Quantum Doubles Research Articles

Local Index Formula for Quantum Double Suspension

Our understanding of the local index formula in noncommutative geometry has stalled for a while because we do not have more than one explicit computation, namely that of Connes for quantum SU(2), and we do not understand the meaning of the various multilinear functionals involved in the formula. In such a situation further progress in understanding necessitates more explicit computations, and here we execute the second explicit computation for the quantum double suspension, a construction inspired by the Toeplitz extension. More specifically, we compute the local index formula for the quantum double suspensions of $$C(\mathcal {S}^2)$$ and the noncommutative 2-torus.

Classification of Pointed Fusion Categories of dimension 8 up to weak Morita equivalence

ABSTRACTIn this paper we give a complete classification of pointed fusion categories over ℂ of global dimension 8. We first classify the equivalence classes of pointed fusion categories of dimension 8, and then we proceed to determine which of these equivalence classes have equivalent categories of modules following the procedure presented in [9, 11]. The results of this paper permit to recover the classification of twisted quantum doubles of groups of order 8 up to gauge equivalence of braided quasi-Hopf algebras that was previously done in [6] and [5].

Quantum Double of Yangian of strange Lie superalgebra Qn and multiplicative formula for universal R-matrix

Main result is the multiplicative formula for universal R-matrix for Quantum Double of Yangian of strange Lie superalgebra Qn type. We introduce the Quantum Double of the Yangian of the strange Lie superalgebra Qn and define its PBW basis. We compute the Hopf pairing for the generators of the Yangian Double. From the Hopf pairing formulas we derive a factorized multiplicative formula for the universal R-matrix of the Yangian Double of the Lie superalgebra Qn. After them we obtain coefficients in this multiplicative formula for universal R-matrix.

The Complete Set of Infinite Volume Ground States for Kitaev’s Abelian Quantum Double Models

We study the set of infinite volume ground states of Kitaev's quantum double model on $\mathbb{Z}^2$ for an arbitrary finite abelian group $G$. It is known that these models have a unique frustration-free ground state. Here we drop the requirement of frustration freeness, and classify the full set of ground states. We show that the ground state space decomposes into $|G|^2$ different charged sectors, corresponding to the different types of abelian anyons (also known as superselection sectors). In particular, all pure ground states are equivalent to ground states that can be interpreted as describing a single excitation. Our proof proceeds by showing that each ground state can be obtained as the weak$^*$ limit of finite volume ground states of the quantum double model with suitable boundary terms. The boundary terms allow for states which represent a pair of excitations, with one excitation in the bulk and one pinned to the boundary, to be included in the ground state space.

Dual pairs of quantum moment maps and doubles of Hopf algebras

Given a dual pair of topological Hopf algebras A,A⁎, under mild conditions there exists a natural associative algebra homomorphism D(A)→H(A) between the corresponding Drinfeld double D(A) and Heisenberg double H(A). We construct this homomorphism using a pair of commuting quantum moment maps, and then use it to provide a homomorphism of certain reflection equation algebras. We also explain how the quantization of the Grothendieck–Springer resolution arises in this context.

Mixed Quantum Double Construction of Subfactors

We generalize the quantum double construction of subfactors to that from arbitrary flat connections on 4-partite graphs and call it the \textit{mixed quantum double construction}. If all the four graphs of the original 4-partite graph are connected, it is easy to see that this construction produces Ocneanu's asymptotic inclusion of both subfactors generated by the original flat connection horizontally and vertically. The construction can be applied for example to the non-standard flat connections which appear in the construction of the Goodman-de la Harpe-Jones subfactors or to those obtained by the composition of flat part of any biunitary connections as in N. Sato's paper~[40]. An easy application shows that the asymptotic inclusions of the Goodman-de la Harpe-Jones subfactors are isomorphic to those of the Jones subfactors of type $A_n$ except for the cases of orbifold type. If two subfactors $A\subset B$ and $A \subset C$ have common $A$-$A$ bimodule systems, we can construct a flat connection in general. Then by applying our construction to the flat connection, we obtain the asymptotic inclusion of both $A\subset B$ and $A \subset C$. We also discuss the case when the original 4-partite graph contains disconnected graphs and give some such examples. General phenomena when disconnected graphs appear are explained by using bimodule systems.

Non-unitary fusion categories and their doubles via endomorphisms

We realise non-unitary fusion categories using subfactor-like methods, and compute their quantum doubles and modular data. For concreteness we focus on generalising the Haagerup–Izumi family of Q-systems. For example, we construct endomorphism realisations of the (non-unitary) Yang–Lee model, and non-unitary analogues of one of the even subsystems of the Haagerup subfactor and of the Grossman–Snyder system. We supplement Izumi's equations for identifying the half-braidings, which were incomplete even in his Q-system setting. We conjecture a remarkably simple form for the modular S and T matrices of the doubles of these fusion categories. We would expect all of these doubles to be realised as the category of modules of a rational VOA and conformal net of factors. We expect our approach will also suffice to realise the non-semisimple tensor categories arising in logarithmic conformal field theories.

