Narrow Results

We describe the generators and prove a number of relations for the construction of a planar algebra from the restricted quantum group ${Ū}_q({𝔰𝔩}_2)$. This is a diagrammatic description of $En{d}_{{Ū}_q({𝔰𝔩}_2)}(X^{\otimes n})$, where $X:={{𝒳}}_2^{+}$ is a two-dimensional ${Ū}_q({𝔰𝔩}_2)$ module.

We develop a general framework to deal with the unitary representations of quantum groups using the language of C*-algebras. Using this framework, we prove that the duality holds in a general context. This extends the framework of the duality theorem using the language of von Neumann algebras previously developed by Masuda and Nakagami.

We show that the left regular representation π_l of a discrete quantum group (A, Δ) has the absorbing property and forms a monoid in the representation category Rep(A, Δ).

Next we show that an absorbing monoid in an abstract tensor *-category gives rise to an embedding functor (or fiber functor) , and we identify conditions on the monoid, satisfied by , implying that E is *-preserving.

As is well-known, from an embedding functor the generalized Tannaka theorem produces a discrete quantum group (A, Δ) such that . Thus, for a C*-tensor category with conjugates and irreducible unit the following are equivalent: (1) is equivalent to the representation category of a discrete quantum group (A, Δ), (2) admits an absorbing monoid, (3) there exists a *-preserving embedding functor .

We identify the quantum isometry groups of spectral triples built on the symmetric groups with length functions arising from the nearest-neighbor transpositions as generators. It turns out that they are isomorphic to certain "doubling" of the group algebras of the respective symmetric groups. We discuss the doubling procedure in the context of regular multiplier Hopf algebras. In the last section we study the dependence of the isometry group of S_n on the choice of generators in the case n = 3. We show that two different choices of generators lead to nonisomorphic quantum isometry groups which exhaust the list of noncommutative noncocommutative semisimple Hopf algebras of dimension 12. This provides noncommutative geometric interpretation of these Hopf algebras.

We showed that there is a complete analogue of a representation of the quantum plane where |q| = 1, with the classical ax+b group. We showed that the Fourier transform of the representation of on has a limit (in the dual corepresentation) toward the Mellin transform of the unitary representation of the ax+b group, and furthermore the intertwiners of the tensor products representation has a limit toward the intertwiners of the Mellin transform of the classical ax+b representation. We also wrote explicitly the multiplicative unitary defining the quantum ax+b semigroup and showed that it defines the corepresentation that is dual to the representation of above, and also correspond precisely to the classical family of unitary representation of the ax+b group.

Thoma's theorem states that a group algebra $C^{\ast }(\mathrm{\Gamma })$ is of type I if and only if $\mathrm{\Gamma }$ is virtually abelian. We discuss here some similar questions for the quantum groups, our main result stating that, under suitable virtually abelianity conditions on a discrete quantum group $\mathrm{\Gamma }$, we have a stationary model of type $\pi :C^{\ast }(\mathrm{\Gamma })\to M_F(C(L))$, with $F$ being a finite quantum group, and with $L$ being a compact group. We discuss then some refinements of these results in the quantum permutation group case, $\widehat{\mathrm{\Gamma }}\subset S_N^{+}$, by restricting the attention to the matrix models which are quasi-flat, in the sense that the images of the standard coordinates, known to be projections, have rank $\le 1$.

For a finite-index ${\mathrm{\text{II}}}_1$ subfactor $N\subset M$, we prove the existence of a universal Hopf ∗-algebra (or, a discrete quantum group in the analytic language) acting on $M$ in a trace-preserving fashion and fixing $N$ pointwise. We call this Hopf ∗-algebra the quantum Galois group for the subfactor and compute it in some examples of interest, notably for arbitrary irreducible finite-index depth-two subfactors. Along the way, we prove the existence of universal acting Hopf algebras for more general structures (tensors in enriched categories), in the spirit of recent work by Agore, Gordienko and Vercruysse.

We classify all the Hopf PBW-deformations of a new type quantum group $U_q({𝔰𝔩}_2^{\ast })$ from which the classical Drinfeld–Jimbo quantum group $U_q({𝔰𝔩}_2)$ can arise as an almost unique nontrivial one. Different from the $U_q({𝔰𝔩}_2)$ case, the category of finite-dimensional $U_q({𝔰𝔩}_2^{\ast })$-modules is non-semisimple. We establish a block decomposition theorem for the category $U_q({𝔰𝔩}_2^{\ast })\text{-}{\mathrm{\text{mod}}}_{\mathrm{\text{wt}}}$ of finite-dimensional weight modules of $U_q({𝔰𝔩}_2^{\ast })$. On the level of tensor category, we show that $U_q({𝔰𝔩}_2^{\ast })\text{-}{\mathrm{\text{mod}}}_{\mathrm{\text{wt}}}$ (respectively, the category $U_q({𝔰𝔩}_2)\text{-}\mathrm{\text{mod}}$ of finite-dimensional $U_q({𝔰𝔩}_2)$-modules) can be realized via (respectively, deformed) preprojective algebras of Dynkin type $𝔸$.

