We discuss the structure of shock singularities of the Burgers-Hopf hierarchy. It is shown that the set of singular solutions defines a stratification of the affine space of the flow parameters in the hierarchy. The stratification is associated with the Birkhoff decomposition of the Grassmannian given by the set of linear spaces spanned by the hierarchy. We then construct integrable hierarchy o...

I give a brief summary of the results reported in [1], in collaboration with G. Amelino-Camelia and F. D’Andrea. I focus on the analysis of the symmetries of κ-Minkowski noncommutative space-time, described in terms of a Weyl map. The commutative-spacetime notion of Lie-algebra symmetries must be replaced by the one of Hopf-algebra symmetries. However, in the Hopf-algebra sense, it is possible ...

A nonlinear car-following model that includes the reaction-time delay of drivers is considered. When investigating the linear stability of the uniform flow solution, boundaries of Hopf bifurcations are determined in the parameter space. Crossing these boundaries, oscillations may appear corresponding to travelling wave solutions. Hopf normal form calculations prove robustly subcritical behavior...

Every C*-algebra gives rise to an effect module and a convex space of states, which are connected via Kadison duality. We explore this duality in several examples, where the C*-algebra is equipped with the structure of a finite-dimensional Hopf algebra. When the Hopf algebra is the function algebra or group algebra of a finite group, the resulting state spaces form convex monoids. We will prove...

Let (A, ∆) be a multiplier Hopf algebra. In general, the underlying algebra A need not have an identity and the coproduct ∆ does not map A into A ⊗ A but rather into its multiplier algebra M (A ⊗ A). In this paper, we study some tools that are frequently used when dealing with such multiplier Hopf algebras and that are typical for working with algebras without identity in this context. The basi...

Suppose that M is a finite-dimensional vector space over a field k and that R : M ⊗M −→M ⊗M is solution to the quantum Yang–Baxter equation(QYBE). The FRT construction [3] is a bialgebra A(R) associated with R in a natural way. There is a quotient of the FRT construction, referred to as the reduced FRT construction and denoted by Ã(R), which seems rather useful in computation [11]. The bialgebr...

A wide class of models, built of the three component unit vector field living in the (3+1) Minkowski space-time, which break explicitly global O(3) symmetry are discussed. The symmetry breaking occurs due to the so-called dielectric function multiplying a standard symmetric term. Integrability conditions are found. Moreover, for some particular forms of the Lagrangian exact toroidal solutions w...

We obtain an R-matrix or matrix representation of the Artin braid group acting in a canonical way on the vector space of every (super)-Lie algebra or braided-Lie algebra. The same result applies for every (super)-Hopf algebra or braided-Hopf algebra. We recover some known representations such as those associated to racks. We also obtain new representations such as a non-trivial one on the ring ...

We consider the metric transformation of measure spaces/pyramids. clarify conditions to obtain convergence sequence transformed spaces from that original sequence, and, conversely, respectively. As an application, we prove spheres and projective with standard Riemannian distance converge a Gaussian space Hopf quotient space, respectively, as dimension diverges infinity.

We provide a new construction of the modes of the Poincaré dodecahedral space S3/I∗. The construction uses the Hopf map, Maxwell’s multipole vectors and orbifolds. In particular, the *235-orbifold serves as a parameter space for the modes of S3/I∗, shedding new light on the geometrical significance of the dimension of each space of k-modes, as well as on the modes themselves.