Showing papers on "Hopf algebra published in 1990"

Hall algebras and quantum groups

Abstract: The Hall algebra of a finitary category encodes its extension structure. The story starts from the work of Steinitz on the module category of an abelian p-group, where the Hall algebra is the algebra of symmetric functions. The theory of Hall algebras is highlighted by Ringel around the 90’s in his seminal work realizing a half of a quantum group via the Hall algebra of quiver representations. Further developments include Lusztig’s canonical bases, cluster categories (Caldero-Keller), higher genus quantum algebras (Burban-Schiffmann, Schiffmann-Vasserot) – just to name a few. One of the goals of the seminar is to introduce Bridgeland’s construction of the quantized enveloping algebras (quantum groups) associated to a symmetric Kac-Moody Lie algebra via Hall algebras of Z/2-graded complexes of quiver representations and several recent progresses around it. In the last part of this seminar, we try to open the window to some applications of Hall algebras to mathematical physics via Hall algebra of curves: for example, they play an important rôle in the proof of AGT conjecture concerning pure N = 2 gauge theory for the group SU(r) ([SV]).

Multiparameter quantum groups and twisted quasitriangular Hopf algebras

Abstract: On montre comment on peut obtenir des groupes quantiques multiparametres a partir des algebres de Hopf torsadees

Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra

Abstract: 0.1. An important role in the theory of modular representations is played by certain finite dimensional Hopf algebras u over Fp (the field with p elements, p = prime). Originally, u was defined (Curtis [3]) as the restricted enveloping algebra of a "simple" Lie algebra over Fp For our purposes, it will be more convenient to define u as follows. Let us fix an indecomposable positive definite symmetric Cartan matrix

Quasitriangular Hopf Algebras and {Yang-Baxter} Equations

Abstract: This is an informal introduction to the theory of quasitriangular Hopf algebras and its connections with physics. Basic properties and applications of Hopf algebras and Yang-Baxter equations are reviewed, with the quantum group Uq(sl2) as a frequent example. The development builds up to the representation theory of quasitriangular Hopf algebras. Much of the abstract representation theory is new, including a formula for the rank of a representation.

Principal homogeneous spaces for arbitrary Hopf algebras

Abstract: LetH be a Hopf algebra over a field with bijective antipode,A a rightH-comodule algebra,B the subalgebra ofH-coinvariant elements and can:A ⊗ B A →A ⊗H the canonical map. ThenA is a faithfully flat (as left or rightB-module) Hopf Galois extension iffA is coflat asH-comodule and can is surjective (Theorem I). This generalizes results on affine quotients of affine schemes by Oberst and Cline, Parshall and Scott to the case of non-commutative algebras. The dual of Theorem I holds and generalizes results of Gabriel on quotients of formal schemes to the case of non-cocommutative coalgebras (Theorem II). Furthermore, in the dual situation, a normal basis theorem is proved (Theorem III) generalizing results of Oberst-Schneider, Radford and Takeuchi.

Central extensions of quantum current groups

Abstract: We describe Hopf algebras which are central extensions of quantum current groups. For a special value of the central charge, we describe Casimir elements in these algebras. New types of generators for quantum current algebra and its central extension for quantum simple Lie groups, are obtained. The application of our construction to the elliptic case is also discussed.

Quantum groups and representations of monoidal categories

Abstract: This paper is intended to make explicit some aspects of the interactions which have recently come to light between the theory of classical knots and links, the theory of monoidal categories, Hopf-algebra theory, quantum integrable systems, the theory of exactly solvable models in statistical mechanics, and quantum field theories. The main results herein show an intimate relation between representations of certain monoidal categories arising from the study of new knot invariants or from physical considerations and quantum groups (that is, Hopf algebras). In particular categories of modules and comodules over Hopf algebras would seem to be much more fundamental examples of monoidal categories than might at first be apparent. This fundamental role of Hopf algebras in monoidal categories theory is also manifest in the Tannaka duality theory of Deligne and Mime [8a], although the relationship of that result and the present work is less clear than might be hoped.

Bialgebra cohomology, deformations, and quantum groups

Abstract: We introduce cohomology and deformation theories for a bialgebra A (over a commutative unital ring k) such that the second cohomology group is the space of infinitesimal deformations. Our theory gives a natural identification between the underlying k-modules of the original and the deformed bialgebra. Certain explicit deformation formulas are given for the construction of quantum groups--i.e., Hopf algebras that are neither commutative nor cocommutative (whether or not they arise from quantum Yang-Baxter operators). These formulas yield, in particular, all GLq(n) and SLq(n) as deformations of GL(n) and SL(n). Using a Hodge decomposition of the underlying cochain complex, we compute our cohomology for GL(n). With this, we show that every deformation of GL(n) is equivalent to one in which the comultiplication is unchanged, not merely on elements of degree one but on all elements (settling in the strongest way a decade-old conjecture) and in which the quantum determinant, as an element of the underlying k-module, is identical with the usual one.

