Finite dimensional algebras and quantum groups pdf download

The text includes in particular infinitedimensional unitary representations for abelian groups, heisenberg groups and sl2, and representation theory of finitedimensional algebras. Our main theorem is a characterization of calgebras that have a separating family of finitedimensional representations. Download online ebook pdf now lie algebras, part 2. Selfextensions for finite symplectic groups via algebraic groups 173 186. Know that ebook versions of most of our titles are still available and may be downloaded immediately after purchase. On finite gkdimensional nichols algebras over abelian groups. Specifically, we prove that the free product of such calgebras again has this property. Finite dimensional approximations and deformations of group. Im a phd student in operator algebraic quantum groups, which is a generalisation of topological groups effectively. Cartan described the semisimple algebras over the complex and real fields and paved the first steps towards the study of non.

This provides a new perspective on the representation theory of such algebras, and leads to several new structures attached to them. This text combines, for the first time in book form, the theories of finite dimensional algebras and quantum groups. Hopf algebras and quantum groups 1st edition stefaan. The finite group w is a coxeter group generated by s1. The coauthor of brauer groups and the cohomology of graded rings and coeditor of rings, hopf algebras, and brauer groups both titles, marcel dekker, inc. The material covered ranges from basic definitions of lie groups to the classification of finite dimensional representations of semisimple lie algebras. The interplay between finite dimensional algebras and lie theory dates back many years. First, we reduce the problem to finite generation of cohomology for finite dimensional nichols algebras of diagonal type. Representations of algebraic groups, quantum groups, and lie.

As we will see, both of these concepts may be characterized as nitedimensional approximations and deformations. Birs workshop on stability conditions and representation theory of finitedimensional algebras casa matematica oaxaca cmo, mexico. Classification of finite dimensional simple lie algebras in prime characteristics 185 198. Later finite dimensional algebras were studied by a large number of mathematicians including b. This volume is based on the proceedings of the hopfalgebras and quantum groups conference at the free university of brussels, belgium. In the first part of the thesis, we study quantum groups associated to a semisimple lie algebra g. The invariants are based on right integrals for these hopf algebras.

Stefaan caenepeel is a professor in the faculty of applied sciences at the free university of brussels, belgium. In this paper we consider twodimensional quantum real weighted projective. The last chapter is devoted to some applications of quivers, including harishchandra modules for sl2. Irreducible quantum group modules with finite dimensional weight.

Hopf algebras, quantum groups and topological eld theory. As a main feature, the book emphasizes the finite dimensional algebra approach to quantum groups. Endomorphism algebras of tensor powers of modules for. Throughout the section, we work over an algebraically closed. On the classification of finitedimensional pointed hopf algebras.

A trip from representations of the kronecker quiver to canonical bases of quantum affine algebras 231 244. Hopf algebras, quantum groups and yangbaxter equations mdpi. There are a wide variety of examples in algebraic lie theory to which this applies. While discussing all classes of finite and infinite dimensional lie algebras and lie superalgebras in terms of their different classes of root systems, the book focuses on kacmoody algebras. Representations of finite dimensional algebras and related. Buy finite dimensional algebras and quantum groups mathematical surveys and. In this paper, we have generally omitted those proofs. The simplest and most important example is the drinfeldjimbo quantum group.

So quantum groups begin with the study of functions on a group as its a generalisation, in the hopf algebra case we have. Hopf algebras, quantum groups and yangbaxter equations. Most of the essential structural results of the theory follow immediately from the structure theory of semisimple algebras, and so this topic occupies a long chapter. Finitedimensional representations of the quantum analog of the enveloping algebra of a complex simple lie algebra. The classical chevalley theorem states that for a cartan subalgebra and w the weyl group of g, the restriction of ginvariant polynomials on g to is an isomorphism onto the winvariant polynomials on, res. In more recent times, these interrelations have become even more strikingly apparent. Infinitedimensional lie groups and algebras in mathematical physics this is an open access article distributed under the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Also the mutliplication is compatible with the vector space structure. Representations of finite dimensional algebras download representations of finite dimensional algebras ebook pdf or read online books in pdf, epub, and mobi format. Basic hopf algebras and quantum groups introduction. On the classification of finite dimensional pointed hopf algebras.

