Showing papers on "Hopf algebra published in 1991"

An introduction to tannaka duality and quantum groups

Abstract: The goal of this paper is to give an account of classical Tannaka duality [C⁄] in such a way as to be accessible to the general mathematical reader, and to provide a key for entry to more recent developments [⁄SR, DM⁄] and quantum groups [⁄D1⁄]. Expertise in neither representation theory nor category theory is assumed. Naively speaking, Tannaka duality theory is the study of the interplay which exists between a group and the category of its representations. The early duality theorems of Tannaka-Krein [Ta, Kr] concentrate on the problem of reconstructing a compact group from the collection of its representations. In the abelian case, this problem amounts to reconstructing the group from its character group, and is the content of the Pontrjagin duality theorem. A good exposition of this theory can be found in the book by Chevalley [C]. In these early developments, there was little or no use of categorical concepts, partly because they did not exist at the time. Moreover, the mathematical community was not yet familiar with category theory, and it was possible to avoid it [BtD]. To Grothendieck we owe the understanding that the process of Tannaka duality can be reversed. In his work to solve the Weil conjectures, he constructed the category of motives as the universal recipient of a Weil cohomology [Kl]. By using a fiber functor from his category of motives to vector spaces, he could construct a pro-algebraic group G. He also conjectured that the category of motives could be recaptured as the category of representations of G. This group is called the Grothendieck Galois group, since it is an extension of the Galois group of _Q⁄⁄/Q . The work spreading from these ideas can be found in [SR, DM]. For other aspects of this question, see [Cb]. An entirely different development came from mathematical physicists working on superselection principles in quantum field theory [DHR] where it was discovered that the superselection structure could be described in terms of a category whose objects are certain endomorphisms of the C*-algebra of local observables, and whose arrows are intertwining operators. Reversing the duality process, they succeeded in constructing a compact group whose representations can be identified with their superselection category [DR]. Another impulse to the development of Tannaka duality comes from the theory of quantum groups. These new mathematical objects were discovered by Jimbo [⁄J⁄] and Drinfel'd [D1] in connection with the work of L.D. Faddeev and his collaborators on the quantum inverse scattering method. V.V. Lyubashenko [Ly] initiated the use of Tannaka duality in the construction of quantum groups; also see K.-H. Ulbrich [U]. We should also mention S.L. Woronowicz [W] in the case of compact quantum groups. Recently, S. Majid [M3] has shown that one can use Tannaka-Krein duality for constructing the quasi-Hopf algebras introduced by Drinfel'd [D2] in connection with the solution of the KnizhnikZamolodchikov equation. The theory of angular momentum in Quantum Physics [BL1] might also provide some extra motivation for studying Tannaka duality. The Racah-Wigner algebra, the 9⁄⁄– ⁄⁄j and 3⁄⁄– ⁄⁄j symbols, and, the Racah and Wigner coefficients, all seem to be about the explicit description of the structures which exist on the category of representations of some

Quasi hope algebras, group cohomology and orbifold models

Abstract: We construct non trivial quasi Hopf algebras associated to any finite group G and any element of H 3 ( G , U (1)). We analyze in details the set of representations of these algebras and show that we recover the main interesting datas attached to particular orbifolds of Rational Conformal Field Theory or equivalently to the topological field theories studied by R. Dijkgraaf and E. Witten. This leads us to the construction of the R-matrix structure in non abelian RCFT orbifold models.

Representations, duals and quantum doubles of monoidal categories

Abstract: [For the entire collection see Zbl 0742.00067.]\par The Tanaka-Krein type equivalence between Hopf algebras and functored monoidal categories provides the heuristic strategy of this paper. The author introduces the notion of a double cross product of monoidal categories as a generalization of double cross product of Hopf algebras, and explains some of the motivation from physics (the representation theory for double quantum groups).\par The Hopf algebra constructions are formulated in terms of monoidal categories $\underline C$ and functors $\underline C\to\underline{\text{Vec}}$ (finite-dimensional vector spaces) and generalized by replacing $\underline{\text{Vec}}$ by another monoidal category $\underline V$. It is interesting to remark that the monoidal category $\underline C$ (functored over a category $\underline V$) has a ``Hopf algebra like structure'', $\underline V$ having the role of the ground field. If $\underline V$ is a quasitensor category, a coadjo!

