Showing papers on "Hopf algebra published in 1995"

Lectures on quantum groups

Abstract: Introduction Gaussian binomial coefficients The quantized enveloping algebra $U_q(\mathfrak s \mathfrak {1}_2)$ Representations of $U_q(\mathfrak{sl}_2)$ Tensor products or: $U_q(\mathfrak{sl}_2)$ as a Hopf algebra The quantized enveloping algebra $U_q(\mathfrak g)$ Representations of $U_q(\mathfrak g)$ Examples of representations The center and bilinear forms $R$-matrices and $k_q[G]$ Braid group actions and PBW type basis Proof of proposition 8.28 Crystal bases I Crystal bases II Crystal bases III References Notations Index.

Algebras and Hopf Algebras in Braided Categories

Abstract: This is an introduction for algebraists to the theory of algebras and Hopf algebras in braided categories. Such objects generalise super-algebras and super-Hopf algebras, aswell as colour-Lie algebras. Basic facts about braided categories C are recalled, the modules and comodules of Hopf algebras in such categories are studied,the notion of `braided -commutative' or `braided-cocommutative' Hopf algebras (braided groups) is reviewed and a fully diagrammatic proof of the reconstruction theorem for a braided group aut(C) is given. The theory has important implications for the theory of quasitriangular Hopf algebras (quantum groups). It also includes important examples such as the degenerate Sklyanin algebra and the quantum plane.

Conformal Field Theory and Integrable Systems Associated to Elliptic Curves

Abstract: It has become clear over the years that quantum groups (i.e., quasitriangular Hopf algebras, see [D]) and their semiclassical counterpart, Poisson Lie groups, are an essential algebraic structure underlying three related subjects: integrable models of statistical mechanics, conformal field theory, and integrable models of quantum field theory in 1+1 dimensions. Still, some points remain obscure from the point of view of Hopf algebras. In particular, integrable models associated with elliptic curves are still poorly understood. We propose her an elliptic version of quantum groups, based on the relation to conformal field theory, which hopefully will be helpful to complete the picture.

Invariants of $3$-manifolds and projective representations of mapping class groups via quantum groups at roots of unity

Abstract: An example of a finite dimensional factorizable ribbon Hopf ℂ-algebra is given by a quotientH=u q (g) of the quantized universal enveloping algebraU q (g) at a root of unityq of odd degree. The mapping class groupM g,1 of a surface of genusg with one hole projectively acts by automorphisms in theH-moduleH *⊗g , ifH * is endowed with the coadjointH-module structure. There exists a projective representation of the mapping class groupM g,n of a surface of genusg withn holes labeled by finite dimensionalH-modulesX 1, ...,X n in the vector space Hom H (X 1 ⊗ ... ⊗X n ,H *⊗g ). An invariant of closed oriented 3-manifolds is constructed. Modifications of these constructions for a class of ribbon Hopf algebras satisfying weaker conditions than factorizability (including most ofu q (g) at roots of unityq of even degree) are described.

Semisimple Hopf algebras of dimension 6, 8

Abstract: We determine the isomorphic classes of 6 or 8 dimensional semisimple Hopf algebrasA over an algebraically closed field such that (dimA)1≠0.

Invariants of 3-manifolds derived from finite dimensional hopf algebras

Abstract: This paper studies invariants of 3-manifolds derived from certain finite dimensional Hopf algebras via regular isotopy invariants of unoriented links in the blackboard framing. The invariants are based on right integrals for these Hopf algebras. It is shown that the resulting class of invariants is definitely distinct from the class of Witten-Reshetikhin-Turaev invariants. The invariant associated with the quantum double of a finite group G is treated in this context, and is shown to count the number of homomorphisms of the fundamental group of the 3-manifold to the given finite group G.

Groupes quantiques et algèbres de battage quantiques

Abstract: Soit U q + la sous-algebre de Hopf «triangulaire superieure» de l'algebre enveloppante quantifiee associee a une matrice de Cartan generalisee symetrisable. On montre que U q + est isomorphe a la sous-algebre de Hopfengendree par les elements de degre 0 et 1 de l'algebre de Hopf cotensorielle associee a un bimodule de Hopf convenable sur l'algebre du groupe Z n . On donne un resultat de classification sur les algebres de Hopf que l'on peut obtenir par cette construction et qui satisfont a une hypothese de croissance-raisonable

Semisimple hopf algebras of dimension 2p

Abstract: We determine the isomorphic classes of 6 or 8 dimensional semisimple Hopf algebrasA over an algebraically closed field such that (dimA)1≠0.

Indecomposable coalgebras, simple comodules, and pointed Hopf algebras

Abstract: We prove that every coalgebra C is a direct sum of coalgebras in such a way that the summands correspond to the connected components of the Ext quiver of the simple comodules of C. This result is used to prove that every pointed Hopf algebra is a crossed product of a group over the indecomposable component of the identity element.

