meeting, lawrence, kansas, 330412012. T he aim of this workshop is to discuss the current state of research in the area of enveloping algebras and their applications, primarily. Request pdf enveloping algebras and geometric representation theory since its inception in the early seventies, the study of enveloping algebras has undergone a significant and continuous.
The universal enveloping algebra of a semisimple lie algebra. A\ x oia has been introduced in 10, 21, 16 which provides a solution of the universal problem for continuous representations of a into bounded. A subrepresentation of a representation vis a subspace u. Let h be a hopf algebra which is a finite module over a central subhopf algebra r. In particular the study of algebras that admit a notion of highest weight modules has turned out to be remarkably fruitful. In particular, their application to the representation theory of associative.
We apply the theory to hecke algebras, reduced universal enveloping algebras of restricted lie algebras, small halfquantum groups, and nichols quantum symmetric algebras. Dennis gaitsgory nick rozenblyum harvard university. Enveloping algebras and geometric representation theory, oberwolfach, germany, 343102012 frg conference. Enveloping algebras and geometric representation theory. Enveloping algebras and geometric representation theory, oberwolfach, germany, 343102012. Lehrer the university of sydney february 2014 1 introduction the temperleylieb algebras arise in the theory of quantum groups as endomorphism algebras of tensor powers of a distinguished representation of the quantised universal enveloping. A representation of an associative algebra a also called a left. In analyzing the representation theory of 9, we shall consider representations concentrated at some character x e spec 9. On the one hand, this example can be treated in an elementary way, simply by writing down a basis and calculating. Since the restricted universal enveloping algebra ul,0 of a finite dimensional restricted lie algebra l is a frobenius algebra, it is possible and indeed more. This paper develops a general theory of canonical bases and how they arise naturally in the context of categorification. Complex semisimple quantum groups and representation theory. International workshop enveloping algebras and representation.
Just as there are many different types of representations for kacmoody algebras and their universal enveloping algebras, so there are many different types of representation for quantum groups. Unbounded operator algebras and representation theory. This book, which is the first systematic exposition of the algebraic approach to representations of lie groups via representations of or modules over the corresponding universal enveloping algebras, turned out to be so well written that even today it remains one of. As an application, we show that lusztigs canonical basis in the whole quantized universal enveloping algebra is given by the classes of the indecomposable 1morphisms in a categorification when the associated lie algebra is of finite type and simply laced. Universal enveloping algebras of lie algebras appeared more than a century ago as one of the major tools in lie theory. In mathematics, a universal enveloping algebra is the most general unital, associative algebra that contains all representations of a lie algebra universal enveloping algebras are used in the representation theory of lie groups and lie algebras. This is an expository survey article on compound du val cdv singularities, with emphasis on recent homological approaches, including.
In addition, the enveloping algebra gives a precise definition for the casimir operators. Identify joint spectrum with a space of galois representations 3. Representations of lie algebras, casimir operators and. Usefi, foxtype problems in enveloping algebras, journal of algebra, 319 2008 24892495. Canonical bases and higher representation theory compositio. Usefi, the isomorphism problem for universal enveloping algebras of lie algebras, algebras and representation theory, 10 2007, no. More generally, we study the theory of bases arising from indecomposable objects in higher representation theory, which we term orthodox and the overlap of this theory with the more classical theory of canonical bases. Itis well known that u qg at roots of unity is not semisimple. Integral bases for the universal enveloping algebras of. Atlantic algebra centre network of ontario lie theorists enveloping algebras and representation theory international workshop august 28 september 1, 2014 memorial university st. The aim of this workshop is to discuss the current state of research in the area of enveloping algebras and their applications, primarily in representation theory.
Introduction to representation theory mit mathematics. Algebraic groups modular lie algebras universal enveloping algebras representations of associative algebras auslanderreiten theory module varieties support varieties, rank varieties, ppoints homological algebra. In recent times, a more general theory, namely that of locally c algebras 7 and of locally w algebras 3, 9, is developed. As is the case for all hopf algebras, u q g has an adjoint representation on itself as a module, with the action being given by. Organizers iain gordon, edinburgh bernard leclerc, caen wolfgang soergel, freiburg. Because casimir operators commute with all elements of a lie. Enveloping algebra lie algebras having triangular decomposition include all kacmoody lie algebras. 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. Mar 27, 2021 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, c algebras and hopf algebras, with particular emphasis on quantum groups. Simple lie superalgebras, which are of particular interest from the standpoint of representation theory. Let 9 be a finite dimensional restricted lie algebra and let x. Universal enveloping algebras are used in the representation theory of lie groups and lie algebras.
Some factorizations in universal enveloping algebras of three dimensional lie algebras and generalizations volume 45 issue 4. Representation theoryquantum field theorygauge theory holy grail of representation theory all hecke operators at almost all p commute langlands program. They find many applications in differential geometry and mathematical physics, and are indispensable to many directions of research in representation theory of lie groups and algebras. Pdf koszul duality and modular representations of semi. In principle, our theory must be modi ed to apply to u qg. For example, verma modules can be constructed as quotients of the universal enveloping algebra.
With the increasing interest in infinitedimensional representations, the role of enveloping algebras becomes even more significant. Representation theory of complex semisimple lie algebras 59 the theorem of the highest weight 59 some multilinear algebra 63 existence of irreducible representations 67 the universal enveloping algebra. Periods of automorphic forms and applications to lfunctions, columbia university, new york, 9169182011. The statement here is a corrected version of cmu, 2. The algebra representations of the algebra u ought to correspond precisely with the lie. Then we have the poincar ebirkho witt pbw isomorphism i pbw. The socalled lie algebras of classical type are psubalgebras of glnf for some positive integer n satisfying certain conditions, e. Similarly, a representation of a lie algebra g on a vector space v is a lie algebra homomorphism. Famous examples are provided by universal enveloping algebras of lie algebras, quantum groups and weyl algebras. A representation of an associative algebra aalso called a left amodule is a vector space v equipped with a homomorphism a. Let g be a nite dimensional lie algebra over a eld kof characteristic zero. Representations of lie algebras, casimir operators and their.
Some factorizations in universal enveloping algebras of. Lie algebras and cocommutative co algebras 219 introduction 219 1. Quite generally, we study this class of algebras from the point of view of poisson geometry, exhibiting connections between their representation theory and some wellknown geometric constructions. A representation of an associative algebra aalso called a left amodule is a vector space v equipped with a homomorphism a endv, i.
Background let rbe a complete discrete valuation ring with uniformiser. We study the ensuing verma module theory, and the associated quotients of these modified quantized enveloping algebras. Introduction a complete locally mconvex algebra a is a topological algebra that is an inverse limit of banach algebras. The aim of this workshop was to discuss the current state of research in the area of enveloping algebras and their applications, primarily in representation theory. In the case when h is ug, the enveloping algebra of a semisimple lie algebra g, a conjecture of humphreys is confirmed. Representation theory princeton math princeton university. The universal enveloping algebra and verma modules. While quivers are useful tools in representation theory, they have their limitations. Representation theory of finite dimensional lie algebras. Kacmoody lie algebras, quantum groups quantized enveloping algebras, quantized coordinate algebras and quantum cluster algebras papers and preprints isomorphisms among quantum grothendieck rings and propagation of positivity, joint work with ryo fujita, david hernandez, and sejin oh, arxiv.
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