Quasi-quantum Groups Research Articles

Finite quasi-quantum groups of rank two

This is a contribution to the structure theory of finite pointed quasi-quantum groups. We classify all finite-dimensional connected graded pointed Majid algebras of rank two which are not twist equivalent to ordinary pointed Hopf algebras.

Quasi-quantum groups obtained from tensor braided Hopf algebras

Let H be a quasi-Hopf algebra, $${}_H^H{{{\mathcal {M}}}}_H^H$$ the category of two-sided two-cosided Hopf modules over H and $${}_H^H{\mathcal YD}$$ the category of left Yetter–Drinfeld modules over H. We show that $${}_H^H{{{\mathcal {M}}}}_H^H$$ admits a braided monoidal structure for which the strong monoidal equivalence $${}_H^H{\mathcal M}_H^H\cong {}_H^H{\mathcal YD}$$ established by the structure theorem for quasi-Hopf bimodules becomes braided monoidal. Using this braided monoidal equivalence, we prove that Hopf algebras within $${}_H^H{{{\mathcal {M}}}}_H^H$$ can be characterized as quasi-Hopf algebras with a projection or as biproduct quasi-Hopf algebras in the sense of Bulacu and Nauwelaerts (J Pure Appl Algebra 174:1–42, 2002) . A particular class of such (braided, quasi-) Hopf algebras is obtained from a tensor product Hopf algebra type construction. Our arguments rely on general categorical facts.

On nondiagonal finite quasi-quantum groups over finite abelian groups

In this paper, we initiate the study of nondiagonal finite quasi-quantum groups over finite abelian groups. We mainly study the Nichols algebras in the twisted Yetter–Drinfeld module category $$_{\mathbb {k}G}^{\mathbb {k}G}{\mathcal {Y}}{\mathcal {D}}^\Phi $$ with $$\Phi $$ a nonabelian 3-cocycle on a finite abelian group G. A complete clarification is obtained for the Nichols algebra B(V) in case V is a simple twisted Yetter–Drinfeld module of nondiagonal type. This is also applied to provide a complete classification of finite-dimensional coradically graded pointed coquasi-Hopf algebras over abelian groups of odd order and confirm partially the generation conjecture of pointed finite tensor categories due to Etingof, Gelaki, Nikshych and Ostrik.

Finite quasi-quantum groups of diagonal type

Abstract In this paper, we give a classification of finite-dimensional radically graded elementary quasi-Hopf algebras of diagonal type, or equivalently, finite-dimensional coradically graded pointed Majid algebras of diagonal type. By a Tannaka–Krein type duality, this determines a big class of pointed finite tensor categories. Some efficient methods of construction are also given.

Representations of the small quasi-quantum group ${\operatorname{Q}}\mathbf{u}_{q}(\mathfrak{sl}_{2})$

The quasi-Frobenius-Lusztig kernel ${\operatorname{Q}}\mathbf{u}_{q}(\mathfrak{sl}_{2})$ associated with $\mathfrak{sl}_{2}$ has been constructed in [9]. In this paper we study the representations of this small quasi-quantum group. We give a complete list of non-isomorphic indecomposables and the tensor product decomposition rules for simples and projectives. A description of the Grothendieck ring is also provided.

Quasi-quantum planes and quasi-quantum groups of dimension $p^3$ and $p^4$

The aim of this paper is to contribute more examples and classification results of finite pointed quasi-quantum groups within the quiver framework initiated in \cite{qha1, qha2}. The focus is put on finite dimensional graded Majid algebras generated by group-like elements and two skew-primitive elements which are mutually skew-commutative. Such quasi-quantum groups are associated to quasi-quantum planes in the sense of nonassociative geomertry \cite{m1, m2}. As an application, we obtain an explicit classification of graded pointed Majid algebras with abelian coradical of dimension $p^3$ and $p^4$ for any prime number $p.$

From projective representations to quasi-quantum groups

This is a contribution to the project of quiver approaches to quasi-quantum groups. We classify Majid bimodules over groups with 3-cocycles by virtue of projective representations. This leads to a theoretic classification of graded pointed Majid algebras over path coalgebras, or equivalently cofree pointed coalgebras, and helps to provide a projective representation-theoretic description of the gauge equivalence of graded pointed Majid algebras. We apply this machinery to construct some concrete examples and obtain a classification of finite-dimensional graded pointed Majid algebras with the set of group-likes equal to the cyclic group of order 2.

