Hopf algebra structure on the differential algebra of the extended q-plane is defined. An algebra of forms which is obtained from the generators of the extended q-plane is introduced and its Hopf algebra structure is given. E-mail: [email protected] E-mail: [email protected]

Let (H,R) be a quasitriangular weak Hopf algebra, and A a quantum commutative weak H-module algebra. We establish the relationship of homological dimensions between weak smash product algebra A#H and A under some conditions. As an application, we consider the case of twisted weak Hopf algebra. Mathematics Subject Classification (2010): 16T05

We extend the Larson–Sweedler theorem for weak Hopf algebras by proving that a finite dimensional weak bialgebra is a weak Hopf algebra iff it possesses a non-degenerate left integral. We establish the autonomous monoidal category of the modules of a weak Hopf algebra A and show the semisimplicity of the unit and the invertible modules of A. We also reveal the connection of these modules to lef...

The category of Yetter-Drinfeld modules YD K over a Hopf algebra K (with bijektive antipode over a field k) is a braided monoidal category. If H is a Hopf algebra in this category then the primitive elements of H do not form an ordinary Lie algebra anymore. We introduce the notion of a (generalized) Lie algebra in YD K such that the set of primitive elements P (H) is a Lie algebra in this sense...

The planar dilatation operator of N = 4 supersymmetric Yang-Mills is the hamiltonian of an integrable spin chain whose length is allowed to fluctuate. We will identify the dynamics of length fluctuations of planar N = 4 Yang-Mills with the existence of an abelian Hopf algebra Z symmetry with non-trivial co-multiplication and antipode. The intertwiner conditions for this Hopf algebra will restri...

Hopf–Galois extensions of rings generalize Galois extensions, with the coaction of a Hopf algebra replacing the action of a group. Galois extensions with respect to a group G are the Hopf–Galois extensions with respect to the dual of the group algebra of G . Rognes recently defined an analogous notion of Hopf–Galois extensions in the category of structured ring spectra, motivated by the fundame...

We study the basic monoidal properties of the category of Hopf modules for a coquasi Hopf algebra. In particular we discuss the so called fundamental theorem that establishes a monoidal equivalence between the category of comodules and the category of Hopf modules. We present a categorical proof of Radford’s S formula for the case of a finite dimensional coquasi Hopf algebra, by establishing a ...

This paper continues our study, begun in [MS], of the relationship between the prime ideals of an algebra A and of a subalgebra R such that R ⊂A is a faithfully flat H -Galois extension, for some finite-dimensional Hopf algebra H . In that paper we defined three basic Krull relations, Incomparability (INC), t-Lying Over (t-LO), and Going Up (GU), analogous to the classical Krull relations for p...

A commutative but not cocommutative graded Hopf algebra HN , based on ordered (planar) rooted trees, is studied. This Hopf algebra is a generalization of the Hopf algebraic structure of unordered rooted trees HC , developed by Butcher in his study of Runge-Kutta methods and later rediscovered by Connes and Moscovici in the context of noncommutative geometry and by Kreimer where it is used to de...