Involutory hopf algebras and 3-manifold invariants

Abstract: We establish a 3-manifold invariant for each finite-dimensional, involutory Hopf algebra. If the Hopf algebra is a group algebra G, the invariant counts homomorphisms from the fundamental group of the manifold to G. The invariant can be viewed as a state model on a Heegaard diagram or a triangulation of the manifold. The computation of the invariant involves tensor products and contractions of the structure tensors of the algebra. We show that every formal expression involving these tensors corresponds to a unique 3-manifold modulo a well-understood equivalence. This raises the possibility of an algorithm which can determine whether two given 3-manifolds are homeomorphic.

Doubles of quasitriangular hopf algebras

Abstract: Let H be a finite dimensional Hopf algebra and D(H) the associated ”quantum double” quasitriangular Hopf algebra of Drin-feld. Previously we showed that as an example of a double cross product of Hopf algebras H H ∗op acting on each other. We now show that if H is itself quasitriangular then D(H) is a semidirect"biproduct"in the sense of Radford: There is an algebra and coalgebra B such that as an ordinary smash product and smash co-product by H. The result arises naturally in the context of representation theoretic considerations for double cross products.

Bicovariant differential calculus on quantum groups SU q (N) and SO q (N)

Abstract: Following Woronowicz's proposal the bicovariant differential calculus on the quantum groupsSU q (N) andSO q (N) is constructed. A systematic construction of bicovariant bimodules by using the $$\hat R_q $$ matrix is presented. The relation between the Hopf algebras generated by the linear functionals relating the left and right multiplication of these bicovariant bimodules, and theq-deformed universal enveloping algebras is given. Imposing the conditions of bicovariance and consistency with the quantum group structure the differential algebras and exterior derivatives are defined. As an application the Maurer-Cartan equations and theq-analogue of the structure constants are formulated.

A Hopf star-algebra of polynomials on the quantum SL(2, R) for a unitary R-matrix

Abstract: A Hopf *-algebra corresponding to a solution of the classical Yang-Baxter equation is introduced, providing a new quantum deformation of SL (2 ℝ).

Hochschild and cyclic homology of quantum groups

Abstract: For an arbitrary complex linear semisimple Lie groupG, we consider Hopf algebras of the deformations of the formal and algebraic functions onG. The Hochschild and cyclic homology of these Hopf algebras are computed when the value of the deformation parameter is generic.

Hopf algebras of symmetric functions and tensor products of symmetric group representations

Abstract: The Hopf algebra structure of the ring of symmetric functions is used to prove a new identity for the internal product, i.e., the operation corresponding to the tensor product of symmetric group representations. From this identity, or by similar techniques which can also involve the λ-ring structure, we derive easy proofs of most known results about this operation. Some of these results are generalized.

Quantum Groups and Homology of Local Systems

Abstract: In this paper we “quantize” results of [SV2], Part II We establish and study the connection between (co)homology of local systems introduced in [SV1], [SV2] and homology of nilpotent subalgebras of certain Hopf algebras very close to Drinfeld-Jimbo q-analogues of Kac-Moody algebras

On $\ast $-representations of the Hopf $\ast $-algebra associated with the quantum group $U_q(n)$

Abstract: The properties of *-representations of the Hopf *-algebra Pol(U q (n)) are investigated. We consider this Hopf *-algebra as a deformation of the algebra of polynomials on the group U(n). The algebraic structure of Pol(U q (n)) is studied in some more detail in order to build *-representations of Pol(U q n)))by means of a Verma module construction. The irreductible *-representations are classified. By use of these *-representations we can complete the Hopf *-algebra Pol(U q (n)) into a type I C*-algebra, which is a quantum group in the sense of Woronowicz

THE QUANTUM SUPERALGEBRA Bq(0|1) AND q-DEFORMED CREATION AND ANNIHILATION OPERATORS

Abstract: We show that the q-deformed creation and annihilation boson operators defined by Biedenharn and Mcfarlane generate the quantum superalgebra B(0|1), denoted also as ospq(1|2), and we give a generalization to q-deformed parabosons and fermions using the infinite dimensional representations of such Hopf algebra.

Elliptic Hopf Algebras

Abstract: An elliptic Hopf algebra is a connected graded cocommutative Hopf algebra that is finitely generated and nilpotent. If (A,m,k) is a local noetherian ring then Ext(A)(k; k) is elliptic if and only A is a complete intersection. Similarly, special conditions are imposed on a simply connected topological space X if H*(OMEGA-X; k) is elliptic. Elliptic Hopf algebras G have finite depth and we show that they are characterized among Hopf algebras of finite depth by any of the following three properties: (i) SIGMA-i less-than-or-equal-to n dim G(i) grows at most polynomially in n; (ii) G is left noetherian; (iii) G is nilpotent.

