Showing papers on "Hopf algebra published in 1993"

Hopf algebras and their actions on rings

Abstract: Definitions and examples Integrals and semisimplicity Freeness over subalgebras Action of finite-dimensional Hopf algebras and smash products Coradicals and filtrations Inner actions Crossed products Galois extensions Duality New constructions from quantum groups Some quantum groups.

Spherical Categories

Abstract: This paper is a study of monoidal categories with duals where the tensor product need not be commutative The motivating examples are categories of representations of Hopf algebras and the motivating application is the definition of 6j-symbols as used in topological field theories We introduce the new notion of a spherical category In the first section we prove a coherence theorem for a monoidal category with duals following MacLane (1963) In the second section we give the definition of a spherical category, and construct a natural quotient which is also spherical In the third section we define spherical Hopf algebras so that the category of representations is spherical Examples of spherical Hopf algebras are involutory Hopf algebras and ribbon Hopf algebras Finally we study the natural quotient in these cases and show it is semisimple

Quantum groups, quantum categories, and quantum field theory

Abstract: and survey of results.- Local quantum theory with braid group statistics.- Superselection sectors and the structure of fusion rule algebras.- Hopf algebras and quantum groups at roots of unity.- Representation theory of U q red (s? 2).- Path representations of the braid groups for quantum groups at roots of unity.- Duality theory for local quantum theories, dimensions and balancing in quantum categories.- The quantum categories with a generator of dimension less than two.

Poisson Lie Groups, Quantum Duality Principle, and the Quantum Double

Abstract: The Heisenberg double of a Hopf algebra may be regarded as a quantum analogue of the cotangent bundle of a Lie group. Quantum duality principle describes relations between a Hopf algebra, its dual, and their Heisenberg double in a way which extends both the theory of coadjoint orbits and the classical Fourier transform. We also describe the twisted Heisenberg double which is relevant for the study of nontrivial deformations of the quantized universal enveloping algebras.

Transmutation theory and rank for quantum braided groups

Abstract: Let f: H1 → H2be any pair of quasitriangular Hopf algebras over k with a Hopf algebra map f between them. We construct in this situation a quasitriangular Hopf algebra B(H1, f, H2) in the braided monoidal category of H1-modules. It consists in the same algebra as H2 with a modified comultiplication and has a quasitriangular structure given by the ratio of those of H1 and H2. This transmutation procedure trades a non-cocommutative Hopf algebra in the category of k-modules for a more cocommutative object in a more non-commutative category. As an application, every Hopf algebra containing the group algebra of ℤ2 becomes transmuted to a super-Hopf algebra.

Remarks on bicovariant differential calculi and exterior Hopf algebras

Abstract: We show that every bicovariant differential calculus over the quantum groupA defines a bialgebra structure on its exterior algebra. Conversely, every exterior bialgebra ofA defines bicovariant bimodule overA. We also study a quasitriangular structure on exterior Hopf algebras in some detail.

Bicovariant quantum algebras and quantum Lie algebras

Abstract: A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from Fun $$(\mathfrak{G}_q )$$ toU q g, given by elements of the pure braid group. These operators—the “reflection matrix”Y≡L + SL − being a special case—generate algebras that linearly close under adjoint actions, i.e. they form generalized Lie algebras. We establish the connection between the Hopf algebra formulation of the calculus and a formulation in compact matrix form which is quite powerful for actual computations and as applications we find the quantum determinant and an orthogonality relation forY inSO q (N).

Quantum symmetry and braid group statistics in $G$-spin models

Abstract: In two-dimensional lattice spin systems in which the spins take values in a finite groupG we find a non-Abelian “parafermion” field of the formorder x disorder that carries an action of the Hopf algebra Open image in new window , the double ofG. This field leads to a “quantization” of the Cuntz algebra and allows one to define amplifying homomorphisms on the Open image in new window subalgebra that create the Open image in new window and generalize the endomorphisms in the Doplicher-Haag-Roberts program. The so-obtained category of representations of the observable algebra is shown to be equivalent to the representation category of Open image in new window . The representation of the braid group generated by the statistics operator and the corresponding statistics parameter are calculated in each sector.

The dual of a quantum group

Abstract: It is shown that the bialgebra (two dimensional pseudo-group) of Woronowicz, with some mild technical conditions, can be embedded into the enveloping algebra of a solvable Lie algebra, with the usual Lie structure and a deformed coproduct. The bialgebra dual of this bialgebra is calculated and found to coincide with U q,q' (sl2) after fixing the center. The (associative) bialgebra dual form is calculated explicitly and found to be a product ofq-exponentials. Implications about quantum transfer matrices are discussed.

Gauss codes, quantum groups and ribbon hopf algebras

Abstract: By relating the diagrammatic foundations of knot theory with the structure of abstract tensors, quantum groups and ribbon Hopf algebras, specific expressions are derived for quantum link invariants. These expressions, when applied to the case of finite dimensional unimodular ribbon Hopf algebras, give rise to invariants of 3-manifolds.

Hopf Algebras of Combinatorial Structures

Abstract: A generalization of the definition of combinatorial species is given by considering functors whose domains are categories of finite sets, with various classes of relations as moronisms. Two cases in particular correspond to species for which one has notions of restriction and quotient of structures. Coalgebras and/or Hopf algebras can be associated to such species, the duals of which provide an algebraic framework for studying invariants of structures.

Colored ribbon hopf algebras and universal invariants of framed links

Abstract: For colored representations of Uq(sl2), we give explicit formulas of universal R-matrices and construct universal invariants of framed links.