Necessity of an energy barrier for self-correction of Abelian quantum doubles

We rigorously establish an Arrhenius law for the mixing time of quantum doubles based on any Abelian group $\mathbb{Z}_d$. We have made the concept of the energy barrier therein mathematically well-defined, it is related to the minimum energy cost the environment has to provide to the system in order to produce a generalized Pauli error, maximized for any generalized Pauli errors, not only logical operators. We evaluate this generalized energy barrier in Abelian quantum double models and find it to be a constant independent of system size. Thus, we rule out the possibility of entropic protection for this broad group of models.

A Remark on CFT Realization of Quantum Doubles of Subfactors: Case Index $${ < 4}$$ < 4

It is well-known that the quantum double $D(N\subset M)$ of a finite depth subfactor $N\subset M$, or equivalently the Drinfeld center of the even part fusion category, is a unitary modular tensor category. Thus should arise in conformal field theory. We show that for every subfactor $N\subset M$ with index $[M:N]<4$ the quantum double $D(N\subset M)$ is realized as the representation category of a completely rational conformal net. In particular, the quantum double of $E_6$ can be realized as a $\mathbb Z_2$-simple current extension of $\mathrm{SU}(2)_{10}\times \mathrm{Spin}(11)_1$ and thus is not exotic in any sense. As a byproduct we obtain a vertex operator algebra for every such subfactor. We obtain the result by showing that if a subfactor $N\subset M $ arises from $\alpha$-induction of completely rational nets $\mathcal A\subset \mathcal B$ and there is a net $\tilde{\mathcal A}$ with the opposite braiding, then the quantum $D(N\subset M)$ is realized by completely rational net. We construct completely rational nets with the opposite braiding of $\mathrm{SU}(2)_k$ and use the well-known fact that all subfactors with index $[M:N]<4$ arise by $\alpha$-induction from $\mathrm{SU}(2)_k$.

The R-matrix of quantum doubles of Nichols algebras of diagonal type

Let H be the quantum double of a Nichols algebra of diagonal type. We compute the R-matrix of 3-tuples of modules for general finite-dimensional highest weight modules over H. We also calculate a multiplicative formula for the universal R-matrix when H is finite dimensional. We show the unicity of a PBW basis (or a Lusztig-type Poincaré-Birkhoff-Witt basis) with a given convex order.

Quantum Fourier Transforms and the Complexity of Link Invariants for Quantum Doubles of Finite Groups

Knot and link invariants naturally arise from any braided Hopf algebra. We consider the computational complexity of the invariants arising from an elementary family of finite-dimensional Hopf algebras: quantum doubles of finite groups (denoted D(G), for a group G). Regarding algorithms for these invariants, we develop quantum circuits for the quantum Fourier transform over D(G); in general, we show that when one can uniformly and efficiently carry out the quantum Fourier transform over the centralizers Z(g) of the elements of G, one can efficiently carry out the quantum Fourier transform over D(G). We apply these results to the symmetric groups to yield efficient circuits for the quantum Fourier transform over D(S_n). With such a Fourier transform, it is straightforward to obtain additive approximation algorithms for the related link invariant. Additionally, we show that certain D(G) invariants (such as D(A_n) invariants) are BPP-hard to additively approximate, SBP-hard to multiplicatively approximate, and #P-hard to exactly evaluate. Finally, we make partial progress on the question of simulating anyonic computation in groups uniformly as a function of the group size. In this direction, we provide efficient quantum circuits for the Clebsch-Gordan transform over D(G) for "fluxon" irreps, i.e., irreps of D(G) characterized by a conjugacy class of G. For general irreps, i.e., those which are associated with a conjugacy class of G and an irrep of a centralizer, we present an efficient implementation under certain conditions such as when there is an efficient Clebsch-Gordan transform over the centralizers. We remark that this also provides a simulation of certain anyonic models of quantum computation, even in circumstances where the group may have size exponential in the size of the circuit.

EXAMPLES OF POINTED COLOR HOPF ALGEBRAS

We present examples of color Hopf algebras, i.e. Hopf algebras in color categories (braided tensor categories with braiding induced by a bicharacter on an abelian group), related with quantum doubles of pointed Hopf algebras. We also discuss semisimple color Hopf algebras.