Finite dimensional nichols algebras over finite cyclic groups. The material covered ranges from basic definitions of lie groups to the classification of finitedimensional representations of semisimple lie algebras. Download pdf representations of finite dimensional algebras. Quasitriangular hopf algebras constitute a very important class of hopf algebras, which were introduced by drinfeld dr1 in order to supply solutions to the quantum yangbaxter equation that arises in mathematical physics. Cuntzpimsner algebras of group representations deaconu, valentin, rocky mountain journal of mathematics, 2018 partial actions of c. More precisely, it investigates the ringelhall al gebra. He suggested the term quantum groups for hopf algebras. On representations of quantum groups and cherednik algebras. A second application is in the realm of soliton equations and their infinitedimensional symmetry groups and algebras. Finite dimensional algebras and quantum groups, by bangming deng, jie du.

Hopf algebras h over an algebraically closed field k. The other side of quantum groups are the purely algebraic versions, that is to say hopf algebras. Highest weight theory for finitedimensional graded algebras. Finite dimensional algebras and quantum groups cover image. Peirce, clifford, weierstrass, dedekind, jordan and frobenius. Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple lie algebra. Get your kindle here, or download a free kindle reading app. In addition to the traditional instructional workshop preceding the conference, there were also workshops on commutative algebra, algebraic geometry and representation theory, finite dimensional algebras, algebraic groups and lie theory, and quantum groups and hall algebras. Finite dimensional algebras and quantum groups, by bangming. Finitedimensional modules of quantum affine algebras. First, the book contains an exposition on the role of finitedimensional semisimple lie algebras and their representations in the standard and grand unified models of elementary particle physics. We show that the cohomology ring of a finitedimensional complex pointed hopf algebra with an abelian group of grouplike elements is finitely generated. Representations of algebras and finite groups 7 preface these notes describe the basic ideas of the theory of representations of nite groups. As in 12, this led us to consider measured quantum spaces measured z algebras or frobenius algebras, see section 2 rather than quantum spaces this framework is in fact very natural, especially when we have a view towards the example of comodule algebras over a finite.

Specifically, we prove that the free product of such c algebras again has this property. Commons attribution cc by license, which allows users to download, copy and build upon. Click download or read online button to representations of finite dimensional algebras book pdf for free now. Lie algebras by shlomo sternberg the campbell baker hausdorff formula, sl2 and its representations, classical simple algebras, engelliecartanweyl, conjugacy of cartan subalgebras, simple finite dimensional algebras, cyclic highest weight modules, serres theorem, and more. This book can serve as a text for an introduction to the theory of quantum groups. The book includes the theory of poisson lie groups quasiclassical version of algebras of functions on quantum groups, a description of representations of algebras of functions, and the theory of quantum weyl groups. It presents stateoftheart papers selected from over 65 participants representing nearly 20 countries and more than 45 lectures on the theory of hopf algebras, including multiplier hopf algebras and. Infinite dimensional lie algebras an introduction progress in. Finite and infinite dimensional lie algebras and applications in physics studies in mathematical physics pt. Infinite dimensional lie groups and algebras in mathematical physics this is an open access article distributed under the creative commons attribution license, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For a finite dimensional semisimple lie algebra \frakg and a root q of unity in a field k, we associate to these data a double quiver. We show that the cohomology ring of a finite dimensional complex pointed hopf algebra with an abelian group of grouplike elements is finitely generated. Our main theorem is a characterization of c algebras that have a separating family of finite dimensional representations. This paper studies invariants of 3manifolds derived from certain finite dimensional hopf algebras via regular isotopy invariants of unoriented links in the blackboard framing.

List of finitedimensional nichols algebras wikipedia. Finite dimensional algebras and quantum groups mathematical. This new notion and the resulting properties of the modules in question led to a number of results concerning the ring structure of the endomorphism algebras of tensor powers of such modules. In this work we seek to investigate when \ nitedimensional approximations of groups exist with particular interest in determining whether group calgebras are quasidiagonal or have property qh. Finite dimensional algebras and quantum groups bangming.