Solutions of the braid equation related to a Hopf algebra

Abstract: Two solutionsT andT′ of the braid equation acting onA ⊗A (whereA is a Hopf algebra) are described. IfA is a cocommutative, thenT=σ. IfA is commutative, thenT′=σ (σ denotes the flip: σ(a ⊗b) =b ⊗a for anya,b ∈A).

Orbifolds, Hopf algebras and the moonshine

Abstract: We describe the modular properties and fusion rules of holomorphic orbifold models by Hopf algebraic techniques, using the representation theory of the orbifold quantum group. We apply this theory to the construction of generalized Thompson series, and discuss its connections with Moonshine.

A two-parametric quantization of sl(2)

Abstract: A two-parametric solution of the constant Yang-Baxter equation is used for the construction of structures corresponding to quantized groups. The corresponding Hopf algebra is generated by five generators.

Representations of Quantum Groups

Abstract: The q - deformation U q (G) of the universal enveloping algebras U(G) of complex simple Lie algebras G arose in the study of the algebraic aspects of quantum integrable systems [Fa, KR1, KS, Sl, S2, S3]. (The definition of U q (G)is below in Section 1.) They provide a powerful tool for the solving of the quantum Yang-Baxter equations. For recent reviews we refer to [FaT, FRT1,2, J4, Ta, Vg]. The algebras U q (G) are called also quantum groups [Dl, D2] or quantum universal enveloping algebras [Re, KiR1]. In [S3] for(G)= sl(2,c) and in [Drl, J1, J2, Dr2] in general it was observed that the algebras U q (G) have the structure of a Hopf algebra. This brought additional mathematical interest in this new algebraic structure (see, e.g., [R1, Wo, R2, Ve, L1]). Recently, inspired by the Knizhnik-Zamolodchikov equations [KZ] Drinfeld has developed a theory of formal deformations and introduced a new notion of quasi-Hopf algebras [Dr3,4].

Hopf algebras and regular isotopy invariants for link diagrams

Abstract: In this paper, we shall consider the following method for obtaining regular isotopy invariants of link diagrams. Given any link diagram L , equip it with a Morse function h , so that the diagram consists entirely of crossings, maxima, minima and vertical arcs. Introduce 2-valent graphical vertices to separate the various segments of the diagram. Given a finite index set I , a state σ for L h is an assignation of one element of I to each graphical vertex. Each segment of the diagram now has a weight associated with it, given in terms of tensor coordinates indexed by the set I by the pictures and, for any state σ, [ L i |σ] denotes the product of the various weights. We then define 〈 L h 〉 to be the sum of [ L h |σ] over all possible states σ,

Quantum Harmonic Oscillator Algebra and Link Invariants

Abstract: The $q$--deformation $U_q (h_4)$ of the harmonic oscillator algebra is defined and proved to be a Ribbon Hopf algebra.Associated with this Hopf algebra we define an infinite dimensional braid group representation on the Hilbert space of the harmonic oscillator, and an extended Yang--Baxter system in the sense of Turaev. The corresponding link invariant is computed in some particular cases and coincides with the inverse of the Alexander--Conway polynomial. The $R$ matrix of $U_q (h_4)$ can be interpreted as defining a baxterization of the intertwiners for semicyclic representations of $SU(2)_q$ at $q=e^{2 \pi i/N}$ in the $N \rightarrow \infty$ limit.Finally we define new multicolored braid group representations and study their relation to the multivariable Alexander--Conway polynomial.

Représentations des groupes quantiques

Abstract: © Association des collaborateurs de Nicolas Bourbaki, 1990-1991, tous droits réservés. L’accès aux archives du séminaire Bourbaki (http://www.bourbaki. ens.fr/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Multi-Parameter Quantum Groups Related to Link-Diagrams*

Abstract: We apply the Faddeev-Reshetikhin-Taktajan method for the construction of Quantum Groups to the Yang-Baxter matrices which are related to the invariants of oriented links in Σ×[0,1], where Σ is a non-trivial 2-dimensional surface. We obtain multi-parameter ribbon Hopf algebras that differ in many respects from their one-parameter counterparts. Among the main differences we mention the existence of a non-central quantum determinant and the fact that the number of independent generators is higher than in the one-parameter case.

Positive Convolution Structures Associated with Quantum Groups

Abstract: Hypergroups originated as abstractions of convolution algebras of measures on locally compact groups, see for instance Jewett [9]. Gelfand pairs and orthogonal systems of special functions which (for certain parameter values) can be interpreted as spherical functions on Gelfand pairs, are good sources of commutative hypergroups.