Arnold showed that the Euler equations of an ideal fluid describe geodesics in the Lie algebra of incompressible vector fields. We will show that helicity induces a splitting of the Lie algebra into two isotropic subspaces, forming a Manin triple. Viewed another way, this shows that there is an infinitesimal quantum group (a.k.a. Lie bi-algebra) underlying classical fluid mechanics.

We present an integral representation to the quantum Knizhnik–Zamolodchikov equation associated with twisted affine symmetry for massless regime |q| = 1. Upon specialization, it leads to a conjectural formula for the correlation function of the Izergin–Korepin model in massless regime |q| = 1. In a limiting case q → -1, our conjectural formula reproduce the correlation function for the Izergin–Korepin model^1,2 at critical point q = -1.

We define a 3-generator algebra obtained by replacing the commutators with anticommutators in the defining relations of the angular momentum algebra. We show that integer spin representations are in one to one correspondence with those of the angular momentum algebra. The half-integer spin representations, on the other hand, split into two representations of dimension . The anticommutator spin algebra is invariant under the action of the quantum group SO_q(3) with q=-1.

In this paper, using a Hopf-algebraic method, we construct deformed Poincaré SUSY algebra in terms of twisted (Hopf) algebra. By adapting this twist deformed super-Poincaré algebra as our fundamental symmetry, we can see the consistency between the algebra and non(anti)commutative relation among (super)coordinates and interpret that symmetry of non(anti)commutative QFT is in fact twisted one. The key point is validity of our new twist element that guarantees non(anti)commutativity of space. It is checked in this paper for case. We also comment on the possibility of noncommutative central charge coordinate. Finally, because our twist operation does not break the original algebra, we can claim that (twisted) SUSY is not broken in contrast to the string inspired SUSY in non(anti)commutative superspace.

We propose a model for random forces in a turbulent incompressible fluid by balancing the energy gain from fluctuations against dissipation by viscosity. This leads to a more singular covariance distribution for the random forces than is ordinarily allowed. We then propose regularization of the fluid system by matrix models. A formula for entropy of a two dimensional fluid is derived and then a vorticity profile of a hurricane that maximizes entropy. A regularization of three dimensional incompressible fluid flow using quantum groups is also proposed.

It is shown that the four quantum trefoil solitons that are described by the irreducible representations of the quantum algebra SL_q(2) [and that may be identified with the four families of elementary fermions (e, μ, τ; ν_eν_μν_τ;d, s, b; u, c, t)] may be built out of three preons, chosen from two charged preons with charges (1/3, -1/3) and two neutral preons. These preons are fermions and are described by the representation of SL_q(2). There are also four bosonic preons described by the and representations of SL_q(2). The knotted standard theory may be replicated at the preon level and the conjectured particles are in principle indirectly observable.

There are suggestive experimental indications that the leptons, neutrinos, and quarks might be composite and that their structure is described by the quantum group SLq(2). Since the hypothetical preons must be very heavy relative to the masses of the leptons, neutrinos, and quarks, there must be a very strong binding field to permit these composite particles to form. Unfortunately there are no experiments direct enough to establish the order of magnitude needed to make the SLq(2) Lagrangian dynamics quantitative. It is possible, however, to parametrize the preon masses and interactions that would be necessary to stabilize the three particle composite representing the leptons, neutrinos, and quarks. In this note we examine possible parametrizations of the masses and the interactions of these hypothetical structures. We also note an alternative view of SLq(2) preons.

The idea that the elementary particles might have the symmetry of knots has had a long history. In any modern formulation of this idea, however, the knot must be quantized. The present review is a summary of a small set of papers that began as an attempt to correlate the properties of quantized knots with empirical properties of the elementary particles. As the ideas behind these papers have developed over a number of years, the model has evolved, and this review is intended to present the model in its current form. The original picture of an elementary fermion as a solitonic knot of field, described by the trefoil representation of SUq(2), has expanded into its present form in which a knotted field is complementary to a composite structure composed of three preons that in turn are described by the fundamental representation of SLq(2). Higher representations of SLq(2) are interpreted as describing composite particles composed of three or more preons bound by a knotted field. This preon model unexpectedly agrees in important detail with the Harari–Shupe model. There is an associated Lagrangian dynamics capable in principle of describing the interactions and masses of the particles generated by the model.

We prove existence of BGG resolution of an irreducible highest weight module over a quantum group, classify morphisms of Verma modules over a quantum group and find formulas for singular vectors in Verma modules. As an application we find cohomology of the quantum group of the type with coefficients in a finite-dimensional module.

For M a finite dimensional complex vector space and A a certain type of (unital) subalgebra of End(M) (including some specific types of physical significance in the field of quantum spin chains) we give an algorithm for constructing the centraliser or commutant B of A on M. We give examples, and discuss the conditions for centralising to be an involution, i.e. A, B a dual pair, and for B and A to be Morita equivalent. A special case of one example shows that H_n(q), U_q(sl₂) act as a dual pair on the tensored vector representation for all q.

We depict the weight diagrams (alias, crystal graphs) of basic and adjoint representations of complex simple Lie algebras/algebraic groups and describe some of their uses.

In this paper, we construct new links invariants from a type I basic Lie superalgebra 𝔤. The construction uses the existence of an unexpected replacement of the vanishing quantum dimension of typical module, by non-trivial "fake quantum dimensions". Using this, we get a multivariable link invariant associated to any one parameter family of irreducible 𝔤-modules.