Structure of Topological Lattice Field Theories in Three Dimensions

Abstract: We construct and classify topological lattice field theories in three dimensions. After defining a general class of local lattice field theories, we impose invariance under arbitrary topology-preserving deformations of the underlying lattice, which are generated by two new local lattice moves. Invariant solutions are in one--to--one correspondence with Hopf algebras satisfying a certain constraint. As an example, we study in detail the topological lattice field theory corresponding to the Hopf algebra based on the group ring $\C[G]$, and show that it is equivalent to lattice gauge theory at zero coupling, and to the Ponzano--Regge theory for $G=$SU(2).

Cross Product Quantisation, Nonabelian Cohomology And Twisting Of Hopf Algebras

Abstract: This is an introduction to work on the generalisation to quantum groups of Mackey's approach to quantisation on homogeneous spaces. We recall the bicrossproduct models of the author, which generalise the quantum double. We describe the general extension theory of Hopf algebras and the nonAbelian cohomology spaces $\CH^2(H,A)$ which classify them. They form a new kind of topological quantum number in physics which is visible only in the quantum world. These same cross product quantisations can also be viewed as trivial quantum principal bundles in quantum group gauge theory. We also relate this nonAbelian cohomology $\CH^2(H,\C )$ to Drinfeld's theory of twisting.

Quantum double construction for graded Hopf algebras

Abstract: A detailed proof of the quantum double construction is given for Zj -graded Hopf algebras, and an explicit formula for the graded universal i2-matrix is obtained in a general fashion.

Duality for multiparametric quantum GL(n)

Abstract: We show that the Hopf algebra Uuq dual to the multiparameter matrix quantum group GLuq(n) may be realized a la Sudbery(1990), i.e. tangent vectors at the identity. Furthermore, we give the Cartan-Weyl basis of Uuq and show that this is consistent with the duality. We show that as a commutation algebra Uuq equivalent to Uu(sl(n, C)) (X) Uu(Z), where Z is one-dimensional and Uu(Z) is a central algebra in Uuq. However, as a co-algebra Uuq cannot be split in this way and depends on all parameters.

A closed expression for the universal R-matrix in a non-standard quantum double

Abstract: In recent papers by the author, a method was developed for constructing quasitriangular Hopf algebras (quantum groups) of the quantum-double type. As a by-product, a novel non-standard example of the quantum double has been found. In this letter, a closed expression (in terms of elementary functions) for the corresponding universal ℛ-matrix is obtained. In reduced form, when the number of generators becomes two instead of four, this quantum group can be interpreted as a deformation of the Lie algebra [x, h]=2h in the context of Drinfeld’s quantization program.

Quantum supergroups of $GL(n|m)$ type: differential forms, Koszul complexes and Berezinians

Abstract: We introduce and study the Koszul complex for a Hecke $R$-matrix. Its cohomologies, called the Berezinian, are used to define quantum superdeterminant for a Hecke $R$-matrix. Their behaviour with respect to Hecke sum of $R$-matrices is studied. Given a Hecke $R$-matrix in $n$-dimensional vector space, we construct a Hecke $R$-matrix in $2n$-dimensional vector space commuting with a differential. The notion of a quantum differential supergroup is derived. Its algebra of functions is a differential coquasitriangular Hopf algebra, having the usual algebra of differential forms as a quotient. Examples of superdeterminants related to these algebras are calculated. Several remarks about Woronowicz's theory are made.

Some aspects of the theory of quantum groups

Abstract: CONTENTSIntroductionChapter I. Basic constructions § 1. Definition of a Hopf algebra § 2. Two constructions of quantum semigroups § 3. Universal coacting and -matrix algebras § 4. The quantum determinant and antipode § 5. The dimension of quantum semigroupsChapter II. Representation theory § 6. Basic concepts of representation theory § 7. The quantum flag space of § 8. The Schur algebra and complete reducibility § 9. Representations of §10. The Frobenius morphismChapter III. Non-commutative differential calculus §11. The non-commutative de Rham complex of an -dimensional vector space §12. Quantum Weyl algebras §13. The de Rham complex of a quantum groupReferences

An R--Matrix Approach to the Quantization of the Euclidean Group E(2)

Abstract: The R-matrices for two different deformations of the Euclidean group E(2), calculated in a two-dimensional representation, are used to determine the deformed Hopf algebra of the representative functions. The duality of the latter with the initial quantum algebras is explicitly proved and the relationship between the two quantum groups is discussed and clarified.

Coideal subalgebras in hopf algebras: freeness, integrals, smash products

Abstract: A. Masuoka and Y. Doi have given several necessary and sufficient conditions for a finite-dimensional Hopf algebra A to be free as a left and a right module over a right coideal subalgebra B. We derive further equivalent conditions: for example that B is a direct summand of A both as a left and a right B-module, or that B contains certain kind of elements called integrals, or that the smash products #(A,B) and #op (A,B) decompose as convolution algebras in a certain way. We study also when semisimplicity of B implies its separability.

Quantum random walks and time reversal

Abstract: Classical random walks and Markov processes are easily described by Hopf algebras. It is also known that groups and Hopf algebras (quantum groups) lead to classical and quantum diffusions. We study here the more primitive notion of a quantum random walk associated with a general Hopf algebra and show that it has a simple physical interpretation in quantum mechanics. This is by means of a representation theorem motivated from the theory of Kac algebras: If H is any Hopf algebra, it may be realized in Lin(H) in such a way that Δh=W(h⊗1)W−1 for an operator W. This W is interpreted as the time evolution operator for the system at time t coupled quantum-mechanically to the system at time t+δ. Finally, for every Hopf algebra there is a dual one, leading us to a duality operation for quantum random walks and quantum diffusions and a notion of the coentropy of an observable. The dual system has its time reversed with respect to the original system, leading us to a novel kind of CTP theorem.