Quivers, Quasi-Quantum Groups and Finite Tensor Categories

We study finite quasi-quantum groups in their quiver setting developed recently by the first author in arXiv:0902.1620 and arXiv:0903.1472. We obtain a classification of finite-dimensional pointed Majid algebras of finite corepresentation type, or equivalently a classification of elementary quasi-Hopf algebras of finite representation type, over the field of complex numbers. By the Tannaka-Krein duality principle, this provides a classification of the finite tensor categories in which every simple object has Frobenius-Perron dimension 1 and there are finitely many indecomposable objects up to isomorphism. Some interesting information of these finite tensor categories is given by making use of the quiver representation theory.

Quasi-quantum groups from strings

Motivated by string theory on the orbifold M/G in presence of a Kalb-Ramond field strength H, we define the operators that lift the group action to the twisted sectors. These operators turn out to generate the quasi-quantum group Dω[G], introduced in the context of conformal field theory by R. Dijkgraaf, V. Pasquier and P. Roche [1], with ω a 3-cocycle determined by a series of cohomological equations in a tricomplex combining de Rham, Cech and group cohomologies. We further illustrate some properties of the quasi-quantum group from a string theoretical point of view. This work is based on [2], from which a full-fledged treatment may be extracted.

The representation-theoretic rank of the doubles of quasi-quantum groups

We compute the representation-theoretic rank of a finite dimensional quasi-Hopf algebra H and of its quantum double D ( H ) , within the rigid braided category of finite dimensional left D ( H ) -modules.

The Quantum Double for Quasitriangular Quasi-Hopf Algebras

Let D(H) be the quantum double associated to a finite dimensional quasi-Hopf algebra H, as in Hausser and Nill ((Hausser, F., Nill, F. (1999a). Diagonal crossed products by duals of quasi-quantum groups. Rev. Math. Phys. 11:553–629) and (Hausser, F., Nill, F. (1999b). Doubles of quasi-quantum groups. Comm. Math. Phys. 199:547–589)). In this note, we first generalize a result of Majid (Majid, S. (1991). Doubles of quasitriangular Hopf algebras. Comm. Algebra 19:3061–3073) for Hopf algebras, and then prove that the quantum double of a finite dimensional quasitriangular quasi-Hopf algebra is a biproduct in the sense of Bulacu and Nauwelaerts (Bulacu, D., Nauwelaerts, E. (2002). Radford's biproduct for quasi-Hopf algebras and bosonization. J. Pure Appl. Algebra 179:1–42.).

Quasitriangular and Ribbon Quasi-Hopf Algebras

Following (Drinfeld, V. G. (1990a). Quasi-Hopf algebras.Leningrad Math. J. 1:1419–1457) a quasi-Hopf algebra has, by definition, its antipode bijective. In this note, we will prove that for a quasitriangular quasi-Hopf algebra with an R-matrix R, this condition is unnecessary and also the condition of invertibility of R. Finally, we will give a characterization for a ribbon quasi-Hopf algebra. This characterization has already been given in Altschuler and Coste (Altschuler, D., Coste, A. (1992). Quasi-quantum groups, knots, three-manifolds and topological field theory. Comm. Math. Phys. 150:83–107.), but with an additional condition. We will prove that this condition is unnecessary.