On some properties of quantum doubles of finite groups

We prove two results about quantum doubles of finite groups over the complex field. The first result is the integrality theorem for higher Frobenius–Schur indicators for wreath product groups SN⋉AN, where A is a finite abelian group. A proof of this result for A=1 appears in a paper by Iovanov, Montgomery, and Mason. The second result is a lower bound for the largest possible number of irreducible representations of the quantum double of a finite group with at most n conjugacy classes. This answers a question asked by Eric Rowell.

Ambidextrous objects and trace functions for nonsemisimple categories

We provide a necessary and sufficient condition for a simple object in a pivotal k \Bbbk -category to be ambidextrous. In turn, these objects imply the existence of nontrivial trace functions in the category. These functions play an important role in low-dimensional topology as well as in studying the category itself. In particular, we prove they exist for factorizable ribbon Hopf algebras, modular representations of finite groups and their quantum doubles, complex and modular Lie (super)algebras, the ( 1 , p ) (1,p) minimal model in conformal field theory, and quantum groups at a root of unity.

The Representations of Quantum Double of Dihedral Groups

Let k be an algebraically closed field of odd characteristic p, and let Dn be the dihedral group of order 2n such that p|2n. Let D(kDn) denote the quantum double of the group algebra kDn. In this paper, we describe the structures of all finite-dimensional indecomposable left D(kDn)-modules, equivalently, of all finite-dimensional indecomposable Yetter-Drinfeld kDn-modules, and classify them.

On the structure and representations of non-balanced quantum doubles

For a field k and two finite groups G and X, when G acts on X from the right by group automorphisms, there is a Hopf algebra structure on k-space (kXop)* ⋇ kG (see Theorem 2.1), called a non-balanced quantum double and denoted by DX(G). In this paper, some Hopf algebra properties of DX(G) are given, the representation types of DX(G) viewed as a k-algebra are discussed, the algebra structure and module category over DX(G) are studied. Since the Hopf algebra structure of non-balanced quantum double DX(G) generalizes the usual quantum double D(G) for a finite group G, all results about DX(G) in this paper can also be used to describe D(G) as a special case and the universal R-matrix of DX(G) provides more solutions of Yang-Baxter equation.

Tensor Products of Modules Over a Certain Class of Quantum Doubles

The quantum doubles of a certain class of rank two pointed Hopf algebras are considered. The socle of the tensor product of two such modules is computed, and formulas similar to the ones in [4] are obtained. Cases when such a tensor product is completely irreducible are also given in the last section.

On the semiduals of local isometry groups in three-dimensional gravity

We use factorisations of the local isometry groups arising in 3D gravity for Lorentzian and Euclidean signatures and any value of the cosmological constant to construct associated bicrossproduct quantum groups via semidualisation. In this way, we obtain quantum doubles of the Lorentz and rotation groups in 3D, as well as κ-Poincaré algebras whose associated r-matrices have spacelike, timelike, and lightlike deformation parameters. We confirm and elaborate the interpretation of semiduality proposed by Majid and Schroers [“q-deformation and semi-dualisation in 3d quantum gravity,” J. Phys. A 42, 425402 (2009)]10.1088/1751-8113/42/42/425402 as the exchange of the cosmological length scale and the Planck mass in the context of 3D quantum gravity. In particular, semiduality gives a simple understanding of why the quantum double of the Lorentz group and the κ-Poincaré algebra with spacelike deformation parameter are both associated with 3D gravity with vanishing cosmological constant, while the κ-Poincaré algebra with a timelike deformation parameter can only be associated with 3D gravity if one takes the Planck mass to be imaginary.

The Quantum Double Model with Boundary: Condensations and Symmetries

Associated to every finite group, Kitaev has defined the quantum double model for every orientable surface without boundary. In this paper, we define boundaries for this model and characterize condensations; that is, we find all quasi-particle excitations (anyons) which disappear when they move to the boundary. We then consider two phases of the quantum double model corresponding to two groups with a domain wall between them, and study the tunneling of anyons from one phase to the other. Using this framework we discuss the necessary and sufficient conditions when two different groups give the same anyon types. As an application we show that in the quantum double model for S_3 (the permutation group over three letters) there is a chargeon and a fluxion which are not distinguishable. This group is indeed a special case of groups of the form of the semidirect product of the additive and multiplicative groups of a finite field, for all of which we prove a similar symmetry.

Toric codes and quantum doubles from two-body Hamiltonians

We present here a procedure to obtain the Hamiltonians of the toric code and Kitaev quantum double models as the low-energy limits of entirely two-body Hamiltonians. Our construction makes use of a new type of perturbation gadget based on error-detecting subsystem codes. The procedure is motivated by a projected entangled pair states (PEPS) description of the target models, and reproduces the target models' behavior using only couplings that are natural in terms of the original Hamiltonians. This allows our construction to capture the symmetries of the target models.