Mappin class group actions on quantum doubles

Abstract: We study representations of the mapping class group of the punctured torus on the double of a finite dimensional possibly non-semisimple Hopf algebra that arise in the construction of universal, extended topological field theories. We discuss how for doubles the degeneracy problem of TQFT's is circumvented. We find compact formulae for theS±1-matrices using the canonical, non-degenerate forms of Hopf algebras and the bicrossed structure of doubles rather than monodromy matrices. A rigorous proof of the modular relations and the computation of the projective phases is supplied using Radford's relations between the canonical forms and the moduli of integrals. We analyze the projectiveSL(2, Z)-action on the center ofUq(sl2) forq anl=2m+1st root of unity. It appears that the 3m+1-dimensional representation decomposes into anm+1-dimensional finite representation and a2m-dimensional, irreducible representation. The latter is the tensor product of the two dimensional, standard representation ofSL(2, Z) and the finite,m-dimensional representation, obtained from the truncated TQFT of the semisimplified representation category ofUq(sl2).

Lie bialgebra contractions and quantum deformations of quasi-orthogonal algebras

Abstract: Lie bialgebra contractions are introduced and classified. A non‐degenerate coboundary bialgebra structure is implemented into all pseudo‐orthogonal real algebras so(p,q) starting from the one corresponding to so(N+1). It allows us to introduce a set of Lie bialgebra contractions which leads to Lie bialgebras of quasi‐orthogonal algebras. This construction is explicitly given for N=2,3,4. All Lie bialgebra contractions studied in this paper define Hopf algebra contractions for the Drinfel’d–Jimbo deformations Uzso(p,q). They are explicitly used to generate new non‐semisimple quantum algebras as it is the case for the Euclidean, Poincare, and Galilean algebras.

On crossed products of Hopf algebras

Abstract: Let B = A#UH denote a crossed product of the associative algebra A with the Hopf algebra H. We investigate the weak dimension and the global dimension of B and show that wdim B < wdim H + wdim A and l.gldim B < r.gldim H + l.gldim A.

Noncommutative Geometry of Finite Groups

Abstract: A finite set can be supplied with a group structure which can then be used to select (classes of) differential calculi on it via the notions of left-, right- and bicovariance. A corresponding framework has been developed by Woronowicz, more generally for Hopf algebras including quantum groups. A differential calculus is regarded as the most basic structure needed for the introduction of further geometric notions like linear connections and, moreover, for the formulation of field theories and dynamics on finite sets. Associated with each bicovariant first order differential calculus on a finite group is a braid operator which plays an important role for the construction of distinguished geometric structures. For a covariant calculus, there are notions of invariance for linear connections and tensors. All these concepts are explored for finite groups and illustrated with examples. Some results are formulated more generally for arbitrary associative (Hopf) algebras. In particular, the problem of extension of a connection on a bimodule (over an associative algebra) to tensor products is investigated, leading to the class of `extensible connections'. It is shown that invariance properties of an extensible connection on a bimodule over a Hopf algebra are carried over to the extension. Furthermore, an invariance property of a connection is also shared by a `dual connection' which exists on the dual bimodule (as defined in this work).

Burnside's theorem for hopf algebras

Abstract: A classical theorem of Burnside asserts that if X is a faithful complex character for the finite group G, then every irreducible character of G is a constituent of some power Xn of X . Fifty years after this appeared, Steinberg generalized it to a result on semigroup algebras K[G] with K an arbitrary field and with G a semigroup, finite or infinite. Five years later, Rieffel showed that the theorem really concerns bialgebras and Hopf algebras. In this note, we simplify and amplify the latter work. Let K be a field and let A be a K-algebra. A map A: A -+ A A is said to be a comultiplication on A if A is a coassociative K-algebra homomorphism. For convenience, we call such a pair (A, A) a b-algebra. Admittedly, this is rather nonstandard notation. One is usually concerned with bialgebras, that is, algebras which are endowed with both a comultiplication A and a counit e: A -K. However, semigroup algebras are not bialgebras in general, and the counit rarely comes into play here. Thus it is useful to have a name for this simpler object. Now a b-algebra homomorphism 6: A -k B is an algebra homomorphism which is compatible with the corresponding comultiplications, and the kernel of such a homomorphism is called a b-ideal. It is easy to see that I is a b-ideal of A if and only if I ' A with A(I) C I A+A +A I. Of course, the b-algebra structure can be used to define the tensor product of A-modules. Specifically, if V and W are left A-modules, then A acts on V X W via a(vow) =A(a)(v?w) for all a e A, v E V, we W. Notice that if I is a b-ideal of A, then the set of all A-modules V with annA V D I is closed under tensor product. Conversely, we have Proposition 1. Let A be a b-algebra and let Y be a family of A-modules closed under tensor product. Then I= n annA V VEY

Quasitriangularity of quantum groups at roots of $1$

Abstract: An important property of a Hopf algebra is its quasitriangularity and it is useful for various applications. This property is investigated for quantum groupssl2 at roots of 1. It is shown that different forms of the quantum groupsl2 at roots of 1 are either quasitriangular or have similar structure which will be called braiding. In the most interesting cases this property means that “braiding automorphism” is a combination of some Poisson transformation and an adjoint transformation with a certain element of the tensor square of the algebra.