Representation theory of Hopf galois extensions

Abstract: LetH be a Hopf algebra over the fieldk andB ⊂A a right faithfully flat rightH-Galois extension. The aim of this paper is to study some questions of representation theory connected with the ring extensionB ⊂A, such as induction and restriction of simple or indecomposable modules. In particular, generalizations are given of classical results of Clifford, Green and Blattner on representations of groups and Lie algebras. The stabilizer of a leftB-module is introduced as a subcoalgebra ofH. Very often the stabilizer is a Hopf subalgebra. The special case whenA is a finite dimensional cocommutative Hopf algebra over an algebraically closed field,B is a normal Hopf subalgebra andH is the quotient Hopf algebra was studied before by Voigt using the language of finite group schemes.

More examples of bicrossproduct and double cross product Hopf algebras

Abstract: We construct examples of bicrossproducts and double cross products of quantum groupsa(R) associated to general matrix solutionsR of the Quantum Yang-Baxter Equations. We also describe iterated double cross products of quantum groups. In the course of constructinga(R) we are led to introduce a suitable notion of mutually dual Hopf algebras and a dual quantum groupŬ(R).

Orthogonal polynomials in connection with quantum groups

Abstract: This is a survey of interpretations of q-hypergeometric orthogonal polynomials on quantum groups. The first half of the paper gives general background on Hopf algebras and quantum groups. The emphasis in the rest of the paper is on the SU(2) quantum group. An interpretation of little q-Jacobi polynomials as matrix elements of its irreducible representations is presented. In the last two sections new results by the author on interpretations of Askey-Wilson polynomials are discussed.

On hopf algebras and rigid monoidal categories

Abstract: LetC be a neutral Tannakian category over a fieldk. By a theorem of Saavedra Rivano there exists a commutative Hopf algebraA overk such thatC is equivalent to the category of finite dimensional rightA-comodules. We review Saavedra Rivano’s construction of the bialgebraA and show thatA has still an antipode if the symmetry condition on the monoidal structure ofC is removed.

Extremal projectors for quantized kac-moody superalgebras and some of their applications

Abstract: A modification of one of the defining relations of quantized Kac-Moody algebras, introduced by Drinfeld and Jimbo, is given. This modification allows one to extend the concept of quantized Kac-Moody algebras to the case of Kac-Moody superalgebras. A q-analogue of the Cartan-Weyl basis is introduced, which has properties similar to the Cartan-Weyl basis of the Kac-Moody (super) algebras. Explicit expressions of the extremal projectors for all quantized Kac-Moody (super) algebras of finite growth are written down. A complete derivation of the Gel'fand-Tsetlin formulae for the quantized gl(n,C)is given.

On Clebsch–Gordan coefficients and matrix elements of representations of the quantum algebra Uq(su2)

Abstract: Clebsch–Gordan coefficients and matrix elements of irreducible representations of the quantum algebra Uq(su2) were considered in several papers. In particular, a few expressions for them were derived. An approach to Clebsch–Gordan coefficients and to matrix elements of representations of Uq(su2) on the base of the theory of basic hypergeometric functions is given. This approach allows one to obtain q‐analogs of all well‐known classical expressions for Clebsch–Gordan coefficients (most of them were absent). New symmetry relations, generating functions, and recurrence formulas for Clebsch–Gordan coefficients of Uq(su2) are obtained. Unlike other papers, Clebsch–Gordan coefficients and matrix elements are considered on the base of minimal theoretical constructions (in fact, without using the notion of a C* algebra and of a Hopf algebra).

Hopf-algebraic structure of combinatorial objects and differential operators

Abstract: A Hopf-algebraic structure on a vector space which has as basis a family of trees is described. Some applications of this structure to combinatorics and to differential operators are surveyed. Some possible future directions for this work are indicated.

The Group of Automorphisms of a Semisimple Hopf Algebra Over a Field of Characteristic 0 is Finite

Abstract: Introduction. Suppose that A is a semisimple Hopf algebra over a field of characteristic 0, or that A is a semisimple cosemisimple involutory Hopf algebra over a field k of characteristic p > dim A. In this paper we prove that the group AutHopf(A) of Hopf algebra automorphisms of A is finite. We show that the semigroup EndHOpf(A) of Hopf algebra endomorphisms of A is also finite as a corollary. Generally the group of Hopf algebra automorphisms of a finite-dimensional Hopf algebra need not be finite. For any positive integer n and any field k, we construct a family of finite-dimensional Hopf algebras over k with automorphism group GLn(k). The fact that AutHopf(A) is finite is a consequence of a theorem of

Representation-theoretic rank and double hopf algebras

Abstract: We compute the intrinsic category-theoretic rank: for quasitriangular Hopf algebras in the case of the quantum double Hopf algebra of Drinfeld. The result is closely related ti recent Hopf algebra invariants of Larson and Radford.