DIAGONAL CROSSED PRODUCTS BY DUALS OF QUASI-QUANTUM GROUPS

A two-sided coaction [Formula: see text] of a Hopf algebra [Formula: see text] on an associative algebra ℳ is an algebra map of the form [Formula: see text] , where (λ,ρ) is a commuting pair of left and right [Formula: see text] -coactions on ℳ, respectively. Denoting the associated commuting right and left actions of the dual Hopf algebra [Formula: see text] on ℳ by ◃ and ▹, respectively, we define the diagonal crossed product[Formula: see text] to be the algebra generated by ℳ and [Formula: see text] with relations given by [Formula: see text] We give a natural generalization of this construction to the case where [Formula: see text] is a quasi-Hopf algebra in the sense of Drinfeld and, more generally, also in the sense of Mack and Schomerus (i.e. where the coproduct Δ is non-unital). In these cases our diagonal crossed product will still be an associative algebra structure on [Formula: see text] extending [Formula: see text], even though the analogue of an ordinary crossed product [Formula: see text] in general is not well defined as an associative algebra. Applications of our formalism include the field algebra constructions with quasi-quantum group symmetry given by G. Mack and V. Schomerus [31, 47] as well as the formulation of Hopf spin chains or lattice current algebras based on truncated quantum groups at roots of unity. In the case [Formula: see text] and λ=ρ=Δ we obtain an explicit definition of the quantum double [Formula: see text] for quasi-Hopf algebras [Formula: see text] , which before had been described in the form of an implicit Tannaka–Krein reconstruction procedure by S. Majid [35]. We prove that [Formula: see text] is itself a (weak) quasi-bialgebra and that any diagonal crossed product [Formula: see text] naturally admits a two-sided [Formula: see text] -coaction. In particular, the above-mentioned lattice models always admit the quantum double [Formula: see text] as a localized cosymmetry, generalizing results of Nill and Szlachányi [42]. A complete proof that [Formula: see text] is even a (weak) quasi-triangular quasi-Hopf algebra will be given in a separate paper [27].

Star-Products and Quasi-Quantum Groups

The purpose of the present paper is to show thatDrinfeld's theories of quasiquantum groups(quasitriangular quasi-Hopf algebras) and of quasi-Liebialgebras can be developed in terms of the starproducts on a simple quasitriangular Poisson–Liegroup.

Dualité tannakienne pour les quasi-groupoïdes quantiques

This paper is the sequel of a previous one [2] where we extended the Tannaka-Krein duality results to the non-commutative situation, i.e. to ‘quantum groupoids’. Here we extend those results to the quasi-monoidal situation, corresponding to ‘quasi-quantum groupoids’ as defined in [3] (‘quasi-’ stands for quasi-associativity a la Drinfeld). More precisely, let B be a commutative algebra over a field k. Given a tensor autonomous category τ,. we define the notion of a quasi-fibre functor ω:τ-proj B (here, ‘quasi-’ means without compatibility to associativity constraints). On the other hand, we define the notion of a transitive quasi-quantum groupoid over B. We then show that the category of tensor autonomous categories equipped with a quasi-fibre functor (with suitable morphisms), is equivalent to the category of transitive quasi-quantum groupoids (5.4.2) Moreover, we classify quasi-fibre functors for a semisimple tensor autonomous category (6.1.2), and give a few examples : a family of quantum groups having the same tensor category of representations as Sl2(C), but with non-isornorphic underlying coalgebras, constructed by means of an R-matrix introduced by Gurevich ([9]) in a manner suggested to the author by Lyubashenko (6.2.1 and 6.2.2), and quasi-quantum groups which cannot be obtained from quantum groups by a Drinfeld twist (6.2.1)

An explicit construction of the quantum group in chiral WZW-models

It is shown how a chiral Wess-Zumino-Witten theory with globally defined vertex operators and a one-to-one correspondence between fields and states can be constructed. The Hilbert space of this theory is the direct sum of tensor products of representations of the chiral algebra and finite dimensional internal parameter spaces. On this enlarged space there exists a natural action of Drinfeld's quasi quantum group $A_{g,t}$, which commutes with the action of the chiral algebra and plays the r\^{o}le of an internal symmetry algebra. The $R$ matrix describes the braiding of the chiral vertex operators and the coassociator $\Phi$ gives rise to a modification of the duality property. For generic $q$ the quasi quantum group is isomorphic to the coassociative quantum group $U_{q}(g)$ and thus the duality property of the chiral theory can be restored. This construction has to be modified for the physically relevant case of integer level. The quantum group has to be replaced by the corresponding truncated quasi quantum group, which is not coassociative because of the truncation. This exhibits the truncated quantum group as the internal symmetry algebra of the chiral WZW model, which therefore has only a modified duality property. The case of $g=su(2)$ is worked out in detail.

