Unlike geometry, spheres in topology have been seen as topological invariants, where their structures are defined spaces. Forgetting, the exact notion of and impossibility embedding one into other, homotopy relates how sphere i dimensions can wrap another n dimensions.

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...

By constructing explicit examples, we show that the core of a group-like element in cocommutative cosemisimple Yetter-Drinfel'd Hopf algebra over group ring finite abelian is not always completely trivial.

Let p be prime. Let R be a discrete valuation ring of characteristic p with field of fractions K. Let C p denote the elementary abelian group of order p. In this paper we use Greither’s approach for classifying short exact sequences of finite R-group schemes to classify R-Hopf orders H in the group ring KC p , reproducing a result of Tossici. We then go further by providing an explicit descript...

We define the Hopf algebra structure on the Grothendieck group of finitedimensional polynomial representations of Uq ĝlN in the limit N → ∞. The resulting Hopf algebra Rep Uq ĝl∞ is a tensor product of its Hopf subalgebras Repa Uqĝl∞, a ∈ C/q. When q is generic (resp., q is a primitive root of unity of order l), we construct an isomorphism between the Hopf algebra Repa Uq ĝl∞ and the algebra of...

We construct, for a field F and a natural number n, algebraic cycles in Bloch's cubical cycle group of codimension n cycles in P 1 F \{1} 2n−1 , which correspond to weight n multiple polylogarithms with generic arguments if F ⊂ C. Moreover, we construct out of them a Hopf subalgebra in the Bloch-Kriz cycle Hopf algebra χ cycle. In the process, we are led to other Hopf algebras built from trees ...

Since the advent of Drinfel’d’s double construction, Hopf algebraic structures have been a centrepiece for many developments in the theory and analysis of integrable quantum systems. An integrable anyonic pairing Hamiltonian will be shown to admit Hopf algebra symmetries for particular values of its coupling parameters. While the integrable structure of the model relates to the well-known six-v...

We investigate two-parameter quantum groups corresponding to the general linear and special linear Lie algebras gln and sln. We show that these quantum groups can be realized as Drinfel’d doubles of certain Hopf subalgebras with respect to Hopf pairings. Using the Hopf pairing, we construct a corresponding R-matrix. The quantum groups have a natural n-dimensional module V . The R-matrix enables...