The algebraic structure of linearly recursive sequences under hadamard product

Abstract: We describe the algebraic structure of linearly recursive sequences under the Hadamard (point-wise) product. We characterize the invertible elements and the zero divisors. Our methods use the Hopf-algebraic structure of this algebra and classical results on Hopf algebras. We show that our criterion for invertibility is effective if one knows a linearly recursive relation for a sequence and certain information about finitely-generated subgroups of the multiplicitive group of the field.

SKEW DERIVATIONS AND Uq(sl(2))

Abstract: This note first describes the basic properties of the skew derivations on the polynomial ring k[X]. As a consequence of these properties it is proved that the q-analogue of the enveloping algebra ofsl(2), Uq(sl(2)), has a unique action on C[X], where "action" means that C[X] is a module algebra in the Hopf algebra sense. This depends on the fact that the generators of a subalgebra of Uq(sl(2)) described by Woronowicz must act via an automorphism, and the skew derivations associated to it.

Skew derivations andU q (sl(2))

Abstract: This note first describes the basic properties of the skew derivations on the polynomial ringk[X]. As a consequence of these properties it is proved that theq-analogue of the enveloping algebra of sl(2),U q(sl(2)), has a unique action on C[X], where “action” means that C[X] is a module algebra in the Hopf algebra sense. This depends on the fact that the generators of a subalgebra ofU q(sl(2)) described by Woronowicz must act via an automorphism, and the skew derivations associated to it.

Irreducible actions and faithful actions of hopf algebras

Abstract: LetH be a Hopf algebra acting on an algebraA. We will examine the relationship betweenA, the ring of invariantsA H, and the smash productA # H. We begin by studying the situation whereA is an irreducibleA # H module and, as an application of our first main theorem, show that ifD is a division ring then [D : D H]≦dimH. We next show that prime rings with central rings of invariants satisfy a polynomial identity under the action of certain Hopf algebras. Finally, we show that the primeness ofA # H is strongly related to the faithfulness of the left and right actions ofA # H onA.

Forms of Hopf algebras and Galois theory

Abstract: The theory of Hopf algebras is closely connected with various applications, in particular to algebraic and formal groups. Although the rst occurence of Hopf algebras was in algebraic topology, they are now found in areas as remote as combinatorics and analysis. Their structure has been studied in great detail and many of their properties are well understood. We are interested in a systematic treatment of Hopf algebras with the techniques of forms and descent. The rst three paragraphs of this paper give a survey of the present state of the theory of forms of Hopf algebras and of Hopf Galois theory especially for separable extensions. It includes many illustrating examples some of which cannot be found in detail in the literature. The last two paragraphs are devoted to some new or partial results on the same eld. There we formulate some of the open questions which should be interesting objects for further study. We assume throughout most of the paper that k is a base eld and do not touch upon the recent beautiful results of Hopf Galois theory for rings of integers in algebraic number elds as developed in [C1].

Quantum groups, filtered rings and Gelfand-Kirillov dimension

Abstract: It is shown here that, for a number of quantum groups, there exists a finite dimensional ‘filtration’ for which the associated graded algebra has a simple form. It follows from this that Gelfand-Kirillov dimension behaves particularly well for these algebras.

Quantum groups as symmetries of chiral conformal algebras

Abstract: Work in progress with D. Buchholz, I. Frenkel and G. Mack and with L. Hadjiivanov and R. Paunov is previewed. A quantum internal symmetry of a minimal chiral conformal model is introduced in such a way that the overall (product) representation of the braid group is trivialized. As an introduction we review (in Secs.1 and 2) basic facts about 2-dimensional conformal QFT and about the quantum enveloping algebra Uq - Uq(sl(2)) and its finite dimensional representations (for qp = -1).

Spectra of smash products

Abstract: LetT=R #H be a smash product whereH is a finite dimensional Hopf algebra. We show that ideals ofT invariant under the dualH* ofH are extended fromH-invariant ideals ofR. This allows us to transport the study of ideals inT to invariant ideals. When the Hopf algebra is pointed the relationship between an ideal and its invariant ideal is shown to be manageable. Restricting to prime ideals, this yields results on the prime spectra ofR andT. We obtain Krull relations forR ⊆T for someH, including Incomparability wheneverH is commutative (or more generally whenH* is pointed after base extension). The results generalize and unify a number of results known in the context of group and restricted Lie actions.

Computing the brauer-long group of a hopf algebra I: The cohomological theory

Abstract: LetH be a commutative, cocommutative and faithfully projective Hopf algebra over a commutative ringR. Using cohomological methods, we obtain a description of a subgroup of Long’s Brauer group ofH-dimodule algebras:Ψ where the mapψ is induced by the smash product, and where BDs (R, H) is the subgroup of the Brauer-Long group consisting of all elements which are split by a faithfully flat extension ofR. As an example, the Brauer-Long group of a free Hopf algebra of rank 2 is computed. The results are also applied to Orzech’s subgroup of the Brauer-Long group.