Quasi quantum group covariant q-oscillators

If q is a pth rooty of unit there exists a quasi-co-associative truncated quantum group algebra U q T(sl 2) whose indecomposable representations ar the physical representations of U q (sl 2), whose co-product yields the truncated tensor product of physical representations of U q (sl 2), and whose R-matrix satisfies quasi Yang-Baxter equations. These truncated quantum group algebras are examples of weak quasi quantum group algebras [2]. For primitive pth roots q, q = e 2 πi/ p , we consider a two-dimensional q-oscillator which admits U q T(sl 2) as a symmetry algebra. Its wave function lie in a space F q T of “functions on the truncated quantum plane”, i.e. of polynomials in noncommuting complex coordinate functions z a , on which multiplication operators Z a and the elements of U q T(sl 2) can act. This illustrates the concept of quasi quantum planes [1]. Due to the truncation, the Hilbert space of states is finite dimensional. The subspaces F R(n) of monomials in z a of nth degree vanish for n⩾ p − 1, and F T(n) carries the (2 J+1)-dimensional irreducible representation of U q T ( sl 2) if n = 2J, J = 0, 1 2 ,…, 1 2 (p−2) . Partial derivatives ∂ a are introduced. We find a ∗-operation on the algebra of multiplication operators, Z i and derivatives ∂ b such that the adjoints Z a ∗ act as differentiation on the truncated quantum plane. Multiplication operators Z a (“creation operators”) and their adjoints (“annihilation operators”) obey q −1 2 - commutation relations. The ∗-operation is used to determine a positive definite scalar product on the truncated quantum plane F q T . Some natural candidates of hamiltonians for the q-oscillators are determined.

Higher algebraic structures and quantization

We derive (quasi-)quantum groups in 2+1 dimensional topological field theory directly from the classical action and the path integral. Detailed computations are carried out for the Chern-Simons theory with finite gauge group. The principles behind our computations are presumably more general. We extend the classical action in a d+1 dimensional topological theory to manifolds of dimension less than d+1. We then ``construct'' a generalized path integral which in d+1 dimensions reduces to the standard one and in d dimensions reproduces the quantum Hilbert space. In a 2+1 dimensional topological theory the path integral over the circle is the category of representations of a quasi-quantum group. In this paper we only consider finite theories, in which the generalized path integral reduces to a finite sum. New ideas are needed to extend beyond the finite theories treated here.

A short introduction to quantum symmetry

In quantum theory, symmetries more general than groups are possible. We give a general definition of a quantum symmetry, such that symmetry operations act on the Hilbert space H> of physical states and notions of unitarity, invariance and covariance are defined. Within this frame, weak quasi quantum groups are described as a natural generalization of group algebras. Consistency with locality distinguishes them from more general quantum symmetries. To find the new kinds of symmetry one should investigate low dimensional quantum systems such as two-dimensional layers.

Action of truncated quantum groups on quasi-quantum planes and a quasi-associative differential geometry and calculus

Ifq is ap th root of unity there exists a quasi-coassociative truncated quantum group algebra whose indecomposable representations are the physical representations ofU q (sl 2), whose coproduct yields the truncated tensor product of physical representations ofU q (sl 2), and whoseR-matrix satisfies quasi-Yang Baxter equations. These truncated quantum group algebras are examples of weak quasitriangular quasi-Hopf algebras (“quasi-quantum group algebras”). We describe a space of “functions on the quasi quantum plane,” i.e. of polynomials in noncommuting complex coordinate functionsz a , on which multiplication operatorsZ a and the elements of can act, so thatz a will transform according to some representation τf of can be made into a quasi-associative graded algebra on which elements of act as generalized derivations. In the special case of the truncatedU q (sl 2) algebra we show that the subspaces of monomials inz a ofn th degree vanish forn≥p−1, and that carries the 2J+ 1 dimensional irreducible representation of ifn=2J, J=0,1/2, ..., 1/2(p−2). Assuming that the representation τf of the quasi-quantum group algebra gives rise to anR-matrix with two eigenvalues, we develop a quasi-associative differential calculus on. This implies construction of an exterior differentiation, a graded algebra of forms and partial derivatives. A quasi-associative generalization of noncommutative differential geometry is introduced by defining a covariant exterior differentiation of forms. It is covariant under gauge transformations.