The Euclidean Hopf algebra U q (e N ) and its fundamental Hilbert-space representations

Abstract: The Euclidean Hopf algebra Uq(eN) dual of Fun(RNq■SOq−1(N)) is constructed by realizing it as a subalgebra of the differential algebra Diff(RNq) on the quantum Euclidean space RNq; in fact, the previous realization [G. Fiore, Commun. Math. Phys. 169, 475–500 (1995)] of Uq−1(so(N)) is extended within Diff(RNq) through the introduction of q derivatives as generators of q translations. The fundamental Hilbert‐space representations of Uq(eN) turn out to be of highest weight type and rather simple ‘‘lattice‐regularized’’ versions of the classical ones. The vectors of a basis of the singlet (i.e., zero‐spin) irrep can be realized as normalizable functions on RNq, going to distributions in the limit q → 1.

Relativistic and Newtonian κ ‐space–times

Abstract: The deformations of the Galilei algebra and their associated noncommutative Newtonian space–times are investigated. This is done by analyzing the possible nonrelativistic limits of an eleven generator (pseudo)extended κ‐Poincare algebra Pκ and their implications for the existence of a first‐order differential calculus. The additional one‐form needed to achieve a consistent calculus on κ‐Minkowski space is shown to be related to the additional central generator entering in the Pκ Hopf algebra. In the process, deformations of the extended Galilei and Galilei algebras are introduced which have, respectively, a cocycle and a bicrossproduct structure.

Invariants of 3-manifolds derived from universal invariants of framed links

Abstract: Reshetikhin and Turaev [ 10 ] gave a method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra (e.g. a quantum group U q ( sl 2 )) using finite-dimensional representations of it. In this paper we give another independent method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra via universal invariants of framed links without using representations of the algebra. For U q ( sl 2 ) these two methods give different invariants of 3-manifolds.

Relativistic and Newtonian kappa spacetimes

Abstract: The deformations of the Galilei algebra and their associated noncommutative Newtonian spacetimes are investigated. This is done by analyzing the possible nonrelativistic limits of an eleven generator (pseudo)extended \kap-Poincar\'e algebra $\tilde{\cal P}_\kappa$ and their implications for the existence of a first order differential calculus. The additional one-form needed to achieve a consistent calculus on \kap-Minkowski space is shown to be related to the additional central generator entering in the $\tilde{\cal P}_\kappa$ Hopf algebra. In the process, deformations of the extended Galilei and Galilei algebras are introduced which have, respectively, a cocycle and a bicrossproduct structure.

Non-standard quantum (1+1) Poincar\'e group: a $T$--matrix approach

Abstract: The Hopf algebra dual form for the non--standard uniparametric deformation of the (1+1) Poincare algebra $iso(1,1)$ is deduced. In this framework, the quantum coordinates that generate $Fun_w(ISO(1,1))$ define an infinite dimensional Lie algebra. A change in the basis of the dual form is obtained in order to compare this deformation to the standard one. Finally, a non--standard quantum Heisenberg group acting on a quantum Galilean plane is obtained.

Quasi-{$*$} Structure on {$q$}-Poincare algebras

Abstract: We use braided groups to introduce a theory of $*$-structures on general inhomogeneous quantum groups, which we formulate as {\em quasi-$*$} Hopf algebras. This allows the construction of the tensor product of unitary representations up to a quantum cocycle isomorphism, which is a novel feature of the inhomogeneous case. Examples include $q$-Poincar\'e quantum group enveloping algebras in $R$-matrix form appropriate to the previous $q$-Euclidean and $q$-Minkowski spacetime algebras $R_{21}x_1x_2=x_2x_1R$ and $R_{21}u_1Ru_2=u_2R_{21}u_1R$. We obtain unitarity of the fundamental differential representations. We show further that the Euclidean and Minkowski Poincar\'e quantum groups are twisting equivalent by a another quantum cocycle.

Induction Functors for the Doi-Koppinen Unified Hopf Modules

Abstract: Let (A, B, D), (A′, B′, D′) be two triples consisting of a Hopf algebra A, an A-comodule algebra B and an A-module coalgebra D. Given α A → A′, β B → B′ and δ D → D′, we define an induction functor between the two corresponding categories of Doi-Koppinen Hopf smodules, and we prove that this functor has a right adjoint; this right adjoint is constructed using the cotensor product. We then investigate when this induction functor and its adjoint are inverse equivalences. We find a necessary and sufficient condition, which turns out to be of Galois-type in some special cases. To be able to prove our result, we have to introduce Doi-Koppinen Hopf bimodules and the bitensor product.