Finite dimensional algebras and quantum groups pdf file

Approximately finitedimensional calgebra wikipedia. Auslander reiten quivers with automorphisms 150 exercises and notes 156 part 2. More precisely, it investigates the ringelhall algebra realization for the positive part of a quantum. Representations of algebraic groups, quantum groups, and lie. Download for offline reading, highlight, bookmark or take notes while you read representation theory of finite groups and associative algebras. Lecture course hopf algebras, quantum groups and topological. A finitedimensional lie algebra arising from a nichols algebra of diagonal type rank 2. Representation theory of finite groups and associative algebras ebook written by charles w. This paper investigates the structure of basic finite dimensional. Click download or read online button to get infinite dimensional lie algebras an introduction progress in. 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 topological field theory. Endomorphism algebras of tensor powers of modules for. Motivated by the appearance of vertex operators in the construction of the moonshine module, borcherds introduced vertex operator algebras as part of his proof of the moonshine conjectures in 1986. Hopf algebras and topological field theories have applications in topology, string theory, quantum gravity and quantum information theory. This text combines, for the first time in book form, the theories of finite dimensional algebras and quantum groups. Apr 24, 2005 quantum groups and double quiver algebras article pdf available in letters in mathematical physics 711 april 2005 with 29 reads how we measure reads. Calgebras and finitedimensional approximations nathanial. Representation theory of finite groups and associative. 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.

Exceptional sequences in hall algebras and quantum groups. 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. The interplay between finite dimensional algebras and lie theory dates back many years. Infinite dimensional lie algebras an introduction progress in. Infinitedimensional lie groups and algebras in mathematical. The resulting quotient algebras are called generalised. This book can serve as a text for an introduction to the theory of quantum groups. Bangming deng, jie du, brian parshall, university of virginia, charlottesville, va and jianpan wang. In these modules we find a maximal submodule and by studying the quotient, we get an explicit description of all finitedimensional irreducible representations corresponding to drinfeld polynomials with distinct zeroes. Knowledge about these algebras can lead to new results about the representations of schur algebras and finite dimensional hecke algebras. Finitedimensional modules of quantum affine algebras. Quantum groups quantized enveloping algebras and related deformations see also 16t20.

Besides their applications to quantum groups and hopf algebras, nichols algebras have many other interesting. Hecke algebras associated with symmetric groups give rise to an interesting class of quasihereditary algebras, the quantum schur algebras. Finite groups and semisimple algebras in quantum mechanics. Given a finitedimensional nichols algebra b, our method provides a hopf algebra h such that b is a braided hopf algebra in the category. Finally, we are indebted to the publishers, springerverlag, for enabling this book to reach such a wide audience in the world of mathematical community.

We present new examples of finitedimensional nichols algebras over fields of positive characteristic. Finite dimensional algebras and quantum groups bok. In mathematics, an approximately finitedimensional af calgebra is a calgebra that is the inductive limit of a sequence of finitedimensional calgebras. Representation theory of finite groups and associative algebras. A representation of the quantum plane establishes the sharpness of this bound, and we prove a bound of 2n.

We study finitedimensional representations of quantum affine algebras using qcharacters. Pbw property, applicable to finite dimensional representations of lie algebras and quantum groups. Mathematical surveys and monographs publication year 2008. Finitedimensional leavitt path algebras article pdf available in journal of pure and applied algebra 2093 june 2007 with 122 reads how we measure reads. It is proved that l is a finitedimensional hopf algebra if and only if there exists an invertible matrix r, solution of the pentagon equation r 12 r r 23 r 23 r 12, such that l. Finitedimensional noncommutative poisson algebras finitedimensional noncommutative poisson algebras kubo, fujio 19961223 00.

The students in that course oleg golberg, sebastian hensel, tiankai liu, alex schwendner, elena yudovina, and dmitry vaintrob co. 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. Infinite dimensional lie algebras an introduction progress. Show full abstract poincarebirkhoffwitt pbw property, applicable to finite dimensional representations of lie algebras and quantum groups. We describe a correspondence or duality between the qcharacters of. Sundar september 14, 2012 throughout h, kstand for nite dimensional hilbert spaces. Quantum groups quantized enveloping algebras and related deformations see also 16t20, 20g42, 81r50, 82b23. Quantum chains of hopf algebras with quantum double cosymmetry. Representations of algebraic groups, quantum groups, and lie algebras.

Asked to write a letter of recommendation for promotion for an abusive former professor one that is paid but cannot be bought. We prove the conjectures from fr and derive some of their corollaries. Examples of finitedimensional pointed hopf algebras in. Endomorphism algebras of tensor powers of modules for quantum. Quantum double finite group algebras and their representations volume 48 issue 2 m. We then describe the associated trigonometric solutions of the quantum yangbaxter equation.

Finite quantum groups over abelian groups of prime. This book systematically studies most of the numerous types of approximation properties that have been important in recent years. Quantum groups quantized enveloping algebras and related deformations. Finitedimensional representations of the quantum lorentz group. Finite dimensional algebras and quantum groups ams bookstore. List of finitedimensional nichols algebras wikipedia. From hopf algebras to topological quantum groups choices. Bialgebra cohomology, pointed hopf algebras tamu math.

Let n srr, thus n is the rank of the grothendieck group k0r of all finite length modules modulo split. Approximate finitedimensionality was first defined and described combinatorially by ola bratteli. Representations of algebraic groups, quantum groups, and. The present lecture notes arose from a representation theory course given by prof. Knowledge about these algebras can lead to new results about the representations of schur algebras and finitedimensional hecke algebras.

It generalizes the construction of quantum function algebras using quantum grassmanian algebras. A is the observable algebra of a generalized quantum spin chain with horder and hdisorder symmetries, where by a duality transformation the role of order and disorder may also appear interchanged. In this article, we give a brief overview of some of these topics. Are all semisimple commutative banach algebras finitedimensional.

Volume 1, the vidavpalmer and gelfandnaimark theorems. On borel subalgebras of quantum groups request pdf. Hopf algebras and topological field theories have applications in representation theory, topology, string theory, quantum gravity and quantum information theory. Quantum groups and double quiver algebras article pdf available in letters in mathematical physics 711 april 2005 with 29 reads how we measure reads. This includes diffeomorphism groups and their natural subgroups like volumepreserving and symplectic transformations, as well as gauge groups and loop groups. No prior knowledge of km lie algebras or of finitedimensional algebraic groups is required, but. Hopf algebras h over an algebraically closed field k. Given a finite dimensional nichols algebra b, our method provides a hopf algebra h such that b is a braided hopf algebra in the category. Elliott gave a complete classification of af algebras using the k 0 functor whose range consists of ordered abelian. Approximate finite dimensionality was first defined and described combinatorially by ola bratteli. This section provides the lecture notes from the course. Kirichenko preface the theory of finite dimensional algebras is one of the oldest branches of modern algebra. In more recent times, these interrelations have become even more strikingly apparent. Classification of finite dimensional irreducible representations of.

This means here that a is a direct sum of matrix algebras. Exceptional sequences in hall algebras and quantum groups volume 117 issue 2 xueqing chen, jie xiao. For finitedimensional ones we show that if they are semisimple as associative algebras then they are standard. Representations of finite dimensional algebras and related. Finite dimensional algebras and quantum groups by bangming. Quantum function algebras from finitedimensional nichols. We shall discuss generic extension monoids associated with. Heisenberg double, pentagon equation, structure and.

Throughout the section, we work over an algebraically closed. These subjects are very closely related to finite dimensional representations of affine hecke algebras, affine quantum groups, nakajimas quiver varieties, and the structure of hall algebras. A second application is in the realm of soliton equations and their infinite dimensional symmetry groups and algebras. Descargar algebras, bialgebras, quantum groups, and. Etingof in march 2004 within the framework of the clay mathematics institute research academy for high school students. An explicit fusion algebra isomorphism for twisted quantum doubles of finite groups.

On the reduced operator algebras of free quantum groups. In particular, we prove that the tensor product of fundamental representations is reducible if and only if at least one of the pairwise normalized rmatrices has a pole. Im a phd student in operator algebraic quantum groups, which is a generalisation of topological groups effectively. First, the book contains an exposition on the role of finite dimensional semisimple lie algebras and their representations in the standard and grand unified models of elementary particle physics. 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. Click download or read online button to get infinite dimensional lie algebras an introduction progress in mathematics book now. Let a be a finite dimensional hopf algebra and h, r a quasitriangular bialgebra. We present new examples of finite dimensional nichols algebras over fields of positive characteristic. Finite dimensional algebras and quantum groups mathematical. Introduction to hopf algebras and representations, spring.

Know that ebook versions of most of our titles are still available and may be downloaded immediately after purchase. Some people would say that any finitedimensional hopf algebra is a finite quantum. Isomorphism classes of small quantum groups in positive. Let r be a finite dimensional algebra over some field k. We describe how to find quantum determinants and antipode formulas from braided vector spaces using the frtconstruction and finitedimensional nichols algebras. An important step in the proof follows from a another result, where we show that a wide family of finite dimensional pointed hopf algebras is generated by grouplike and skewprimitive elements, giving additional support to a conjecture in our previous paper finite quantum groups and cartan matrices, adv. Quantum groups via finitedimensional algebras school of. The study of hopf algebras sometimes also known as quantum groups is a very active field, relating algebra, representation theory and mathematical physics.

The study of the pentagon equation leads to results on the structure and classification of finite quantum groups. A second application is in the realm of soliton equations and their infinitedimensional symmetry groups and algebras. We describe how to find quantum determinants and antipode formulas from braided vector spaces using the frtconstruction and finite dimensional nichols algebras. Introduction to hopf algebras and representations, spring 2011. Special emphasis in this class will be on complex finitedimensional hopf algebras. In mathematics, an approximately finite dimensional af calgebra is a calgebra that is the inductive limit of a sequence of finite dimensional c algebras. We construct finitedimensional representations of the quantum affine algebra associated to the affine lie algebra. Hopf algebra h and its dual h we construct the infinite crossed product and study its superselection sectors in the framework of algebraic quantum field theory. If q is a primitive lth root of unity of k, then one can define the small quantum group uqsu2k, which is a finitedimensional quotient of uqsu2k of dimension l3. Capproximation theory has provided the foundation for many of the most important conceptual breakthroughs and applications of operator algebras.

It is also a very standard knowledge the power of finite groups and their representations. Finite dimensional algebras and quantum groups about this title. Special emphasis in this class will be on complex finite dimensional hopf algebras. We study finite dimensional representations of quantum affine algebras using qcharacters. The other side of quantum groups are the purely algebraic versions, that is to say hopf algebras. Lecture notes introduction to representation theory. Langlands duality for finitedimensional representations of quantum affine algebras edward frenkel1 and david hernandez2 to v. Irreducible representations of quantum affine algebras, doctoral thesis in mathematical sciences 2000. The corresponding braided vector spaces are not of diagonal type, admit a realization as yetterdrinfeld modules over finite abelian groups and are analogous to braidings over fields of characteristic zero whose nichols algebras have finite gelfandkirillov dimension described in arxiv.